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    <title>Blake Cole</title>
    <link>https://www.blakecole.com/</link>
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    <description>Blake Cole</description>
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      <title>Blake Cole</title>
      <link>https://www.blakecole.com/</link>
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      <title>Example Talk</title>
      <link>https://www.blakecole.com/event/example/</link>
      <pubDate>Sat, 01 Jun 2030 13:00:00 +0000</pubDate>
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      <description>&lt;div class=&#34;px-4 py-3 mb-6 rounded-md bg-primary-100 dark:bg-primary-900&#34;&gt;
&lt;pre&gt;&lt;code&gt;&amp;lt;div class=&amp;quot;flex&amp;quot;&amp;gt;
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  &amp;lt;span class=&amp;quot;dark:text-neutral-300&amp;quot;&amp;gt;Click on the &amp;lt;strong&amp;gt;Slides&amp;lt;/strong&amp;gt; button above to view the built-in slides feature.&amp;lt;/span&amp;gt;
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&lt;p&gt;Slides can be added in a few ways:&lt;/p&gt;
&lt;ul&gt;
&lt;li&gt;&lt;strong&gt;Create&lt;/strong&gt; slides using Hugo Blox Builder&amp;rsquo;s &lt;a href=&#34;https://docs.hugoblox.com/reference/content-types/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;&lt;em&gt;Slides&lt;/em&gt;&lt;/a&gt; feature and link using &lt;code&gt;slides&lt;/code&gt; parameter in the front matter of the talk file&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Upload&lt;/strong&gt; an existing slide deck to &lt;code&gt;static/&lt;/code&gt; and link using &lt;code&gt;url_slides&lt;/code&gt; parameter in the front matter of the talk file&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Embed&lt;/strong&gt; your slides (e.g. Google Slides) or presentation video on this page using &lt;a href=&#34;https://docs.hugoblox.com/reference/markdown/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;shortcodes&lt;/a&gt;.&lt;/li&gt;
&lt;/ul&gt;
&lt;p&gt;Further event details, including &lt;a href=&#34;https://docs.hugoblox.com/reference/markdown/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;page elements&lt;/a&gt; such as image galleries, can be added to the body of this page.&lt;/p&gt;
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    <item>
      <title>Geometric programming for aerodynamically-actuated wingsail design optimization</title>
      <link>https://www.blakecole.com/publication/jrnl-ieee-joe-2025/</link>
      <pubDate>Sat, 26 Jul 2025 00:00:00 +0000</pubDate>
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    <item>
      <title>Reverse Engineering an LED Display</title>
      <link>https://www.blakecole.com/post/3-led-timer/</link>
      <pubDate>Wed, 16 Apr 2025 08:18:13 -0400</pubDate>
      <guid>https://www.blakecole.com/post/3-led-timer/</guid>
      <description>


&lt;details class=&#34;print:hidden xl:hidden&#34; open&gt;
  &lt;summary&gt;Table of Contents&lt;/summary&gt;
  &lt;div class=&#34;text-sm&#34;&gt;
  &lt;nav id=&#34;TableOfContents&#34;&gt;
  &lt;ul&gt;
    &lt;li&gt;&lt;a href=&#34;#backstory&#34;&gt;Backstory&lt;/a&gt;&lt;/li&gt;
    &lt;li&gt;&lt;a href=&#34;#topics-not-covered&#34;&gt;Topics Not Covered&lt;/a&gt;&lt;/li&gt;
    &lt;li&gt;&lt;a href=&#34;#methodology&#34;&gt;Methodology&lt;/a&gt;&lt;/li&gt;
    &lt;li&gt;&lt;a href=&#34;#inspect-hardware&#34;&gt;Inspect Hardware&lt;/a&gt;
      &lt;ul&gt;
        &lt;li&gt;&lt;a href=&#34;#power-supply&#34;&gt;Power Supply&lt;/a&gt;&lt;/li&gt;
        &lt;li&gt;&lt;a href=&#34;#microcontrollers&#34;&gt;Microcontrollers&lt;/a&gt;&lt;/li&gt;
        &lt;li&gt;&lt;a href=&#34;#shift-registers&#34;&gt;Shift Registers&lt;/a&gt;&lt;/li&gt;
      &lt;/ul&gt;
    &lt;/li&gt;
    &lt;li&gt;&lt;a href=&#34;#modify-hardware&#34;&gt;Modify Hardware&lt;/a&gt;
      &lt;ul&gt;
        &lt;li&gt;&lt;a href=&#34;#signal-injection&#34;&gt;Signal Injection&lt;/a&gt;&lt;/li&gt;
        &lt;li&gt;&lt;a href=&#34;#a-small-setback&#34;&gt;A Small Setback&lt;/a&gt;&lt;/li&gt;
        &lt;li&gt;&lt;a href=&#34;#adding-garnishes&#34;&gt;Adding Garnishes&lt;/a&gt;&lt;/li&gt;
      &lt;/ul&gt;
    &lt;/li&gt;
    &lt;li&gt;&lt;a href=&#34;#write-software&#34;&gt;Write Software&lt;/a&gt;
      &lt;ul&gt;
        &lt;li&gt;&lt;a href=&#34;#timer-logic&#34;&gt;Timer Logic&lt;/a&gt;&lt;/li&gt;
        &lt;li&gt;&lt;a href=&#34;#numeric-digits&#34;&gt;Numeric Digits&lt;/a&gt;&lt;/li&gt;
      &lt;/ul&gt;
    &lt;/li&gt;
    &lt;li&gt;&lt;a href=&#34;#build-controller&#34;&gt;Build Controller&lt;/a&gt;
      &lt;ul&gt;
        &lt;li&gt;&lt;a href=&#34;#final-stretch&#34;&gt;Final Stretch&lt;/a&gt;&lt;/li&gt;
        &lt;li&gt;&lt;a href=&#34;#stress-testing&#34;&gt;Stress Testing&lt;/a&gt;&lt;/li&gt;
      &lt;/ul&gt;
    &lt;/li&gt;
    &lt;li&gt;&lt;a href=&#34;#reflection&#34;&gt;Reflection&lt;/a&gt;&lt;/li&gt;
  &lt;/ul&gt;
&lt;/nav&gt;
  &lt;/div&gt;
&lt;/details&gt;

&lt;h2 id=&#34;backstory&#34;&gt;Backstory&lt;/h2&gt;
&lt;p&gt;Last September, my brother came to me with a conundrum: the controller for a high-power LED timer he had been using to run our local community surfing competitions had been stolen.  To make matters worse, the original manufacturer wanted more than $1,000 to replace it &amp;ndash; a sum which far exceeded the modest budget of the nonprofit organization for which he volunteers, the &lt;a href=&#34;https://southbayboardriders.org/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;South Bay Boardriders Club (SBBC)&lt;/a&gt;.  With competitions scheduled to resume in just a few months, they were at a loss for what to do.&lt;/p&gt;






&lt;figure style=&#34;text-align: center; margin-top: 50px; margin-bottom: 50px;&#34;&gt;

  &lt;a data-fancybox=&#34;&#34; href=&#34;sbbc.png&#34; style=&#34;display:inline-block; margin:0; padding:0; line-height:0;&#34; data-caption=&#34;LED timer in action at a surf competition in Manhattan Beach, CA.&#34;&gt;

&lt;img src=&#34;sbbc.png&#34; style=&#34;display:block; margin:0; padding:0;&#34; alt=&#34;LED timer in action at a surf competition in Manhattan Beach, CA.&#34; &gt;
&lt;/a&gt;


&lt;figcaption style=&#34;text-align: left;&#34; data-pre=&#34;Figure &#34; data-post=&#34;:&#34; &gt;
  
  &lt;p&gt;
    LED timer in action at a surf competition in Manhattan Beach, CA.
    
    
    
  &lt;/p&gt; 
&lt;/figcaption&gt;

&lt;/figure&gt;

&lt;p&gt;I told them I couldn&amp;rsquo;t make any promises, but I&amp;rsquo;d have a look.  I proceeded to take apart the LED display box and inspect the copper traces on the manufacturer&amp;rsquo;s printed circuit board (PCB) using my backpacking headlamp.  I identified the power source, microcontroller, and an array of shift registers.  I realized that if I could carefully remove the manufacturer&amp;rsquo;s microcontroller, and replace it with one of my own, I could inject serial data into the shift registers to control each of the 30 LED channels.&lt;/p&gt;
&lt;p&gt;I didn&amp;rsquo;t want to replicate the prior controller architecture, I wanted to make it better.  So, I made it wireless and waterproof.  I also added custom toggle buttons to improve the end-user&amp;rsquo;s experience.  I soldered the requisite wires into through holes on the manufacturer&amp;rsquo;s PCB, covered the joints with conformal coating, and ran a sequence of tests to identify any bugs in my software.  The controller worked flawlessly, and my total bill of materials was an order of magnitude less than the manufacturer&amp;rsquo;s quote &amp;ndash; about $135.&lt;/p&gt;






&lt;figure style=&#34;text-align: center; margin-top: 50px; margin-bottom: 50px;&#34;&gt;

  &lt;a data-fancybox=&#34;&#34; href=&#34;finished_product.png&#34; style=&#34;display:inline-block; margin:0; padding:0; line-height:0;&#34; data-caption=&#34;The finished product.&#34;&gt;

&lt;img src=&#34;finished_product.png&#34; style=&#34;display:block; margin:0; padding:0;&#34; alt=&#34;&amp;lt;center&amp;gt;The finished product.&amp;lt;/center&amp;gt;&#34; width=&#34;450&#34; &gt;
&lt;/a&gt;


&lt;figcaption style=&#34;text-align: left;&#34; data-pre=&#34;Figure &#34; data-post=&#34;:&#34; &gt;
  
  &lt;p&gt;
    &lt;center&gt;The finished product.&lt;/center&gt;
    
    
    
  &lt;/p&gt; 
&lt;/figcaption&gt;

&lt;/figure&gt;

&lt;h2 id=&#34;topics-not-covered&#34;&gt;Topics Not Covered&lt;/h2&gt;
&lt;p&gt;This project involved a few microcontroller flourishes that I chose not to cover:&lt;/p&gt;
&lt;ul&gt;
&lt;li&gt;Wireless communication using the &lt;a href=&#34;https://www.espressif.com/en/solutions/low-power-solutions/esp-now&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;ESP-NOW&lt;/a&gt; protocol&lt;/li&gt;
&lt;li&gt;Reducing power consumption using &lt;a href=&#34;https://docs.espressif.com/projects/esp-idf/en/stable/esp32/api-reference/system/sleep_modes.html&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;deep-sleep&lt;/a&gt; mode&lt;/li&gt;
&lt;li&gt;External hardware interrupts&lt;/li&gt;
&lt;li&gt;Shift register control&lt;/li&gt;
&lt;/ul&gt;
&lt;p&gt;I decided not to cover these topics in detail because fantastic resources already exist online.  Instead, I decided to focus mostly on the narrative journey, and provide some insight into my thought process as I worked through the problem.&lt;/p&gt;
&lt;p&gt;If you&amp;rsquo;re interested in learning more about microcontrollers and hobby electronics, I&amp;rsquo;d highly recommend checking out the &lt;a href=&#34;https://www.youtube.com/@Dronebotworkshop&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;DroneBot Workshop&lt;/a&gt; channel on YouTube.  Bill is a fantastic educator, and one of my favorite engineering content creators.&lt;/p&gt;
&lt;p&gt;If you are curious about my specific implementation, feel free to &lt;a href=&#34;../../#contact&#34;&gt;reach out&lt;/a&gt;.&lt;/p&gt;
&lt;h2 id=&#34;methodology&#34;&gt;Methodology&lt;/h2&gt;
&lt;p&gt;One of the most important lessons I learned in graduate school was the value of breaking large, complex problems into a sequence of simpler ones with measurable outcomes.  This framework permits incremental progress, and reduces the likelihood that you&amp;rsquo;ll waste a bunch of time on a problem you can&amp;rsquo;t actually solve.  This was my first guiding principle:&lt;/p&gt;








  


&lt;div class=&#34;px-4 py-3 mb-6 rounded-md bg-primary-100 dark:bg-primary-900&#34;&gt;
  
    
      &lt;div class=&#34;mb-2 font-bold text-primary-600 dark:text-primary-300&#34;&gt;
        Guiding Principle #1: Incremental Progress
      &lt;/div&gt;
    
    &lt;div class=&#34;dark:text-neutral-300&#34;&gt;Break problems into a series of sequential hurdles with measurable outcomes.&lt;/div&gt;
  
&lt;/div&gt;


&lt;p&gt;When I first got my hands on the LED timer, I had no idea whether I&amp;rsquo;d actually be able to fix it.  There were so many uncertainties:&lt;/p&gt;
&lt;ul&gt;
&lt;li&gt;What is the power and control architecture inside the box?&lt;/li&gt;
&lt;li&gt;Will I be able to make sense of the manufacturer&amp;rsquo;s PCB?&lt;/li&gt;
&lt;li&gt;Could I modify the manufacturer&amp;rsquo;s PCB without destroying it?&lt;/li&gt;
&lt;li&gt;Is the power system functional?&lt;/li&gt;
&lt;li&gt;Could I write a software program that mimics countdown timer logic?&lt;/li&gt;
&lt;li&gt;How might I allow the end-user to control the timer in an intuitive manner?&lt;/li&gt;
&lt;/ul&gt;
&lt;p&gt;I took a deep breath, and started breaking down the problem, working backwards from my desired endpoint.&lt;/p&gt;
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&lt;div class=&#34;mermaid&#34;&gt;
  
flowchart BT
    A[LED Power] --&gt; B[LED Control]
    B --&gt; C[Timer Logic]
    C --&gt; D[Interactivity]
    D --&gt; E[Wired Control]
    E --&gt; F[Wireless Control]


&lt;/div&gt;


&lt;p&gt;I knew I wanted to build a wireless controller with an array of buttons that would allow an end-user to set, start, stop, and reset the LED timer; but, before fussing over wireless capabilities, I needed to make sure I could build a wired controller, and write a simple countdown timer program; before I could do that, I needed figure out a way to illuminate each of the 30 LED channels independently; before that, I needed to figure out how the LEDs were being controlled; and before that, I needed to figure out how the LEDs were being powered.&lt;/p&gt;
&lt;p&gt;Soon, a mental map of the problem began to take shape:&lt;/p&gt;
&lt;div class=&#34;markmap&#34; style=&#34;height: 500px;&#34;&gt;

&lt;pre&gt;- Fix Display
  - Inspect Hardware
    - Power Supply
    - Shift Registers
    - Microcontroller
    - PCB Traces
  - Modify Hardware
    - ?
  - Write Software
    - Timer Logic
    - Button Logic
    - Sleep Mode
    - Wireless
  - Build Controller
    - Enclosure
    - Microcontroller
    - Battery
    - Buttons
    - Charging Port&lt;/pre&gt;
&lt;/div&gt;

&lt;h2 id=&#34;inspect-hardware&#34;&gt;Inspect Hardware&lt;/h2&gt;
&lt;p&gt;I knew needed to make sense of the electronic architecture of the display box &amp;ndash; if I couldn&amp;rsquo;t figure out how the display was &lt;em&gt;supposed&lt;/em&gt; to work, well then I obviously wouldn&amp;rsquo;t be able to modify it without wrecking it.  So, I opened the four access panels on the box, and started poking around.&lt;/p&gt;






&lt;figure style=&#34;text-align: center; margin-top: 50px; margin-bottom: 50px;&#34;&gt;

  &lt;a data-fancybox=&#34;&#34; href=&#34;blank_timer.jpeg&#34; style=&#34;display:inline-block; margin:0; padding:0; line-height:0;&#34; data-caption=&#34;My first look inside the display box; after nearly going blind, I draped a towel over the front of the box to attenuate the LED intensity.&#34;&gt;

&lt;img src=&#34;blank_timer.jpeg&#34; style=&#34;display:block; margin:0; padding:0;&#34; alt=&#34;My first look inside the display box; after nearly going blind, I draped a towel over the front of the box to attenuate the LED intensity.&#34; &gt;
&lt;/a&gt;


&lt;figcaption style=&#34;text-align: left;&#34; data-pre=&#34;Figure &#34; data-post=&#34;:&#34; &gt;
  
  &lt;p&gt;
    My first look inside the display box; after nearly going blind, I draped a towel over the front of the box to attenuate the LED intensity.
    
    
    
  &lt;/p&gt; 
&lt;/figcaption&gt;

&lt;/figure&gt;

&lt;p&gt;My high-level goal at this point was to figure out where I might be able to break into the usual chain of command in a minimally-invasive manner.  This was another one of my guiding principles:&lt;/p&gt;








  


&lt;div class=&#34;px-4 py-3 mb-6 rounded-md bg-primary-100 dark:bg-primary-900&#34;&gt;
  
    
      &lt;div class=&#34;mb-2 font-bold text-primary-600 dark:text-primary-300&#34;&gt;
        Guiding Principle #2: Do No (Unnecessary) Harm
      &lt;/div&gt;
    
    &lt;div class=&#34;dark:text-neutral-300&#34;&gt;If you can&amp;rsquo;t avoid making hardware modifications, be judicious.&lt;/div&gt;
  
&lt;/div&gt;


&lt;p&gt;The manufacturer&amp;rsquo;s PCB was gorgeous: they included capacitors and diodes in the power circuits to ensure clean, unidirectional current flow; all of the soldered joints were covered in conformal coating; and the wire harnessess were all professional-grade.  Somebody clearly put a lot of thought into the layout, and I really didn&amp;rsquo;t want to mess with it.  Instead, I was hoping I might be able to overwrite the manufacturer&amp;rsquo;s firmware, and leave the PCB completely intact.  Turns out, that wasn&amp;rsquo;t going to be possible, but more on that &lt;a href=&#34;https://www.blakecole.com/post/3-led-timer/#modify-hardware&#34;&gt;later&lt;/a&gt;.&lt;/p&gt;






&lt;figure style=&#34;text-align: center; margin-top: 50px; margin-bottom: 50px;&#34;&gt;

  &lt;a data-fancybox=&#34;&#34; href=&#34;first_look_pcb.jpeg&#34; style=&#34;display:inline-block; margin:0; padding:0; line-height:0;&#34; data-caption=&#34;Initial inspection of the manufacturer&amp;rsquo;s printed circuit board (PCB), with annonymous finger pointing at the microcontroller.&#34;&gt;

&lt;img src=&#34;first_look_pcb.jpeg&#34; style=&#34;display:block; margin:0; padding:0;&#34; alt=&#34;Initial inspection of the manufacturer&amp;#39;s printed circuit board (PCB), with annonymous finger pointing at the microcontroller.&#34; width=&#34;500&#34; &gt;
&lt;/a&gt;


&lt;figcaption style=&#34;text-align: left;&#34; data-pre=&#34;Figure &#34; data-post=&#34;:&#34; &gt;
  
  &lt;p&gt;
    Initial inspection of the manufacturer&amp;rsquo;s printed circuit board (PCB), with annonymous finger pointing at the microcontroller.
    
    
    
  &lt;/p&gt; 
&lt;/figcaption&gt;

&lt;/figure&gt;

&lt;p&gt;Before anything else, I wanted to make sure that the display box was still capable of delivering power to the LEDs.&lt;/p&gt;
&lt;h3 id=&#34;power-supply&#34;&gt;Power Supply&lt;/h3&gt;
&lt;p&gt;When inspecting a new piece of hardware, I generally like to start with the power system &amp;ndash; it is the least ambiguous and easiest to identify.  The external power cable is typically connected to a &lt;em&gt;power supply&lt;/em&gt;, which contains a &lt;a href=&#34;https://en.wikipedia.org/wiki/Rectifier&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;rectifier&lt;/a&gt; that converts alternating current (AC) to direct current (DC).  Once I found the power supply, I identified positive voltage and ground wires, and traced them to the PCB.  The output voltage is almost always specified on the power supply, but it is nevertheless good practice to verify it with a multimeter.&lt;/p&gt;






&lt;figure style=&#34;text-align: center; margin-top: 50px; margin-bottom: 50px;&#34;&gt;

  &lt;a data-fancybox=&#34;&#34; href=&#34;power_supply.png&#34; style=&#34;display:inline-block; margin:0; padding:0; line-height:0;&#34; data-caption=&#34;A 27V power supply.&#34;&gt;

&lt;img src=&#34;power_supply.png&#34; style=&#34;display:block; margin:0; padding:0;&#34; alt=&#34;A 27V power supply.&#34; width=&#34;400&#34; &gt;
&lt;/a&gt;


&lt;figcaption style=&#34;text-align: left;&#34; data-pre=&#34;Figure &#34; data-post=&#34;:&#34; &gt;
  
  &lt;p&gt;
    A 27V power supply.
    
    
    
  &lt;/p&gt; 
&lt;/figcaption&gt;

&lt;/figure&gt;

&lt;p&gt;I plugged in the display box, and immediately heard the power supply fan roar to life.  A default pattern of LEDs illuminated.  So far so good.  The power supply was outputting 27 VDC, which I found somewhat curious &amp;ndash; most devices run on 5V, 12V, or 24V.  Turns out, there was a very good reason for this.  I noticed that each LED channel was composed of 13 individual LEDs, connected in series.  All LEDs are designed to operate within a nominal voltage range, and a quick search on &lt;a href=&#34;https://www.digikey.com/en/products/detail/cree-led/C5SMF-RJF-CT0W0BB2/4794052&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;DigiKey&lt;/a&gt; indicated that these LEDs preferred to run at 2.1V.&lt;/p&gt;
&lt;p&gt;What is 27 Volts, divided by 13 LEDs?  &lt;strong&gt;2.08 Volts per LED&lt;/strong&gt;.  Not a coincidence.&lt;/p&gt;
&lt;p&gt;I also noticed a secondary pair of power wires which were routed from the main power supply to the PCB via a 5V &lt;a href=&#34;https://en.wikipedia.org/wiki/Battery_eliminator_circuit&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;battery eliminator circuit&lt;/a&gt;, or BEC.  A BEC is a small onboard voltage regulator that draws power from a device’s main power source, and steps it down to a stable, lower voltage to safely run sensitive electronics like microcontrollers.  I made a mental note of the 5V input on the PCB &amp;ndash; eventually, I would probably need to find a place to tap into this voltage source to power my own microcontroller.&lt;/p&gt;
&lt;h3 id=&#34;microcontrollers&#34;&gt;Microcontrollers&lt;/h3&gt;
&lt;p&gt;Have you ever wondered how modern consumer applicances work?&lt;/p&gt;
&lt;p&gt;What actually happens when you push the buttons on your microwave oven, or turn the dial on your washing machine?  How do these actions actually control the device?   If you aren&amp;rsquo;t an electrical engineer, you might assume that whatever is going on behind the scenes is far beyond your understanding.  Well, I can assure you, it isn&amp;rsquo;t!&lt;/p&gt;
&lt;p&gt;Hiding behind almost every electronic device is a single key component: a &lt;em&gt;&lt;strong&gt;microcontroller&lt;/strong&gt;&lt;/em&gt;.  A microcontroller is essentially a simple computer that continuously measures electrical signals (inputs) to sense its environment, makes decisions based on those inputs, and generates electrical signals (outputs) to control actuators or communicate with other devices.  If an electronic device is a football team, the microcontroller is the quarterback: it coordinates all the peripherals (i.e., sensors, actuators, displays) using predefined communication protocols.&lt;/p&gt;






&lt;figure style=&#34;text-align: center; margin-top: 50px; margin-bottom: 50px;&#34;&gt;

  &lt;a data-fancybox=&#34;&#34; href=&#34;arduino.png&#34; style=&#34;display:inline-block; margin:0; padding:0; line-height:0;&#34; data-caption=&#34;An ATmega328P microcontroller (outlined in red) on an Arduino UNO printed circuit board.&#34;&gt;

&lt;img src=&#34;arduino.png&#34; style=&#34;display:block; margin:0; padding:0;&#34; alt=&#34;An [ATmega328P](https://www.digikey.com.au/en/products/detail/microchip-technology/ATMEGA328P-PU/1914589) microcontroller (outlined in red) on an [Arduino UNO](https://store-usa.arduino.cc/products/arduino-uno-rev3?queryID=f87920403a6ad2e9dba7b0fe935fd940) printed circuit board.&#34; width=&#34;400&#34; &gt;
&lt;/a&gt;


&lt;figcaption style=&#34;text-align: left;&#34; data-pre=&#34;Figure &#34; data-post=&#34;:&#34; &gt;
  
  &lt;p&gt;
    An &lt;a href=&#34;https://www.digikey.com.au/en/products/detail/microchip-technology/ATMEGA328P-PU/1914589&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;ATmega328P&lt;/a&gt; microcontroller (outlined in red) on an &lt;a href=&#34;https://store-usa.arduino.cc/products/arduino-uno-rev3?queryID=f87920403a6ad2e9dba7b0fe935fd940&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Arduino UNO&lt;/a&gt; printed circuit board.
    
    
    
  &lt;/p&gt; 
&lt;/figcaption&gt;

&lt;/figure&gt;

&lt;p&gt;It really is as simple as that!  Once you start looking, you&amp;rsquo;ll see these everwhere &amp;ndash; in your kitchen, in your car, and even on your body.  Indeed, smartwatches contain &lt;em&gt;very&lt;/em&gt; small microcontrollers that read data from embedded sensors and specify which pixels to illuminate on the watch face.  Basically any time you see a device with buttons, switches, or LED indicators, you can be reasonably sure that there is a microcontroller nearby.&lt;/p&gt;
&lt;p&gt;This fact has an exciting corollary: &lt;em&gt;&lt;strong&gt;if you know how to program microcontrollers, you can control almost any electronic system&lt;/strong&gt;&lt;/em&gt;.  This is honestly insane.  Microcontrollers allow you to build everything from home security networks to automated drip-irrigation systems for your houseplants.  You can design party costumes with embedded LEDs that react to light, sound, and heat signatures.  You can build custom weather stations.  You could design a remote-controlled actuator to open and close a window, or deploy a retractable sun shade.  The possibilities are functionally infinite.&lt;/p&gt;
&lt;p&gt;If you want to learn how to do this, there is no shortage of tutorials online (check out &lt;a href=&#34;https://www.youtube.com/@Dronebotworkshop&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;DroneBot Workshop&lt;/a&gt;).  If you like working on little personal projects, I promise it is worth your time.&lt;/p&gt;
&lt;p&gt;Without an understanding of microcontrollers, there is no chance I would have been able to build a controller for this LED display.  Again, this would be like trying to run a play in football without a quarterback &amp;ndash; the center would hike the ball into the clear blue sky, and the play would be over.&lt;/p&gt;






&lt;figure style=&#34;text-align: center; margin-top: 50px; margin-bottom: 50px;&#34;&gt;

  &lt;a data-fancybox=&#34;&#34; href=&#34;first_look_mcu.jpeg&#34; style=&#34;display:inline-block; margin:0; padding:0; line-height:0;&#34; data-caption=&#34;Computer, enhance!  A close-up image of the OEM microcontroller.  The alphanumeric code on this chip helped me identify the microcontroller, and find its corresponding data sheet.  The chip pinout diagram allowed me to zero in on a subset of ports that could be used for LED control.&#34;&gt;

&lt;img src=&#34;first_look_mcu.jpeg&#34; style=&#34;display:block; margin:0; padding:0;&#34; alt=&#34;Computer, enhance!  A close-up image of the OEM microcontroller.  The alphanumeric code on this chip helped me identify the microcontroller, and find its corresponding [data sheet](https://www.digikey.ch/htmldatasheets/production/33029/0/0/1/mc68h-s-c705c8a.html).  The chip pinout diagram allowed me to zero in on a subset of ports that could be used for LED control.&#34; width=&#34;500&#34; &gt;
&lt;/a&gt;


&lt;figcaption style=&#34;text-align: left;&#34; data-pre=&#34;Figure &#34; data-post=&#34;:&#34; &gt;
  
  &lt;p&gt;
    Computer, enhance!  A close-up image of the OEM microcontroller.  The alphanumeric code on this chip helped me identify the microcontroller, and find its corresponding &lt;a href=&#34;https://www.digikey.ch/htmldatasheets/production/33029/0/0/1/mc68h-s-c705c8a.html&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;data sheet&lt;/a&gt;.  The chip pinout diagram allowed me to zero in on a subset of ports that could be used for LED control.
    
    
    
  &lt;/p&gt; 
&lt;/figcaption&gt;

&lt;/figure&gt;

&lt;p&gt;After I had located the microcontroller on the PCB, I knew I would need to identify the pins that were being used to control the LEDs.  Once I figured this out, I could either:&lt;/p&gt;
&lt;ul&gt;
&lt;li&gt;Leave the OEM microcontroller in place, overwrite the firmware, and send commands to these pins.&lt;/li&gt;
&lt;li&gt;Yank out the OEM microcontroller, replace it with one of my own, and inject my own signals (the specific nature of these signals was still TBD).&lt;/li&gt;
&lt;/ul&gt;
&lt;p&gt;There were a few ways of going about this.  The easiest was probably to connect each pin to an oscilloscope and look for a digital signal, but I didn&amp;rsquo;t have an oscilloscope.  Back to the drawing board.  The alphanumeric code printed on the microcontroller indicated that it was a &lt;a href=&#34;https://www.digikey.ch/htmldatasheets/production/33029/0/0/1/mc68h-s-c705c8a.html&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;MC68HSC705C8A&lt;/a&gt;, manufactured by NXP Semiconductors.  I pulled up the pinout diagram, and began a process of elimination.&lt;/p&gt;






&lt;figure style=&#34;text-align: center; margin-top: 50px; margin-bottom: 50px;&#34;&gt;

  &lt;a data-fancybox=&#34;&#34; href=&#34;mcu_pinout.png&#34; style=&#34;display:inline-block; margin:0; padding:0; line-height:0;&#34; data-caption=&#34;Using this pinout diagram, I was able to eliminate a number of pins from consideration; moreover, it revealed the location of the microcontroller&amp;rsquo;s designated serial output ports &amp;ndash; these became my prime suspects.&#34;&gt;

&lt;img src=&#34;mcu_pinout.png&#34; style=&#34;display:block; margin:0; padding:0;&#34; alt=&#34;Using this pinout diagram, I was able to eliminate a number of pins from consideration; moreover, it revealed the location of the microcontroller&amp;#39;s designated serial output ports -- these became my prime suspects.&#34; &gt;
&lt;/a&gt;


&lt;figcaption style=&#34;text-align: left;&#34; data-pre=&#34;Figure &#34; data-post=&#34;:&#34; &gt;
  
  &lt;p&gt;
    Using this pinout diagram, I was able to eliminate a number of pins from consideration; moreover, it revealed the location of the microcontroller&amp;rsquo;s designated serial output ports &amp;ndash; these became my prime suspects.
    
    
    
  &lt;/p&gt; 
&lt;/figcaption&gt;

&lt;/figure&gt;

&lt;p&gt;Once I had narrowed my search to five or six pins, I used my multimeter to check if any of them exhibited any electrical activity &amp;ndash; I wouldn&amp;rsquo;t be able to resolve the shape of digital signals, but I&amp;rsquo;d at least be able to see the voltage jumping around.  After a few rounds of this, I had a fairly good sense of which GPIO pins were being used for LED control.&lt;/p&gt;
&lt;p&gt;The precise means by which the LEDs were being controlled was still somewhat unclear to me, but I was making progress.  I knew the command signals originated from the microcontroller, and I knew which ports were being used to issue those commands.  I knew that a high-power LED display like this one couldn&amp;rsquo;t be powered directly by a microcontroller, so I expected to see an array of &lt;a href=&#34;https://en.wikipedia.org/wiki/Relay&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;relays&lt;/a&gt; or other electronic switches further downstream.  But, confusingly, I didn&amp;rsquo;t see anything on the PCB that looked like a relay &amp;ndash; much less 30 of them.  Something else was going on.&lt;/p&gt;
&lt;h3 id=&#34;shift-registers&#34;&gt;Shift Registers&lt;/h3&gt;
&lt;p&gt;Microcontrollers have a limited number of so-called general purpose input-output (GPIO) pins.  These GPIO pins are the physical ports where the microcontroller interfaces with sensors, actuators, and displays.  For most simple projects, the handful of GPIO ports built into the microcontroller are more than sufficient; however, some electronic devices require hundreds (e.g., automotive control), thousands, or even millions (e.g., 4K LED display) of outputs.&lt;/p&gt;
&lt;p&gt;When the number of required outputs exceeds the number of GPIO pins on the microcontroller, a device called a &lt;a href=&#34;https://en.wikipedia.org/wiki/Shift_register&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;shift register&lt;/a&gt; can come in handy.  From a functional perspective, shift registers are incredibly simple: they have a sequence of memory slots that can hold a single binary bit (i.e., either a &lt;code&gt;0&lt;/code&gt; or a &lt;code&gt;1&lt;/code&gt;); when a new bit is introduced at the &amp;ldquo;front end&amp;rdquo;, all the other values shift over by one slot, and the oldest value gets bumped out.  Importantly, the state of each memory slot can be read simultaneously.  You can think of a shift register as a marquee billboard, wherein new tiles can only be added from the lefthand side: each time a new tile is added, all the other tiles shift to the right, and the oldest tile is knocked off the end.&lt;/p&gt;






&lt;figure style=&#34;text-align: center; margin-top: 50px; margin-bottom: 50px;&#34;&gt;

  &lt;a data-fancybox=&#34;&#34; href=&#34;spyder_marquee.jpeg&#34; style=&#34;display:inline-block; margin:0; padding:0; line-height:0;&#34; data-caption=&#34;A shift register can be thought of as a simple marquee billboard, wherein new tiles can only be added from the left, and each tile can only display a 0 or 1 [photo: Spyder Surf].&#34;&gt;

&lt;img src=&#34;spyder_marquee.jpeg&#34; style=&#34;display:block; margin:0; padding:0;&#34; alt=&#34;A shift register can be thought of as a simple marquee billboard, wherein new tiles can only be added from the left, and each tile can only display a ```0``` or ```1``` [photo: [Spyder Surf](https://spydersurf.com/)].&#34; width=&#34;500&#34; &gt;
&lt;/a&gt;


&lt;figcaption style=&#34;text-align: left;&#34; data-pre=&#34;Figure &#34; data-post=&#34;:&#34; &gt;
  
  &lt;p&gt;
    A shift register can be thought of as a simple marquee billboard, wherein new tiles can only be added from the left, and each tile can only display a &lt;code&gt;0&lt;/code&gt; or &lt;code&gt;1&lt;/code&gt; [photo: &lt;a href=&#34;https://spydersurf.com/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Spyder Surf&lt;/a&gt;].
    
    
    
  &lt;/p&gt; 
&lt;/figcaption&gt;

&lt;/figure&gt;

&lt;p&gt;So what&amp;rsquo;s the big deal?&lt;/p&gt;
&lt;p&gt;A Serial-In, Parallel-Out (SIPO) shift register effectively transforms a single GPIO pin into many.  How is this possible?  A microcontroller can transmit individual bits (i.e., &lt;code&gt;0&lt;/code&gt;s and &lt;code&gt;1&lt;/code&gt;s) to a shift register hundreds of thousands of times per second; the shift register then makes the data contained in those bits available in parallel, effectively instantaneously.  A &lt;code&gt;clock&lt;/code&gt; signal (generally a square wave) determines when the shift register reads a new value, and a &lt;code&gt;latch&lt;/code&gt; signal determines when the data will be sent to the output channels.  The following animation, created by &lt;a href=&#34;https://lastminuteengineers.com/74hc595-shift-register-arduino-tutorial/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Last Minute Engineers&lt;/a&gt;, is an absolutely brilliant demonstration of how shift registers work.&lt;/p&gt;






&lt;figure style=&#34;text-align: center; margin-top: 50px; margin-bottom: 50px;&#34;&gt;

  &lt;a data-fancybox=&#34;&#34; href=&#34;shift_register.gif&#34; style=&#34;display:inline-block; margin:0; padding:0; line-height:0;&#34; data-caption=&#34;Animation showing the inner workings of a shift register.  Every time the clock signal goes from high (1) to low (0), a new bit is read into the shift register; every time the latch signal goes from high to low, the full set of stored bits is transfered to the storage register, where they can be read simultaneously on each output channel  [source: Last Minute Engineers].&#34;&gt;

&lt;img src=&#34;shift_register.gif&#34; style=&#34;display:block; margin:0; padding:0;&#34; alt=&#34;Animation showing the inner workings of a shift register.  Every time the ```clock``` signal goes from high (```1```) to low (```0```), a new bit is read into the ```shift register```; every time the ```latch``` signal goes from high to low, the full set of stored bits is transfered to the ```storage register```, where they can be read simultaneously on each ```output``` channel  [source: [Last Minute Engineers](https://lastminuteengineers.com/74hc595-shift-register-arduino-tutorial/)].&#34; &gt;
&lt;/a&gt;


&lt;figcaption style=&#34;text-align: left;&#34; data-pre=&#34;Figure &#34; data-post=&#34;:&#34; &gt;
  
  &lt;p&gt;
    Animation showing the inner workings of a shift register.  Every time the &lt;code&gt;clock&lt;/code&gt; signal goes from high (&lt;code&gt;1&lt;/code&gt;) to low (&lt;code&gt;0&lt;/code&gt;), a new bit is read into the &lt;code&gt;shift register&lt;/code&gt;; every time the &lt;code&gt;latch&lt;/code&gt; signal goes from high to low, the full set of stored bits is transfered to the &lt;code&gt;storage register&lt;/code&gt;, where they can be read simultaneously on each &lt;code&gt;output&lt;/code&gt; channel  [source: &lt;a href=&#34;https://lastminuteengineers.com/74hc595-shift-register-arduino-tutorial/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Last Minute Engineers&lt;/a&gt;].
    
    
    
  &lt;/p&gt; 
&lt;/figcaption&gt;

&lt;/figure&gt;

&lt;p&gt;So, if you wanted to control the &lt;code&gt;ON/OFF&lt;/code&gt; state of 8 LEDs, rather than wiring each LED to its own GPIO pin, you could connect the LEDs to an 8-bit shift register.  If you wanted to turn on the first three LEDs, you would send the following serial signal to the shift register: &lt;code&gt;1 1 1 0 0 0 0 0&lt;/code&gt;.&lt;/p&gt;






&lt;figure style=&#34;text-align: center; margin-top: 50px; margin-bottom: 50px;&#34;&gt;

  &lt;a data-fancybox=&#34;&#34; href=&#34;sipo_shift_register.png&#34; style=&#34;display:inline-block; margin:0; padding:0; line-height:0;&#34; data-caption=&#34;Diagram of a 4-bit shift register [source: Wikimedia Commons].&#34;&gt;

&lt;img src=&#34;sipo_shift_register.png&#34; style=&#34;display:block; margin:0; padding:0;&#34; alt=&#34;Diagram of a 4-bit shift register [source: [Wikimedia Commons](https://en.wikipedia.org/wiki/File:4-Bit_SIPO_Shift_Register.svg)].&#34; &gt;
&lt;/a&gt;


&lt;figcaption style=&#34;text-align: left;&#34; data-pre=&#34;Figure &#34; data-post=&#34;:&#34; &gt;
  
  &lt;p&gt;
    Diagram of a 4-bit shift register [source: &lt;a href=&#34;https://en.wikipedia.org/wiki/File:4-Bit_SIPO_Shift_Register.svg&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Wikimedia Commons&lt;/a&gt;].
    
    
    
  &lt;/p&gt; 
&lt;/figcaption&gt;

&lt;/figure&gt;

&lt;p&gt;What if you wanted to control more than 8 LEDs?  Perhaps many more?  One of the awesome things about shift registers is that they can be connected in series; in this configuration, the oldest bit in the leading register spills over and becomes the newest bit in the following register.&lt;/p&gt;
&lt;p&gt;I was starting to suspect that this was probably how the LEDs on my display were being controlled: the microcontroller sends 32 bits of serial data to the first of four 8-bit shift registers connected in series; once the first shift register is full, it begins passing bits to the second, and so on, like a beautiful cascading waterfall.  This would explain the small number of active GPIO pins on the microcontroller.&lt;/p&gt;






&lt;figure style=&#34;text-align: center; margin-top: 50px; margin-bottom: 50px;&#34;&gt;

  &lt;a data-fancybox=&#34;&#34; href=&#34;waterfall.png&#34; style=&#34;display:inline-block; margin:0; padding:0; line-height:0;&#34; data-caption=&#34;Shift registers connected in series allow data to cascade from one to another [Source: DALL-E 3].&#34;&gt;

&lt;img src=&#34;waterfall.png&#34; style=&#34;display:block; margin:0; padding:0;&#34; alt=&#34;Shift registers connected in series allow data to cascade from one to another [Source: DALL-E 3].&#34; width=&#34;400&#34; &gt;
&lt;/a&gt;


&lt;figcaption style=&#34;text-align: left;&#34; data-pre=&#34;Figure &#34; data-post=&#34;:&#34; &gt;
  
  &lt;p&gt;
    Shift registers connected in series allow data to cascade from one to another [Source: DALL-E 3].
    
    
    
  &lt;/p&gt; 
&lt;/figcaption&gt;

&lt;/figure&gt;

&lt;p&gt;However, I still couldn&amp;rsquo;t explain the lack of relays: you can&amp;rsquo;t illuminate high-power LEDs using weak digital signals like these.&lt;/p&gt;
&lt;p&gt;Well, it turns out the shift registers used in this particular application are somewhat special &amp;ndash; they are so-called &lt;em&gt;sinking current&lt;/em&gt; shift registers.  According to the &lt;a href=&#34;https://www.ti.com/lit/ds/symlink/tpic6a595.pdf&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;TPIC6A595 data sheet&lt;/a&gt;, &amp;ldquo;Outputs are low-side, open-drain DMOS transistors with output ratings of 50V and a 350mA continuous sink-current capability.  When data in the output buffers is low, the DMOS-transistor outputs are off.  When data is high, the DMOS-transistor outputs have sink current capability.&amp;rdquo;&lt;/p&gt;
&lt;p&gt;This effectively means that the shift registers are able to act like relays, dictating the flow of high-voltage current to each LED channel.  When the shift register output is &lt;code&gt;low&lt;/code&gt;, the corresponding LED channel is allowed to float at 27 VDC (0 V differential voltage), and will not illuminate; when the shift register output goes &lt;code&gt;high&lt;/code&gt;, the channel is grounded, and the LEDs illuminate.&lt;/p&gt;






&lt;figure style=&#34;text-align: center; margin-top: 50px; margin-bottom: 50px;&#34;&gt;

  &lt;a data-fancybox=&#34;&#34; href=&#34;shift_reg_pinout.png&#34; style=&#34;display:inline-block; margin:0; padding:0; line-height:0;&#34; data-caption=&#34;More handwritten notes.  At this point, I had mostly figured out which GPIO pins the microcontroller was using to drive the shift registers; however, just to make sure, I used my backpacking headlamp to follow the copper traces from the first shift register&amp;rsquo;s inputs (SER IN, RCK, and SRCK) back to their respective sources at the microcontroller.  The digital output pins D0 - D7 are each connected to a single LED channel.&#34;&gt;

&lt;img src=&#34;shift_reg_pinout.png&#34; style=&#34;display:block; margin:0; padding:0;&#34; alt=&#34;More handwritten notes.  At this point, I had mostly figured out which GPIO pins the microcontroller was using to drive the shift registers; however, just to make sure, I used my backpacking headlamp to follow the copper traces from the first shift register&amp;#39;s inputs (```SER IN```, ```RCK```, and ```SRCK```) back to their respective sources at the microcontroller.  The digital output pins ```D0``` - ```D7``` are each connected to a single LED channel.&#34; &gt;
&lt;/a&gt;


&lt;figcaption style=&#34;text-align: left;&#34; data-pre=&#34;Figure &#34; data-post=&#34;:&#34; &gt;
  
  &lt;p&gt;
    More handwritten notes.  At this point, I had mostly figured out which GPIO pins the microcontroller was using to drive the shift registers; however, just to make sure, I used my backpacking headlamp to follow the copper traces from the first shift register&amp;rsquo;s inputs (&lt;code&gt;SER IN&lt;/code&gt;, &lt;code&gt;RCK&lt;/code&gt;, and &lt;code&gt;SRCK&lt;/code&gt;) back to their respective sources at the microcontroller.  The digital output pins &lt;code&gt;D0&lt;/code&gt; - &lt;code&gt;D7&lt;/code&gt; are each connected to a single LED channel.
    
    
    
  &lt;/p&gt; 
&lt;/figcaption&gt;

&lt;/figure&gt;

&lt;p&gt;Gradually, a more complete picture was coming into focus: the main power supply provides 27 VDC to illuminate the LEDs, while the BEC provides 5 VDC to power the microcontroller and shift registers.  Each of the four white connectors beneath the microcontroller are responsible for delivering power to a single LED panel; each panel has either 7 or 8 channels, and each channel has 13 discrete LEDs.  There is a copper trace connecting each of the four shift registers in series, allowing bits to cascade from one to the next.  The microcontroller sends serial commands to the shift registers, and the shift registers dictate which LED channels are illuminated.&lt;/p&gt;
&lt;p&gt;Knowing this, I began to formulate a plan to control each LED channel independently.&lt;/p&gt;
&lt;h2 id=&#34;modify-hardware&#34;&gt;Modify Hardware&lt;/h2&gt;
&lt;p&gt;I soon realized that I wouldn&amp;rsquo;t be able to repurpose the existing microcontroller.  For one, there was no obvious means of communicating with it &amp;ndash; no USB port, no exposed UART serial pins.  Further, I didn&amp;rsquo;t have any experience with this specific microcontroller, so I didn&amp;rsquo;t know which application programming interfaces (APIs) I could utilize.  So, I figured, if I was going to write my own firmware anyway, I might as well remove the existing microcontroller and use one I was already familiar with.&lt;/p&gt;






&lt;figure style=&#34;text-align: center; margin-top: 50px; margin-bottom: 50px;&#34;&gt;

  &lt;a data-fancybox=&#34;&#34; href=&#34;inject_serial.jpeg&#34; style=&#34;display:inline-block; margin:0; padding:0; line-height:0;&#34; data-caption=&#34;Having removed the OEM microcontroller, I was able to use jumper wires to inject serial signals from my microcontroller.  Eventually, I would develop a more robust wiring scheme, but this approach worked for proof of concept purposes.&#34;&gt;

&lt;img src=&#34;inject_serial.jpeg&#34; style=&#34;display:block; margin:0; padding:0;&#34; alt=&#34;Having removed the OEM microcontroller, I was able to use jumper wires to inject serial signals from my microcontroller.  Eventually, I would develop a more robust wiring scheme, but this approach worked for *proof of concept* purposes.&#34; &gt;
&lt;/a&gt;


&lt;figcaption style=&#34;text-align: left;&#34; data-pre=&#34;Figure &#34; data-post=&#34;:&#34; &gt;
  
  &lt;p&gt;
    Having removed the OEM microcontroller, I was able to use jumper wires to inject serial signals from my microcontroller.  Eventually, I would develop a more robust wiring scheme, but this approach worked for &lt;em&gt;proof of concept&lt;/em&gt; purposes.
    
    
    
  &lt;/p&gt; 
&lt;/figcaption&gt;

&lt;/figure&gt;

&lt;h3 id=&#34;signal-injection&#34;&gt;Signal Injection&lt;/h3&gt;
&lt;p&gt;Using a small flathead screwdriver, I gently removed the microcontroller from its 40-pin DIP IC socket adaptor.  Admittedly, this was a little freaky; once the microcontroller was removed, we had a football team with no quarterback.  When I plugged in the display, a completely random sequence of LED channels was illuminated.  This was disheartening, but not altogether surprising.  I pressed on.&lt;/p&gt;
&lt;p&gt;I took a quick break from hardware hacking to figure out how to control shift registers using a microcontroller; a few Google searches and tutorials later, I felt like I was ready to give it a crack.  I used jumper wires to connect my microcontroller to the appropriate pins on the socket adaptor, and deployed my code.&lt;/p&gt;
&lt;p&gt;Nothing happend.  This made sense.  The display box was unplugged. &lt;em&gt;Sigh&lt;/em&gt;.&lt;/p&gt;
&lt;p&gt;With my microcontroller dutifully sending serial signals to the shift registers (I hoped), I took a deep breath, grabbed the display box power cable, and plugged it in.&lt;/p&gt;
&lt;p&gt;It worked.&lt;/p&gt;
&lt;p&gt;It f**king worked!&lt;/p&gt;

    &lt;div style=&#34;position: relative; padding-bottom: 56.25%; height: 0; overflow: hidden;&#34;&gt;
      &lt;iframe allow=&#34;accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share&#34; allowfullscreen=&#34;allowfullscreen&#34; loading=&#34;eager&#34; referrerpolicy=&#34;strict-origin-when-cross-origin&#34; src=&#34;https://www.youtube.com/embed/7D-akG85p3o?autoplay=0&amp;amp;controls=1&amp;amp;end=0&amp;amp;loop=0&amp;amp;mute=0&amp;amp;start=0&#34; style=&#34;position: absolute; top: 0; left: 0; width: 100%; height: 100%; border:0;&#34; title=&#34;YouTube video&#34;&gt;&lt;/iframe&gt;
    &lt;/div&gt;

&lt;p&gt;I can&amp;rsquo;t really express the excitement I felt in this moment.  When I saw intelligible numeric digits dancing across each panel, I knew I had basically  won the game &amp;ndash; I had independent control of the LED channels.  I was going to beat this thing.  Everything else &amp;ndash; writing the timer logic, enabling wireless communication, adding external interrupts &amp;ndash; all these tasks would take time and effort, but I knew I could do them.&lt;/p&gt;
&lt;p&gt;As it turned out, I wasn&amp;rsquo;t out of the woods, not even close &amp;ndash; many painful setbacks were yet to come.  But I got used to them.  I&amp;rsquo;d throw a temper tantrum (sorry Mom and Dad), settle down, and figure out what to do next.&lt;/p&gt;
&lt;h3 id=&#34;a-small-setback&#34;&gt;A Small Setback&lt;/h3&gt;
&lt;p&gt;At some point two of the LED channels on one of the panels inexplicably went dark, making the afflicted digit unreadable.  This was infuriating &amp;ndash; I had come so far!  Nevertheless, there was no way I could deliver the display to the South Bay Boardriders Club in this condition.  This was sort of an &lt;em&gt;all-or-nothing&lt;/em&gt; game.  So, I started tinkering.&lt;/p&gt;
&lt;p&gt;First, I used a small flathead screwdriver to gently scrape the conformal coating off the LED solder joints on the broken panel.  I then powered up the display, and used my microcontroller to send the digit &lt;code&gt;8&lt;/code&gt; to this panel (or, in serial speak, &lt;code&gt;11111111&lt;/code&gt;) to ensure that all 7 LED channels were receiving power.  The two broken channels remained dark.  Using a 3V power supply, I applied voltage across each LED in the broken channels to identify the duds; this was a little sketchy, as the max rated voltage for each LED was 2.6V.  I was effectively risking destroying good LEDs in order to find the bad ones, but I figured they could probably handle a momentary hit of excess voltage.  LEDs are pretty sturdy.  Next, I used desoldering braid to remove the solder from the anode and cathode of each broken LED, and popped them out of the board.  I ordered a few replacement LEDs from DigiKey, and waited a few days.  Sometimes it is good to take a little break!&lt;/p&gt;








  


&lt;div class=&#34;px-4 py-3 mb-6 rounded-md bg-primary-100 dark:bg-primary-900&#34;&gt;
  
    
      &lt;div class=&#34;mb-2 font-bold text-primary-600 dark:text-primary-300&#34;&gt;
        Guiding Principle #3: Prioritize Sanity
      &lt;/div&gt;
    
    &lt;div class=&#34;dark:text-neutral-300&#34;&gt;Take a break, avoid a headache!&lt;/div&gt;
  
&lt;/div&gt;


&lt;p&gt;This principle was flagrantly violated many times over the course of this project, but so it goes.&lt;/p&gt;
&lt;p&gt;Anyway, when my replacement LEDs arrived, I soldered them in place, applied conformal coating, and tested the panel &amp;ndash; good as new.  Another sigh of relief.&lt;/p&gt;








  
  


&lt;div class=&#34;gallery&#34;&gt;

  
  
  
  
    
    
    
    
    
  &lt;a data-fancybox=&#34;gallery-led-panel&#34; href=&#34;https://www.blakecole.com/post/3-led-timer/led-panel/IMG_5712.jpeg&#34; &gt;
  &lt;img src=&#34;https://www.blakecole.com/post/3-led-timer/led-panel/IMG_5712_hu17002878301467713175.jpeg&#34; alt=&#34;&#34;&gt;
  &lt;/a&gt;
  
    
    
    
    
    
  &lt;a data-fancybox=&#34;gallery-led-panel&#34; href=&#34;https://www.blakecole.com/post/3-led-timer/led-panel/IMG_5807.jpeg&#34; &gt;
  &lt;img src=&#34;https://www.blakecole.com/post/3-led-timer/led-panel/IMG_5807_hu1487689399782247760.jpeg&#34; alt=&#34;&#34;&gt;
  &lt;/a&gt;
  

  
&lt;/div&gt;

&lt;h3 id=&#34;adding-garnishes&#34;&gt;Adding Garnishes&lt;/h3&gt;
&lt;p&gt;Having overcome the first hurdle &amp;ndash; and the biggest hurdle &amp;ndash; I went to work on all the little flourishes.  I cleaned up the wiring and found better, more permanent places on the PCB to inject my signals; I wrote code to handle the &lt;a href=&#34;https://www.blakecole.com/post/3-led-timer/#timer-logic&#34;&gt;countdown timer logic&lt;/a&gt;; I designed a state machine that could be controlled using toggle buttons, and wrote code that would recognize each button press as a non-blocking external interrupt; finally, I enabled wireless communication between the microcontroller I had installed inside the LED display box and a secondary external microcontroller outfitted with a series of toggle buttons.&lt;/p&gt;
&lt;p&gt;The external microcontroller &amp;ndash; which would be housed inside a waterproof enclosure and handle user commands &amp;ndash; would be battery powered, so I enabled &lt;a href=&#34;https://docs.espressif.com/projects/esp-idf/en/stable/esp32/api-reference/system/sleep_modes.html&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;deep-sleep mode&lt;/a&gt; to conserve power when it was not in use.  The end-user could wake the microcontroller by pressing a button.&lt;/p&gt;






&lt;figure style=&#34;text-align: center; margin-top: 50px; margin-bottom: 50px;&#34;&gt;

  &lt;a data-fancybox=&#34;&#34; href=&#34;esp32.jpeg&#34; style=&#34;display:inline-block; margin:0; padding:0; line-height:0;&#34; data-caption=&#34;Somewhat inconveniently, my favorite microcontroller uses 3.3V logic, but the TPIC6A595 shift register requires 5V logic.  So, I soldered a logic-level converter onto a breadboard alongside my microcontroller to clean up the wiring a bit.  The green and yellow wires carry the clock and latch signals, while the blue wire carries the serial input signal.  The microcontroller is powered by a USB-C cable, which taps into the 5V source provided by the BEC.&#34;&gt;

&lt;img src=&#34;esp32.jpeg&#34; style=&#34;display:block; margin:0; padding:0;&#34; alt=&#34;Somewhat inconveniently, my favorite microcontroller uses 3.3V logic, but the TPIC6A595 shift register requires 5V logic.  So, I soldered a logic-level converter onto a breadboard alongside my microcontroller to clean up the wiring a bit.  The green and yellow wires carry the ```clock``` and ```latch``` signals, while the blue wire carries the ```serial input``` signal.  The microcontroller is powered by a USB-C cable, which taps into the 5V source provided by the BEC.&#34; width=&#34;500&#34; &gt;
&lt;/a&gt;


&lt;figcaption style=&#34;text-align: left;&#34; data-pre=&#34;Figure &#34; data-post=&#34;:&#34; &gt;
  
  &lt;p&gt;
    Somewhat inconveniently, my favorite microcontroller uses 3.3V logic, but the TPIC6A595 shift register requires 5V logic.  So, I soldered a logic-level converter onto a breadboard alongside my microcontroller to clean up the wiring a bit.  The green and yellow wires carry the &lt;code&gt;clock&lt;/code&gt; and &lt;code&gt;latch&lt;/code&gt; signals, while the blue wire carries the &lt;code&gt;serial input&lt;/code&gt; signal.  The microcontroller is powered by a USB-C cable, which taps into the 5V source provided by the BEC.
    
    
    
  &lt;/p&gt; 
&lt;/figcaption&gt;

&lt;/figure&gt;

&lt;p&gt;The frame of the LED display box is made entirely of sheet metal; very sturdy, but also completely unamenable to radio wave transmission.  I needed a way to get my wireless commands into this stubborn &lt;a href=&#34;https://en.wikipedia.org/wiki/Faraday_cage&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Faraday Cage&lt;/a&gt;, so I added an external antenna.&lt;/p&gt;






&lt;figure style=&#34;text-align: center; margin-top: 50px; margin-bottom: 50px;&#34;&gt;

  &lt;a data-fancybox=&#34;&#34; href=&#34;ext_antenna.jpeg&#34; style=&#34;display:inline-block; margin:0; padding:0; line-height:0;&#34; data-caption=&#34;The display box itself is made of galvanized steel sheet metal, and thus constitutes a Faraday Cage; accordingly, I installed an external antenna to permit wireless communication.&#34;&gt;

&lt;img src=&#34;ext_antenna.jpeg&#34; style=&#34;display:block; margin:0; padding:0;&#34; alt=&#34;The display box itself is made of galvanized steel sheet metal, and thus constitutes a [Faraday Cage](https://en.wikipedia.org/wiki/Faraday_cage); accordingly, I installed an external antenna to permit wireless communication.&#34; width=&#34;500&#34; &gt;
&lt;/a&gt;


&lt;figcaption style=&#34;text-align: left;&#34; data-pre=&#34;Figure &#34; data-post=&#34;:&#34; &gt;
  
  &lt;p&gt;
    The display box itself is made of galvanized steel sheet metal, and thus constitutes a &lt;a href=&#34;https://en.wikipedia.org/wiki/Faraday_cage&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Faraday Cage&lt;/a&gt;; accordingly, I installed an external antenna to permit wireless communication.
    
    
    
  &lt;/p&gt; 
&lt;/figcaption&gt;

&lt;/figure&gt;


    &lt;div style=&#34;position: relative; padding-bottom: 56.25%; height: 0; overflow: hidden;&#34;&gt;
      &lt;iframe allow=&#34;accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share&#34; allowfullscreen=&#34;allowfullscreen&#34; loading=&#34;eager&#34; referrerpolicy=&#34;strict-origin-when-cross-origin&#34; src=&#34;https://www.youtube.com/embed/DhhVJPRAjXo?autoplay=0&amp;amp;controls=1&amp;amp;end=0&amp;amp;loop=0&amp;amp;mute=0&amp;amp;start=0&#34; style=&#34;position: absolute; top: 0; left: 0; width: 100%; height: 100%; border:0;&#34; title=&#34;YouTube video&#34;&gt;&lt;/iframe&gt;
    &lt;/div&gt;

&lt;h2 id=&#34;write-software&#34;&gt;Write Software&lt;/h2&gt;
&lt;p&gt;Most of the code I used for this project was adapted from similar stuff I wrote in grad school.  This is one of the really cool things about software: you can gradually build up a quiver of increasingly advanced capabilities over time, and deploy them as needed in your future projects.  Consequently, each successive project gets easier: the effort required to finish your $n^{th}$ project is approximately $\mathcal{O}(e^{-n})$.&lt;/p&gt;
&lt;h3 id=&#34;timer-logic&#34;&gt;Timer Logic&lt;/h3&gt;
&lt;p&gt;Have you ever wondered how a digital clock works?&lt;/p&gt;
&lt;p&gt;If you answered &lt;em&gt;yes&lt;/em&gt; to that, I am both impressed and a little concerned.&lt;/p&gt;
&lt;p&gt;I certainly hadn&amp;rsquo;t given it much thought before embarking on this project; but, once I started working out the algorithm, I became increasingly absorbed.  It was really fun to break down the basic behavior of a timer &amp;ndash; something we have all used a million times &amp;ndash; and describe it programmatically: &lt;em&gt;Okay, so we decrement the one&amp;rsquo;s place of the seconds digit until we get to zero, but then what happens?  We add nine to it, and decrement the ten&amp;rsquo;s place of the seconds digit&amp;hellip;&lt;/em&gt;&lt;/p&gt;
&lt;p&gt;And so on and so forth.&lt;/p&gt;
&lt;p&gt;I&amp;rsquo;m sure there&amp;rsquo;s probably a fancier way to implement this logic, but there was no premium on computational efficiency for this particular task &amp;ndash; it only runs once per second (which is like a million years in computer time), and involves only scalar arithmetic operations.  Nested &lt;em&gt;if-statements&lt;/em&gt; are interpretable and get the job done:&lt;/p&gt;
&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-matlab&#34; data-lang=&#34;matlab&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;%% Set initial timer digit values&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;m1&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;;&lt;/span&gt;     &lt;span class=&#34;c&#34;&gt;% digit 1: minute (10)&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;m2&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;;&lt;/span&gt;     &lt;span class=&#34;c&#34;&gt;% digit 2: minute (1)&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;s1&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;;&lt;/span&gt;     &lt;span class=&#34;c&#34;&gt;% digit 3: second (10)&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;s2&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;;&lt;/span&gt;     &lt;span class=&#34;c&#34;&gt;% digit 4: second (1)&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;%% Timer Active (decrement once per second)&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;while&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;~&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;m1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;==&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;0&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;&amp;amp;&amp;amp;&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;m2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;==&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;0&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;&amp;amp;&amp;amp;&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;s1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;==&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;0&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;&amp;amp;&amp;amp;&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;s2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;==&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;if&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;s2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;&amp;gt;&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;s2&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;s2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;;&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;else&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;s2&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;s2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;9&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;;&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;k&#34;&gt;if&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;s1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;&amp;gt;&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;            &lt;span class=&#34;n&#34;&gt;s1&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;s1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;;&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;k&#34;&gt;else&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;            &lt;span class=&#34;n&#34;&gt;s1&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;s1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;;&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;            &lt;span class=&#34;k&#34;&gt;if&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;m2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;&amp;gt;&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;                &lt;span class=&#34;n&#34;&gt;m2&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;m2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;;&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;            &lt;span class=&#34;k&#34;&gt;else&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;                &lt;span class=&#34;n&#34;&gt;m2&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;m2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;9&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;;&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;                &lt;span class=&#34;k&#34;&gt;if&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;m1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;&amp;gt;&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;                    &lt;span class=&#34;n&#34;&gt;m1&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;m1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;;&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;                &lt;span class=&#34;k&#34;&gt;else&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;                    &lt;span class=&#34;n&#34;&gt;m1&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;m1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;;&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;                &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;            &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;fprintf&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#39;%i%i:%i%i\n&amp;#39;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;m1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;m2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;s1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;s2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;);&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;pause&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;);&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;
    &lt;div style=&#34;position: relative; padding-bottom: 56.25%; height: 0; overflow: hidden;&#34;&gt;
      &lt;iframe allow=&#34;accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share&#34; allowfullscreen=&#34;allowfullscreen&#34; loading=&#34;eager&#34; referrerpolicy=&#34;strict-origin-when-cross-origin&#34; src=&#34;https://www.youtube.com/embed/usOYB9nmdMs?autoplay=0&amp;amp;controls=1&amp;amp;end=0&amp;amp;loop=0&amp;amp;mute=0&amp;amp;start=0&#34; style=&#34;position: absolute; top: 0; left: 0; width: 100%; height: 100%; border:0;&#34; title=&#34;YouTube video&#34;&gt;&lt;/iframe&gt;
    &lt;/div&gt;

&lt;p&gt;The logic for &lt;code&gt;stopwatch mode&lt;/code&gt; was mostly the same, with a few sign changes.&lt;/p&gt;
&lt;h3 id=&#34;numeric-digits&#34;&gt;Numeric Digits&lt;/h3&gt;
&lt;p&gt;In order to display digits on each of the four panels of the LED display, I needed to figure out what sequences of bits corresponded to the numbers &lt;code&gt;0&lt;/code&gt; through &lt;code&gt;9&lt;/code&gt;.  This wasn&amp;rsquo;t particularly challenging once I realized that the LED channels were arranged in a counter-clockwise fashion around each digit.  The following table shows how to express each digit using either binary or hexidecimal serial commands.&lt;/p&gt;






&lt;figure style=&#34;text-align: center; margin-top: 50px; margin-bottom: 50px;&#34;&gt;

  &lt;a data-fancybox=&#34;&#34; href=&#34;digit_binary_hex.png&#34; style=&#34;display:inline-block; margin:0; padding:0; line-height:0;&#34; data-caption=&#34;A useful cheat sheet that illustrates how to create numeric digits on a 7-segment display, using either binary or hex serial commands.&#34;&gt;

&lt;img src=&#34;digit_binary_hex.png&#34; style=&#34;display:block; margin:0; padding:0;&#34; alt=&#34;A useful cheat sheet that illustrates how to create numeric digits on a 7-segment display, using either binary or hex serial commands.&#34; &gt;
&lt;/a&gt;


&lt;figcaption style=&#34;text-align: left;&#34; data-pre=&#34;Figure &#34; data-post=&#34;:&#34; &gt;
  
  &lt;p&gt;
    A useful cheat sheet that illustrates how to create numeric digits on a 7-segment display, using either binary or hex serial commands.
    
    
    
  &lt;/p&gt; 
&lt;/figcaption&gt;

&lt;/figure&gt;

&lt;h2 id=&#34;build-controller&#34;&gt;Build Controller&lt;/h2&gt;
&lt;p&gt;This stage of the project was notably less exciting for me.  I knew I had already overcome the biggest challenges, and the ones that remained were all fairly pedestrian: wire cutting, soldering, and repurposing code I had already written for other projects.  At this point, I had been working on this project off-and-on for a few weeks, and I was pretty eager to wrap things up.&lt;/p&gt;
&lt;p&gt;I think most engineering projects follow an arc like this.  At first you&amp;rsquo;re really excited, eager to tackle a new and interesting problem.  Gradually, you realize how much work will be involved.  You wonder whether you can even solve the problem.  You experience a setback or two.  You spiral into disillusionment.  But, eventually, you experience a breakthrough!  You become momentarily reinvigorated, only to realize that there is still an absurd amount of work left to do.  You have already invested far too much time to quit now, so you grind out the remainder of the project.&lt;/p&gt;
&lt;p&gt;One of my favorite engineering content creators, &lt;a href=&#34;https://www.youtube.com/@rctestflight&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Daniel Riley (@rctestflight)&lt;/a&gt;, illustrated a similar narrative arc which he called &lt;em&gt;The Project Curve.&lt;/em&gt;&lt;/p&gt;






&lt;figure style=&#34;text-align: center; margin-top: 50px; margin-bottom: 50px;&#34;&gt;

  &lt;a data-fancybox=&#34;&#34; href=&#34;project_curve.png&#34; style=&#34;display:inline-block; margin:0; padding:0; line-height:0;&#34; data-caption=&#34;The Project Curve, featuring the Trough of Despair [source: rctestflight].&#34;&gt;

&lt;img src=&#34;project_curve.png&#34; style=&#34;display:block; margin:0; padding:0;&#34; alt=&#34;The Project Curve, featuring the Trough of Despair [source: [rctestflight](https://www.youtube.com/@rctestflight)].&#34; &gt;
&lt;/a&gt;


&lt;figcaption style=&#34;text-align: left;&#34; data-pre=&#34;Figure &#34; data-post=&#34;:&#34; &gt;
  
  &lt;p&gt;
    The Project Curve, featuring the Trough of Despair [source: &lt;a href=&#34;https://www.youtube.com/@rctestflight&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;rctestflight&lt;/a&gt;].
    
    
    
  &lt;/p&gt; 
&lt;/figcaption&gt;

&lt;/figure&gt;

&lt;h3 id=&#34;final-stretch&#34;&gt;Final Stretch&lt;/h3&gt;
&lt;p&gt;In any case, I still had a fair bit of work to do.  This was my basic sequence of tasks:&lt;/p&gt;
&lt;ol&gt;
&lt;li&gt;Permanently install internal microcontroller, solder connectors.&lt;/li&gt;
&lt;li&gt;Apply conformal coating to any soldered joints.&lt;/li&gt;
&lt;li&gt;Implement timer logic wirelessly on external microcontroller.&lt;/li&gt;
&lt;li&gt;Handle button presses using external interrupts (non-blocking).&lt;/li&gt;
&lt;li&gt;Design state machine (i.e. &lt;em&gt;if Button_A pressed, then&amp;hellip;&lt;/em&gt;).&lt;/li&gt;
&lt;li&gt;Test all buttons.&lt;/li&gt;
&lt;li&gt;Cut holes in enclosure, install buttons and pass-throughs.&lt;/li&gt;
&lt;/ol&gt;






&lt;figure style=&#34;text-align: center; margin-top: 50px; margin-bottom: 50px;&#34;&gt;

  &lt;a data-fancybox=&#34;&#34; href=&#34;button_chaos.jpeg&#34; style=&#34;display:inline-block; margin:0; padding:0; line-height:0;&#34; data-caption=&#34;Before you solder your buttons into a fancy waterproof enclosure, it is good to make sure they work!  This is what the chaotic prototyping stage looks like.&#34;&gt;

&lt;img src=&#34;button_chaos.jpeg&#34; style=&#34;display:block; margin:0; padding:0;&#34; alt=&#34;Before you solder your buttons into a fancy waterproof enclosure, it is good to make sure they work!  This is what the chaotic prototyping stage looks like.&#34; &gt;
&lt;/a&gt;


&lt;figcaption style=&#34;text-align: left;&#34; data-pre=&#34;Figure &#34; data-post=&#34;:&#34; &gt;
  
  &lt;p&gt;
    Before you solder your buttons into a fancy waterproof enclosure, it is good to make sure they work!  This is what the chaotic prototyping stage looks like.
    
    
    
  &lt;/p&gt; 
&lt;/figcaption&gt;

&lt;/figure&gt;

&lt;p&gt;The external controller itself was nothing fancy &amp;ndash; I ordered a waterproof enclosure big enough to hold a microcontroller, a LiPo battery, and the inevitable tangled mess of wires connecting the buttons to the microcontroller.  I drilled nine holes in the enclosure lid for the buttons, and one in the side wall of the base for a USB-C charging port.  I soldered 3.5mm bullet connectors to the button wires to facilitate connections, wired everything up, slapped the lid on, and bada bing bada boom we had a controller.&lt;/p&gt;
&lt;h3 id=&#34;stress-testing&#34;&gt;Stress Testing&lt;/h3&gt;
&lt;p&gt;When you are building a device that will eventually be used by someone else, you have to think very carefully about all the ways in which things could go wrong.  Or, you can simply go nuts and subject your device to unadulterated chaos.  I took the latter approach.&lt;/p&gt;
&lt;p&gt;I slammed my fingers on the controller buttons in the most absurd sequences; I pressed and held multiple buttons at the same time, while simultaneously clicking others &amp;ndash; &lt;em&gt;what would happen if the user attempted to increment and decrement a digit simultaneously while holding down the start button?&lt;/em&gt;  I needed to make sure it wasn&amp;rsquo;t possible to accidentally push the state machine into some unforseen and irrecoverable corner.  Once I was satisfied that no combination of button presses would result in catastrophe, I gave the controller a gentle pat and a knowing nod of respect.&lt;/p&gt;

    &lt;div style=&#34;position: relative; padding-bottom: 56.25%; height: 0; overflow: hidden;&#34;&gt;
      &lt;iframe allow=&#34;accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share&#34; allowfullscreen=&#34;allowfullscreen&#34; loading=&#34;eager&#34; referrerpolicy=&#34;strict-origin-when-cross-origin&#34; src=&#34;https://www.youtube.com/embed/_EQp3Tdwu8k?autoplay=0&amp;amp;controls=1&amp;amp;end=0&amp;amp;loop=0&amp;amp;mute=0&amp;amp;start=0&#34; style=&#34;position: absolute; top: 0; left: 0; width: 100%; height: 100%; border:0;&#34; title=&#34;YouTube video&#34;&gt;&lt;/iframe&gt;
    &lt;/div&gt;

&lt;p&gt;I subjected the display box to some abuse as well:&lt;/p&gt;

    &lt;div style=&#34;position: relative; padding-bottom: 56.25%; height: 0; overflow: hidden;&#34;&gt;
      &lt;iframe allow=&#34;accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share&#34; allowfullscreen=&#34;allowfullscreen&#34; loading=&#34;eager&#34; referrerpolicy=&#34;strict-origin-when-cross-origin&#34; src=&#34;https://www.youtube.com/embed/QYc6ZZx7gSI?autoplay=0&amp;amp;controls=1&amp;amp;end=0&amp;amp;loop=0&amp;amp;mute=0&amp;amp;start=0&#34; style=&#34;position: absolute; top: 0; left: 0; width: 100%; height: 100%; border:0;&#34; title=&#34;YouTube video&#34;&gt;&lt;/iframe&gt;
    &lt;/div&gt;

&lt;p&gt;Sometimes, it is both useful and appropriate to be a bit of a menace!&lt;/p&gt;
&lt;h2 id=&#34;reflection&#34;&gt;Reflection&lt;/h2&gt;
&lt;p&gt;This project took me about 40 hours over four weeks to complete &amp;ndash; not a strictly optimal use of my time, but illustrative of the commitment required to reverse engineer a modern device if you have a sufficiently sophisticated technical background.&lt;/p&gt;
&lt;p&gt;Plus, it felt good to do something nice for my brother &amp;ndash; he always looks out for me, so I was happy to be in a position to reciprocate.&lt;/p&gt;
&lt;p&gt;Six months have passed since I delivered this LED display and wireless controller to the &lt;a href=&#34;https://southbayboardriders.org/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;South Bay Boardriders Club&lt;/a&gt;, and apparently it is still working.  I am stoked to have had the opportunity to leverage my engineering skills to help out, and learn some new tricks in the process.&lt;/p&gt;
&lt;p&gt;Thanks for reading.&lt;/p&gt;
</description>
    </item>
    
    <item>
      <title>Open Water Swim Route Optimization Using Reversible Jump Markov Chain Monte Carlo (RJ-MCMC)</title>
      <link>https://www.blakecole.com/post/2-mcmc-mv-swim/</link>
      <pubDate>Sun, 23 Mar 2025 00:00:00 +0000</pubDate>
      <guid>https://www.blakecole.com/post/2-mcmc-mv-swim/</guid>
      <description>


&lt;details class=&#34;print:hidden xl:hidden&#34; open&gt;
  &lt;summary&gt;Table of Contents&lt;/summary&gt;
  &lt;div class=&#34;text-sm&#34;&gt;
  &lt;nav id=&#34;TableOfContents&#34;&gt;
  &lt;ul&gt;
    &lt;li&gt;&lt;a href=&#34;#should-you-read-this&#34;&gt;Should You Read This?&lt;/a&gt;&lt;/li&gt;
    &lt;li&gt;&lt;a href=&#34;#recapitulation&#34;&gt;Recapitulation&lt;/a&gt;&lt;/li&gt;
    &lt;li&gt;&lt;a href=&#34;#need-for-speed&#34;&gt;Need for Speed&lt;/a&gt;&lt;/li&gt;
    &lt;li&gt;&lt;a href=&#34;#markov-chain-monte-carlo&#34;&gt;Markov Chain Monte Carlo&lt;/a&gt;
      &lt;ul&gt;
        &lt;li&gt;&lt;a href=&#34;#history&#34;&gt;History&lt;/a&gt;&lt;/li&gt;
        &lt;li&gt;&lt;a href=&#34;#algorithm&#34;&gt;Algorithm&lt;/a&gt;&lt;/li&gt;
        &lt;li&gt;&lt;a href=&#34;#python-demo&#34;&gt;Python Demo&lt;/a&gt;&lt;/li&gt;
      &lt;/ul&gt;
    &lt;/li&gt;
    &lt;li&gt;&lt;a href=&#34;#problem-setup&#34;&gt;Problem Setup&lt;/a&gt;
      &lt;ul&gt;
        &lt;li&gt;&lt;a href=&#34;#goal&#34;&gt;Goal&lt;/a&gt;&lt;/li&gt;
        &lt;li&gt;&lt;a href=&#34;#constraints&#34;&gt;Constraints&lt;/a&gt;&lt;/li&gt;
        &lt;li&gt;&lt;a href=&#34;#assumptions&#34;&gt;Assumptions&lt;/a&gt;&lt;/li&gt;
        &lt;li&gt;&lt;a href=&#34;#parameters&#34;&gt;Parameters&lt;/a&gt;&lt;/li&gt;
        &lt;li&gt;&lt;a href=&#34;#methodology&#34;&gt;Methodology&lt;/a&gt;&lt;/li&gt;
        &lt;li&gt;&lt;a href=&#34;#useful-functions&#34;&gt;Useful Functions&lt;/a&gt;&lt;/li&gt;
      &lt;/ul&gt;
    &lt;/li&gt;
    &lt;li&gt;&lt;a href=&#34;#results&#34;&gt;Results&lt;/a&gt;
      &lt;ul&gt;
        &lt;li&gt;&lt;a href=&#34;#caseys-solution&#34;&gt;Casey&amp;rsquo;s Solution&lt;/a&gt;&lt;/li&gt;
        &lt;li&gt;&lt;a href=&#34;#my-solution&#34;&gt;My Solution&lt;/a&gt;&lt;/li&gt;
      &lt;/ul&gt;
    &lt;/li&gt;
    &lt;li&gt;&lt;a href=&#34;#flourishes&#34;&gt;Flourishes&lt;/a&gt;
      &lt;ul&gt;
        &lt;li&gt;&lt;a href=&#34;#gaussian-bump&#34;&gt;Gaussian Bump&lt;/a&gt;&lt;/li&gt;
        &lt;li&gt;&lt;a href=&#34;#simulated-annealing&#34;&gt;Simulated Annealing&lt;/a&gt;&lt;/li&gt;
      &lt;/ul&gt;
    &lt;/li&gt;
  &lt;/ul&gt;
&lt;/nav&gt;
  &lt;/div&gt;
&lt;/details&gt;

&lt;h2 id=&#34;should-you-read-this&#34;&gt;Should You Read This?&lt;/h2&gt;
&lt;p&gt;Life is short &amp;ndash; there are mountains to climb, oceans to explore, TikTok reels to enjoy and forget about mere seconds later.&lt;/p&gt;
&lt;p&gt;Continue reading if:&lt;/p&gt;
&lt;ul&gt;
&lt;li&gt;You have contemplated swimming to Martha&amp;rsquo;s Vineyard, and want to know how it can be done optimally (in theory).*&lt;/li&gt;
&lt;li&gt;You want to learn about the history and implementation of &lt;em&gt;Markov Chain Monte Carlo&lt;/em&gt; (MCMC) optimization methods.&lt;/li&gt;
&lt;li&gt;You need to optimize a system defined by an unknown number of parameters (sometimes this is called &lt;em&gt;model selection&lt;/em&gt;).  We will accomplish this task using a cool extension of MCMC called &lt;em&gt;Reversible Jump MCMC&lt;/em&gt; (RJ-MCMC).&lt;/li&gt;
&lt;li&gt;You want to extend my &lt;a href=&#34;https://github.com/blakecole/mv-mcmc&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;RJ-MCMC repository on GitHub&lt;/a&gt;.&lt;/li&gt;
&lt;/ul&gt;
&lt;p&gt;*Please don&amp;rsquo;t be an idiot.  Be safe.  Do your homework.  Don&amp;rsquo;t get hit by a ferry. &lt;br&gt;  Or, if you prefer legalese:








  
  
  


&lt;div class=&#34;px-4 py-3 mb-6 rounded-md bg-yellow-100 dark:bg-yellow-900&#34;&gt;
  
    &lt;div class=&#34;flex&#34;&gt;
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      &lt;/span&gt;
      &lt;span class=&#34;dark:text-neutral-300&#34;&gt;&lt;strong&gt;Disclaimer:&lt;/strong&gt;&lt;br&gt;
The content provided on this website/blog is for informational purposes only and does not constitute professional advice. While I strive to ensure accuracy, I make no representations or warranties regarding its completeness or applicability. Readers are solely responsible for how they use or interpret the information presented. I disclaim any liability for actions taken based on the content of this website.&lt;/span&gt;
    &lt;/div&gt;
  
&lt;/div&gt;

&lt;/p&gt;
&lt;h2 id=&#34;recapitulation&#34;&gt;Recapitulation&lt;/h2&gt;
&lt;p&gt;In August 2022, I published a lengthy blog post describing a &lt;a href=&#34;../1-adcirc-mv-swim/&#34;&gt;very nerdy plan to swim across Vineyard Sound&lt;/a&gt;.  As the crow flies, Martha’s Vineyard and Woods Hole are separated by only 3.3 miles; however, the sea separating these two landmasses is infamously tempestuous, with tidally-driven currents routinely exceeding 5 knots.  My goal was to leverage NOAA&amp;rsquo;s &lt;a href=&#34;https://adcirc.org/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;ADvanced CIRCulation (ADCIRC) model&lt;/a&gt; to predict these currents, and determine a sensible combination of swim parameters &amp;ndash; departure time, speed, and heading &amp;ndash; that yielded reasonable trajectories across Vineyard Sound.&lt;/p&gt;






&lt;figure style=&#34;text-align: center; margin-top: 50px; margin-bottom: 50px;&#34;&gt;

  &lt;a data-fancybox=&#34;&#34; href=&#34;mv_tides.gif&#34; style=&#34;display:inline-block; margin:0; padding:0; line-height:0;&#34; &gt;

&lt;img src=&#34;mv_tides.gif&#34; style=&#34;display:block; margin:0; padding:0;&#34; &gt;
&lt;/a&gt;

&lt;/figure&gt;

&lt;p&gt;Specifically, I looked for solutions that were:&lt;/p&gt;
&lt;ul&gt;
&lt;li&gt;&lt;strong&gt;Safe.&lt;/strong&gt;  The swim route must avoid ferry traffic.&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Timely.&lt;/strong&gt;  The swim must not exceed 2.5 hours, to avoid the strongest currents.&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Practical.&lt;/strong&gt;  The swimmer must arrive near the intended destination.&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Efficient.&lt;/strong&gt;  The swimmer should not fight the current, if possible.&lt;/li&gt;
&lt;/ul&gt;
&lt;p&gt;My simple guess-and-check approach worked well enough &amp;ndash; the swim was a success, and we arrived safely on shore near our intended destination, a little less than two hours after we departed.  However, inquisitive minds soon began to wonder: &lt;em&gt;&amp;ldquo;Could it have been done better?&amp;rdquo;&lt;/em&gt;&lt;/p&gt;






&lt;figure style=&#34;text-align: center; margin-top: 50px; margin-bottom: 50px;&#34;&gt;

  &lt;a data-fancybox=&#34;&#34; href=&#34;mv_swim_finish.jpg&#34; style=&#34;display:inline-block; margin:0; padding:0; line-height:0;&#34; data-caption=&#34;Swimmers arriving on the shore of Martha&amp;rsquo;s Vineyard, near Lake Tashmoo.&#34;&gt;

&lt;img src=&#34;mv_swim_finish.jpg&#34; style=&#34;display:block; margin:0; padding:0;&#34; alt=&#34;Swimmers arriving on the shore of Martha&amp;#39;s Vineyard, near Lake Tashmoo.&#34; &gt;
&lt;/a&gt;


&lt;figcaption style=&#34;text-align: left;&#34; data-pre=&#34;Figure &#34; data-post=&#34;:&#34; &gt;
  
  &lt;p&gt;
    Swimmers arriving on the shore of Martha&amp;rsquo;s Vineyard, near Lake Tashmoo.
    
    
    
  &lt;/p&gt; 
&lt;/figcaption&gt;

&lt;/figure&gt;

&lt;h2 id=&#34;need-for-speed&#34;&gt;Need for Speed&lt;/h2&gt;
&lt;p&gt;In the days and weeks after I published the original &lt;a href=&#34;https://www.blakecole.com/post/1-adcirc-mv-swim/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;post&lt;/a&gt;, I received a great deal of positive feedback on my work, and was admittedly somewhat pleased with myself.  I had used my brain to plan a logistically complex adventure which my body executed, and that felt very good.  I was still indulgently basking in self-satisfaction when I received a message from my good friend and colleague, &lt;a href=&#34;https://www.caseyhandmer.com/home&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Casey Handmer&lt;/a&gt;, saying something along the lines of,&lt;/p&gt;
&lt;blockquote&gt;
&lt;p&gt;You should optimize the swim.&lt;/p&gt;
&lt;/blockquote&gt;
&lt;p&gt;I knew he was right, but absolutely did not want to admit it.  Admitting it would mean more work for me, and probably not just a little more &amp;ndash; I suspected I would need to familiarize myself with some super fancy gradient-free, nonlinear, nonconvex optimization routine.  At the time, I was already five years into my PhD program, and could feel the institutional pressure rising: I needed to defend my thesis soon, or risk being kicked out of the program.  I bristled at Casey&amp;rsquo;s nonchalance, which seemed to suggest that this problem &amp;ndash; which I didn&amp;rsquo;t even know how to structure &amp;ndash; could be solved easily in an afternoon.  Suppressing my indignation, I informed Casey that I wasn&amp;rsquo;t going to optimize the swim, and that I needed to focus on my doctoral research.  He replied quickly, and asked me to send him the tidal current simulation data so that he could do it himself.&lt;/p&gt;
&lt;p&gt;This is classic Casey Handmer.  For those of you who do not know him, that sucks &amp;ndash; he is certainly one of the most fascinating people I have ever met.  He seems to crave technical problems like most people crave sweets or other sensory indulgences.  A physicist named Stanislaw Ulam (more on him &lt;a href=&#34;https://www.blakecole.com/post/2-mcmc-mv-swim/#history&#34;&gt;later&lt;/a&gt;) once remarked, &amp;ldquo;the mathematicians know a great deal about very little, and the physicists [know] very little about a great deal.&amp;rdquo;  Casey knows basically everything about everything.   At the age of 19, he traveled to Vietnam by himself, and hitchhiked through China and Russia before arriving in Italy; he speaks multiple languages, sings, and plays the trombone; when he isn&amp;rsquo;t riding his unicycle, he likes to fly airplanes or climb mountains; he also has a PhD in theoretical physics from Caltech, and founded &lt;a href=&#34;https://www.terraformindustries.com/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Terraform Industries&lt;/a&gt;, a company which aims to mitigate climate change by &lt;a href=&#34;https://terraformindustries.wordpress.com/2022/07/24/terraform-industries-whitepaper/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;synthesizing natural gas from sunlight and air&lt;/a&gt;.  He is a caring husband and father of three, and yet still somehow finds time to crank out wide-ranging technical &lt;a href=&#34;https://caseyhandmer.wordpress.com/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;articles&lt;/a&gt; on a weekly basis.  He is also Australian.  Go figure.&lt;/p&gt;






&lt;figure style=&#34;text-align: center; margin-top: 50px; margin-bottom: 50px;&#34;&gt;

  &lt;a data-fancybox=&#34;&#34; href=&#34;casey_handmer.jpg&#34; style=&#34;display:inline-block; margin:0; padding:0; line-height:0;&#34; data-caption=&#34;Casey Handmer, in the desert somewhere, presumably before he had children.&#34;&gt;

&lt;img src=&#34;casey_handmer.jpg&#34; style=&#34;display:block; margin:0; padding:0;&#34; alt=&#34;Casey Handmer, in the desert somewhere, presumably before he had children.&#34; &gt;
&lt;/a&gt;


&lt;figcaption style=&#34;text-align: left;&#34; data-pre=&#34;Figure &#34; data-post=&#34;:&#34; &gt;
  
  &lt;p&gt;
    Casey Handmer, in the desert somewhere, presumably before he had children.
    
    
    
  &lt;/p&gt; 
&lt;/figcaption&gt;

&lt;/figure&gt;

&lt;p&gt;The rate at which Casey solves problems is astounding.  This ability is rooted in an inclination to find first-order solutions quickly, and then decide whether higher fidelity approaches are warranted.  All good engineers seem to operate this way.  Not me.  I frequently succumb to the siren song of complexity, and spend hours deliberating which state-of-the-art technique I&amp;rsquo;m going to use to solve a problem.  This an ill-advised strategy: you could spend half a lifetime figuring out the best approach to a problem, and never get around to actually solving it.  Plus, you can&amp;rsquo;t really know the &lt;em&gt;best&lt;/em&gt; approach to a problem unless you have already solved it a few times (or talked to someone who has), so you might as well start with something simple &amp;ndash; dare I say, something &lt;em&gt;random&lt;/em&gt;.  Such was Casey&amp;rsquo;s reasoning when he decided to utilize a relatively simple stochastic methodology to determine the best possible swim route from Woods Hole to Martha&amp;rsquo;s Vineyard.&lt;/p&gt;
&lt;h2 id=&#34;markov-chain-monte-carlo&#34;&gt;Markov Chain Monte Carlo&lt;/h2&gt;
&lt;p&gt;There is power in randomness &amp;ndash; this is the fundamental premise of so-called &lt;em&gt;Monte Carlo&lt;/em&gt; methods.  How can this be?  Randomness is inherently chaotic, and chaos is bad, right?  Isn&amp;rsquo;t the foremost goal of science and engineering to distill order from chaos?&lt;/p&gt;
&lt;p&gt;Sort of.  Another way to think about randomness is as means of observing reality in an unbiased manner.  Imagine being a journalist who is told to write an article about one of ten equally important subjects &amp;ndash; how should they choose which one to cover?  It obviously wouldn&amp;rsquo;t be fair if they simply picked their favorite topic, so instead, they might randomly select one of the subjects.  By doing so, they ensure that each subject is treated with equal importance, free from any personal bias.  This randomness introduces fairness into the process, enabling a more objective approach to gathering information.  In much the same way, Monte Carlo methods use randomness to explore possible solutions to a problem, systematically sampling different outcomes to estimate the most likely result (or discover the best solution).  While it may seem counterintuitive, this stochastic process can be surprisingly powerful, as it allows us to explore a broad range of scenarios and capture complex behaviors that deterministic methods might miss.&lt;/p&gt;
&lt;h3 id=&#34;history&#34;&gt;History&lt;/h3&gt;
&lt;p&gt;Monte Carlo methods trace their origins to the 1940s, when they were developed as part of the Manhattan Project to aid in the &lt;a href=&#34;https://www.lanl.gov/media/publications/actinide-research-quarterly/1123-hitting-the-jackpot-the-birth-of-the-monte-carlo-method&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;design of nuclear weapons&lt;/a&gt;.  Pioneered by scientists such as &lt;a href=&#34;https://en.wikipedia.org/wiki/Stanis%C5%82aw_Ulam&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Stanislaw Ulam&lt;/a&gt; and &lt;a href=&#34;https://en.wikipedia.org/wiki/John_von_Neumann&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;John von Neumann&lt;/a&gt;, these methods were used to model complex physical processes, including neutron diffusion in fusion reactions.  The fundamental idea was to use random sampling to approximate solutions to otherwise intractable mathematical problems, leveraging the power of early computers to perform large-scale simulations.  The name &lt;em&gt;Monte Carlo&lt;/em&gt; was inspired by the famous casino in Monaco, reflecting the role of chance in the method.&lt;/p&gt;






&lt;figure style=&#34;text-align: center; margin-top: 50px; margin-bottom: 50px;&#34;&gt;

  &lt;a data-fancybox=&#34;&#34; href=&#34;mc_pi_estimation.png&#34; style=&#34;display:inline-block; margin:0; padding:0; line-height:0;&#34; data-caption=&#34;A canoncial demonstration of the Monte Carlo approach involves the estimation of $\pi$ without geometrical relationships.  As points are randomly scattered inside a unit square, some fall within an inscribed unit circle, and some do not; as the number of points approaches infinity, the fraction of points that land inside the circle approaches $\pi/4$ [source: Iván Mauricio Cely Toro].&#34;&gt;

&lt;img src=&#34;mc_pi_estimation.png&#34; style=&#34;display:block; margin:0; padding:0;&#34; alt=&#34;A canoncial demonstration of the Monte Carlo approach involves the estimation of $\pi$ without geometrical relationships.  As points are randomly scattered inside a unit square, some fall within an inscribed unit circle, and some do not; as the number of points approaches infinity, the fraction of points that land inside the circle approaches $\pi/4$ [source: [Iván Mauricio Cely Toro](https://mauriciocely.github.io/blog/2020/08/05/estimating-pi-using-the-monte-carlo-method/)].&#34; &gt;
&lt;/a&gt;


&lt;figcaption style=&#34;text-align: left;&#34; data-pre=&#34;Figure &#34; data-post=&#34;:&#34; &gt;
  
  &lt;p&gt;
    A canoncial demonstration of the Monte Carlo approach involves the estimation of $\pi$ without geometrical relationships.  As points are randomly scattered inside a unit square, some fall within an inscribed unit circle, and some do not; as the number of points approaches infinity, the fraction of points that land inside the circle approaches $\pi/4$ [source: &lt;a href=&#34;https://mauriciocely.github.io/blog/2020/08/05/estimating-pi-using-the-monte-carlo-method/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Iván Mauricio Cely Toro&lt;/a&gt;].
    
    
    
  &lt;/p&gt; 
&lt;/figcaption&gt;

&lt;/figure&gt;

&lt;p&gt;A major advance came in the 1950s with the introduction of the Metropolis algorithm by Nicholas Metropolis and his colleagues Marshall Rosenbluth, Arianna Rosenbluth, Augusta Teller, and Edward Teller at Los Alamos National Laboratory.  This method combined Monte Carlo techniques with the mathematics of Markov Chains &amp;ndash; theoretical frameworks that describe sequences of possible events in which the probability of each event depends only on the state attained in the previous event, not the full history.  This &lt;em&gt;memoryless&lt;/em&gt; property of Markov Chains makes them especially useful for exploring complex parameter spaces in a controlled (yet random) fashion.  In the Metropolis algorithm, a Markov Chain is constructed in such a way that, over time, it produces samples from a desired probability distribution &amp;ndash; or, equivalently, yields a set of parameters that minimizes a cost function (more on this equivalency in the &lt;a href=&#34;https://www.blakecole.com/post/2-mcmc-mv-swim/#algorithm&#34;&gt;Algorithm&lt;/a&gt; section).&lt;/p&gt;
&lt;p&gt;Why would anyone want to do this?  Well, in short, there quite a few ways in which nuclear fusion does not happen, and Nicholas Metropolis and his colleagues wanted very badly for it to happen.  I would say more, but that&amp;rsquo;s literally all I know, and even if I did know more, I probably wouldn&amp;rsquo;t say more &amp;ndash; apologies in advance to Iran, South Korea, and Saudi Arabia.&lt;/p&gt;






&lt;figure style=&#34;text-align: center; margin-top: 50px; margin-bottom: 50px;&#34;&gt;

  &lt;a data-fancybox=&#34;&#34; href=&#34;markov_chain.png&#34; style=&#34;display:inline-block; margin:0; padding:0; line-height:0;&#34; data-caption=&#34;A diagram representing a two-state Markov process. The numbers are the probability of changing from one state to another state [source: Wikipedia Commons].&#34;&gt;

&lt;img src=&#34;markov_chain.png&#34; style=&#34;display:block; margin:0; padding:0;&#34; alt=&#34;A diagram representing a two-state Markov process. The numbers are the probability of changing from one state to another state [source: [Wikipedia Commons](https://en.wikipedia.org/wiki/Markov_chain)].&#34; width=&#34;200&#34; &gt;
&lt;/a&gt;


&lt;figcaption style=&#34;text-align: left;&#34; data-pre=&#34;Figure &#34; data-post=&#34;:&#34; &gt;
  
  &lt;p&gt;
    A diagram representing a two-state Markov process. The numbers are the probability of changing from one state to another state [source: &lt;a href=&#34;https://en.wikipedia.org/wiki/Markov_chain&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Wikipedia Commons&lt;/a&gt;].
    
    
    
  &lt;/p&gt; 
&lt;/figcaption&gt;

&lt;/figure&gt;

&lt;p&gt;The Metropolis algorithm was later generalized by W.K. Hastings in 1970, resulting in the &lt;a href=&#34;https://en.wikipedia.org/wiki/Metropolis%E2%80%93Hastings_algorithm&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Metropolis-Hastings algorithm&lt;/a&gt;, a flexible and widely-used approach that underpins modern Markov Chain Monte Carlo (MCMC) techniques.  By using carefully designed Markov Chains to guide random sampling, MCMC methods are able to efficiently explore high-dimensional solution spaces.&lt;/p&gt;
&lt;p&gt;&lt;a href=&#34;https://academic.oup.com/biomet/article/82/4/711/252058&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Reversible Jump MCMC&lt;/a&gt; (RJ-MCMC) is an extension of traditional MCMC methods, developed by Peter J. Green in 1995.  Standard MCMC methods like Metropolis–Hastings typically operate in parameter spaces with a fixed dimension &amp;ndash; that is, the number of parameters doesn&amp;rsquo;t change during optimization.  In contrast, RJ-MCMC introduces special moves that alter the dimension of the parameter space by dynamically adding or removing parameters (effectively &amp;ldquo;jumping&amp;rdquo; between models with different complexities).  This flexibility makes RJ-MCMC ideal for problems where the best number of parameters (like the optimal number of discrete swim headings) is not known ahead of time.&lt;/p&gt;
&lt;h3 id=&#34;algorithm&#34;&gt;Algorithm&lt;/h3&gt;
&lt;p&gt;If you haven&amp;rsquo;t taken a probability course in a while (or even if you have), you might get bogged down in the mathematical language used to describe the &lt;a href=&#34;https://en.wikipedia.org/wiki/Metropolis%E2%80%93Hastings_algorithm&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Metropolis-Hastings algorithm&lt;/a&gt;.  If you peruse its Wikipedia page, you will find no shortage of obscure remarks that assume considerable familiarity with probability theory:&lt;/p&gt;
&lt;blockquote&gt;
&lt;p&gt;The Metropolis–Hastings algorithm can draw samples from any probability distribution with probability density $P(x)$, provided that we know a function $f(x)$ proportional to the density $P$ and the values of $f(x)$ can be calculated.&lt;/p&gt;
&lt;/blockquote&gt;
&lt;p&gt;What?&lt;/p&gt;
&lt;p&gt;How does that relate to optimization?&lt;/p&gt;
&lt;p&gt;I wondered the same thing at first &amp;ndash; do not dismay.&lt;/p&gt;
&lt;p&gt;The Metropolis–Hastings algorithm was originally developed to approximate probability distributions by generating a sequence of samples such that, after many iterations, the frequency of samples at any point, $x$, approximates the probability density, $P(x)$.&lt;/p&gt;
&lt;p&gt;Very clever people eventually figured out that this same method could be used to minimize a cost function by defining a target probability distribution that preferentially selects good solutions, and rejects bad ones.  Suppose you have a cost function, $C(x),$ that you want to minimize.  You could then define a target probability distribution, $P(x)$, where solutions with lower costs have higher probabilities:&lt;/p&gt;
$$
P(x) \propto e^{-\beta C(x)},
$$&lt;p&gt;
where $\beta$ is a positive (&lt;em&gt;inverse cooling&lt;/em&gt;) parameter controlling the sharpness of the distribution.  If you run the Metropolis–Hastings algorithm with this distribution, the generated &amp;ldquo;samples&amp;rdquo; &amp;ndash; which represent sets of parameters, $x$, in this instance &amp;ndash; will preferentially come from low-cost regions of the parameter space, because these samples have a higher probability of being selected.&lt;/p&gt;
&lt;p&gt;If none of &lt;em&gt;that&lt;/em&gt; made sense to you, that&amp;rsquo;s okay &amp;ndash; I have great news: the Metropolis–Hastings algorithm is shockingly easy to implement, even if you do not fully understand the underlying theory.  This means you (yes, &lt;em&gt;you&lt;/em&gt;!) can optimize all sorts of cool things in your life.&lt;/p&gt;
&lt;p&gt;These are the key steps:&lt;/p&gt;








  


&lt;div class=&#34;px-4 py-3 mb-6 rounded-md bg-primary-100 dark:bg-primary-900&#34;&gt;
  
    
      &lt;div class=&#34;mb-2 font-bold text-primary-600 dark:text-primary-300&#34;&gt;
        Metropolis-Hastings in Plain English
      &lt;/div&gt;
    
    &lt;div class=&#34;dark:text-neutral-300&#34;&gt;&lt;ol&gt;
&lt;li&gt;Define a cost function to evaluate the quality of a prospective solution.&lt;/li&gt;
&lt;li&gt;Propose an initial solution, and compute its cost.&lt;/li&gt;
&lt;li&gt;Introduce a random perturbation to the previous solution, and compute the new cost.
&lt;ul&gt;
&lt;li&gt;If the perturbation results in a &lt;strong&gt;lower&lt;/strong&gt; cost, accept the solution.&lt;/li&gt;
&lt;li&gt;If the perturbation results in a &lt;strong&gt;higher&lt;/strong&gt; cost, accept the solution with probability $\alpha$, where $\alpha$ is an &lt;em&gt;acceptance probability&lt;/em&gt; that depends on the difference between the previous cost and the current cost.  The acceptance probability can be systematically adjusted over time (this is called &lt;em&gt;simulated annealing&lt;/em&gt;).&lt;/li&gt;
&lt;li&gt;If the perturbation results in the &lt;strong&gt;lowest&lt;/strong&gt; cost thus far, store the solution.&lt;/li&gt;
&lt;/ul&gt;
&lt;/li&gt;
&lt;li&gt;Repeat &lt;strong&gt;Step 3&lt;/strong&gt; until a convergence criterion (or iteration limit) is met.&lt;/li&gt;
&lt;/ol&gt;
&lt;/div&gt;
  
&lt;/div&gt;


&lt;p&gt;Or, if you&amp;rsquo;re more of a flowchart kind of person:
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&lt;div class=&#34;mermaid&#34;&gt;
  
flowchart TD
    A[Initial Guess] --&gt; B[Compute Cost]
    B --&gt; C[Modify Solution]
    C --&gt; D[Compute Cost &amp; Acceptance Probability]
    D -- $C(x_k) &lt; C(x_{k-1})$ --&gt; E[Accept New Solution]
    D -- $C(x_k) \geq C(x_{k-1})$ --&gt; F{Roll Dice}
    F -- $\mathcal{U}(0,1) &lt; \alpha_k$ --&gt; E
    F -- $\mathcal{U}(0,1) \geq \alpha_k$ --&gt; G[Keep Old Solution]
    G --&gt; H[Current Solution]
    E --&gt; H[Current Solution]
    H --&gt; C

&lt;/div&gt;

&lt;/p&gt;
&lt;p&gt;Why periodically accept worse (higher cost) solutions?&lt;/p&gt;
&lt;p&gt;Imagine you are hiking in the mountains on a foggy day, searching for the highest peak.  If you simply climb upwards at every step, you might find yourself atop a small hill and assume you&amp;rsquo;ve reached the highest point (these little hilltops are called a &lt;em&gt;local maxima&lt;/em&gt; or &lt;em&gt;local extrema&lt;/em&gt; in nerd speak).  In order to locate the highest peak &amp;ndash; the &lt;em&gt;global maximum&lt;/em&gt; &amp;ndash; you might need to occasionally descend to lower elevations, even if it seems counterproductive in the moment.&lt;/p&gt;






&lt;figure style=&#34;text-align: center; margin-top: 50px; margin-bottom: 50px;&#34;&gt;

  &lt;a data-fancybox=&#34;&#34; href=&#34;hiker.png&#34; style=&#34;display:inline-block; margin:0; padding:0; line-height:0;&#34; data-caption=&#34;Sometimes you have to go lower to get higher [source: DALL-E 16.20.12].&#34;&gt;

&lt;img src=&#34;hiker.png&#34; style=&#34;display:block; margin:0; padding:0;&#34; alt=&#34;Sometimes you have to go lower to get higher [source: DALL-E 16.20.12].&#34; &gt;
&lt;/a&gt;


&lt;figcaption style=&#34;text-align: left;&#34; data-pre=&#34;Figure &#34; data-post=&#34;:&#34; &gt;
  
  &lt;p&gt;
    Sometimes you have to go lower to get higher [source: DALL-E 16.20.12].
    
    
    
  &lt;/p&gt; 
&lt;/figcaption&gt;

&lt;/figure&gt;

&lt;p&gt;To me, the fact that this algorithm actually works (most of the time) is nothing short of astounding &amp;ndash; it is basically &amp;ldquo;guess-and-check&amp;rdquo; on steroids.  And yet, despite its simplicity, this technique has revolutionized the fields of nuclear physics, economics, genetics, and climate science, just to name a few.  It also paved the way for modern machine learning techniques.  The power of this approach cannot be attributed to luck or magic; it is underwritten by &lt;em&gt;randomness&lt;/em&gt;, and the properties inherent thereto.&lt;/p&gt;
&lt;h3 id=&#34;python-demo&#34;&gt;Python Demo&lt;/h3&gt;
&lt;p&gt;Below is a simple, fully-functional 1-D implementation of the Metropolis–Hastings algorithm.  Feel free to &lt;a href=&#34;https://github.com/blakecole/mv-mcmc/blob/main/src/mcmc_demo_1d.py&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;download&lt;/a&gt; it, and modify it as you see fit.  I would encourage you to play around with the initial guess (&lt;code&gt;initial&lt;/code&gt;) and standard deviation (&lt;code&gt;proposal_std&lt;/code&gt;) parameters to get a sense of how these influence solution convergence.&lt;/p&gt;
&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-python&#34; data-lang=&#34;python&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;kn&#34;&gt;import&lt;/span&gt; &lt;span class=&#34;nn&#34;&gt;random&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;kn&#34;&gt;import&lt;/span&gt; &lt;span class=&#34;nn&#34;&gt;math&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;kn&#34;&gt;import&lt;/span&gt; &lt;span class=&#34;nn&#34;&gt;matplotlib.pyplot&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;as&lt;/span&gt; &lt;span class=&#34;nn&#34;&gt;plt&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;def&lt;/span&gt; &lt;span class=&#34;nf&#34;&gt;metropolis_hastings&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;cost_func&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;initial&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;iterations&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;proposal_std&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;):&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;s2&#34;&gt;&amp;#34;&amp;#34;&amp;#34;
&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;s2&#34;&gt;    Metropolis-Hastings algorithm for cost function minimization.
&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;s2&#34;&gt;
&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;s2&#34;&gt;    Args:
&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;s2&#34;&gt;        cost_func: The cost function to minimize. It should take a single parameter.
&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;s2&#34;&gt;        initial: Initial value for the parameter.
&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;s2&#34;&gt;        iterations: Number of iterations to run.
&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;s2&#34;&gt;        proposal_std: Standard deviation of the Gaussian proposal distribution.
&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;s2&#34;&gt;
&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;s2&#34;&gt;    Returns:
&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;s2&#34;&gt;        A tuple (best, best_cost, samples) where:
&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;s2&#34;&gt;          - best is the best found parameter value.
&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;s2&#34;&gt;          - best_cost is the corresponding cost.
&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;s2&#34;&gt;          - samples is a list of the parameter values over the iterations.
&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;s2&#34;&gt;    &amp;#34;&amp;#34;&amp;#34;&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;current&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;initial&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;current_cost&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;cost_func&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;current&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;best&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;current&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;best_cost&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;current_cost&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;samples&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;current&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;best_cost_history&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;best_cost&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;for&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;i&lt;/span&gt; &lt;span class=&#34;ow&#34;&gt;in&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;range&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;iterations&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;):&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;c1&#34;&gt;# Propose a new candidate using a Gaussian random step.&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;candidate&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;current&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;+&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;random&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;gauss&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;proposal_std&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;candidate_cost&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;cost_func&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;candidate&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;c1&#34;&gt;# Calculate acceptance probability.&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;c1&#34;&gt;# For minimization, lower cost is better.&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;c1&#34;&gt;# If the candidate cost is lower, accept it outright.&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;c1&#34;&gt;# Otherwise, accept it with probability exp(-(candidate_cost - current_cost)).&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;k&#34;&gt;if&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;candidate_cost&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;&amp;lt;&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;current_cost&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;:&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;            &lt;span class=&#34;n&#34;&gt;accept&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;kc&#34;&gt;True&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;k&#34;&gt;else&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;:&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;            &lt;span class=&#34;n&#34;&gt;accept_prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;math&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;exp&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;candidate_cost&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;current_cost&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;            &lt;span class=&#34;n&#34;&gt;accept&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;random&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;random&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;()&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;&amp;lt;&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;accept_prob&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;k&#34;&gt;if&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;accept&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;:&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;            &lt;span class=&#34;n&#34;&gt;current&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;candidate&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;            &lt;span class=&#34;n&#34;&gt;current_cost&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;candidate_cost&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;            &lt;span class=&#34;c1&#34;&gt;# Update best if the new candidate is better.&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;            &lt;span class=&#34;k&#34;&gt;if&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;candidate_cost&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;&amp;lt;&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;best_cost&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;:&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;                &lt;span class=&#34;n&#34;&gt;best&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;candidate&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;                &lt;span class=&#34;n&#34;&gt;best_cost&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;candidate_cost&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;samples&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;append&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;current&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;best_cost_history&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;append&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;best_cost&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;best&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;best_cost&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;samples&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;best_cost_history&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c1&#34;&gt;# Example usage&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;if&lt;/span&gt; &lt;span class=&#34;vm&#34;&gt;__name__&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;==&lt;/span&gt; &lt;span class=&#34;s1&#34;&gt;&amp;#39;__main__&amp;#39;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;:&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;c1&#34;&gt;# Define a simple cost function, e.g., a parabola with a minimum at x = 2.&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;def&lt;/span&gt; &lt;span class=&#34;nf&#34;&gt;cost&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;):&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;**&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;c1&#34;&gt;# Run the Metropolis-Hastings algorithm.&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;best&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;best_cost&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;samples&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;best_cost_history&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;metropolis_hastings&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;cost_func&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;cost&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;                                                                      &lt;span class=&#34;n&#34;&gt;initial&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;100&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;       &lt;span class=&#34;c1&#34;&gt;# Starting value&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;                                                                      &lt;span class=&#34;n&#34;&gt;iterations&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;250&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;c1&#34;&gt;# Number of iterations&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;                                                                      &lt;span class=&#34;n&#34;&gt;proposal_std&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;c1&#34;&gt;# Step size&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;nb&#34;&gt;print&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;sa&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;s2&#34;&gt;&amp;#34;Best x found:   &lt;/span&gt;&lt;span class=&#34;si&#34;&gt;{&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;best&lt;/span&gt;&lt;span class=&#34;si&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;s2&#34;&gt;.5f&lt;/span&gt;&lt;span class=&#34;si&#34;&gt;}&lt;/span&gt;&lt;span class=&#34;s2&#34;&gt;&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;nb&#34;&gt;print&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;sa&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;s2&#34;&gt;&amp;#34;Cost at best x: &lt;/span&gt;&lt;span class=&#34;si&#34;&gt;{&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;best_cost&lt;/span&gt;&lt;span class=&#34;si&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;s2&#34;&gt;.8f&lt;/span&gt;&lt;span class=&#34;si&#34;&gt;}&lt;/span&gt;&lt;span class=&#34;s2&#34;&gt;&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;c1&#34;&gt;# Plot convergence: best cost over iterations.&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;plt&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;figure&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;figsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;10&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;plt&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;best_cost_history&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;plt&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;s2&#34;&gt;&amp;#34;Iteration&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;plt&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ylabel&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;s2&#34;&gt;&amp;#34;Best Cost&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;plt&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;title&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;s2&#34;&gt;&amp;#34;Convergence of Metropolis-Hastings&lt;/span&gt;&lt;span class=&#34;se&#34;&gt;\n&lt;/span&gt;&lt;span class=&#34;s2&#34;&gt;&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;plt&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;grid&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;kc&#34;&gt;True&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;plt&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;show&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;()&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;If you run the code as is, you should find that the optimizer rapidly finds the smallest value of the cost function $C(x) = (x-2)^2$ at&amp;hellip;&lt;br&gt; &amp;hellip;drumroll, please&amp;hellip;&lt;br&gt; $x=2$.&lt;br&gt;Even for an absurd initial guess like $x_0 = 100$, the Metropolis-Hastings algorithm converges to the true solution in about 250 iterations.&lt;/p&gt;






&lt;figure style=&#34;text-align: center; margin-top: 50px; margin-bottom: 50px;&#34;&gt;

  &lt;a data-fancybox=&#34;&#34; href=&#34;mcmc_demo_convergence.png&#34; style=&#34;display:inline-block; margin:0; padding:0; line-height:0;&#34; data-caption=&#34;Convergence of the Metropolis-Hastings algorithm for $C(x) = (x-2)^2$, with $x_0 = 100$.&#34;&gt;

&lt;img src=&#34;mcmc_demo_convergence.png&#34; style=&#34;display:block; margin:0; padding:0;&#34; alt=&#34;Convergence of the Metropolis-Hastings algorithm for $C(x) = (x-2)^2$, with $x_0 = 100$.&#34; &gt;
&lt;/a&gt;


&lt;figcaption style=&#34;text-align: left;&#34; data-pre=&#34;Figure &#34; data-post=&#34;:&#34; &gt;
  
  &lt;p&gt;
    Convergence of the Metropolis-Hastings algorithm for $C(x) = (x-2)^2$, with $x_0 = 100$.
    
    
    
  &lt;/p&gt; 
&lt;/figcaption&gt;

&lt;/figure&gt;

&lt;p&gt;This is a fun example, but it really doesn&amp;rsquo;t demonstrate the true power of this technique; after all, this trivial problem can be solved using basic calculus.  What makes the MCMC approach so fascinating is its generality: it can explore parameter spaces of varying dimension, and the cost function need not be differentiable or continuous.&lt;/p&gt;
&lt;h2 id=&#34;problem-setup&#34;&gt;Problem Setup&lt;/h2&gt;
&lt;h3 id=&#34;goal&#34;&gt;Goal&lt;/h3&gt;
&lt;p&gt;Depart Nobska Beach between 2:00a and 11:00a on July 30th, 2022 and cross Vineyard Sound as fast as possible.&lt;/p&gt;
&lt;h3 id=&#34;constraints&#34;&gt;Constraints&lt;/h3&gt;
&lt;p&gt;&lt;strong&gt;None&lt;/strong&gt;.  We&amp;rsquo;re prioritizing &lt;em&gt;speed&lt;/em&gt;, and speed alone &amp;ndash; ferries, ruffled feathers, and private beaches be damned (&lt;em&gt;especially&lt;/em&gt; the private beaches, those can go straight to hell).  Any heading sequence that delivers the swimmer to Martha&amp;rsquo;s Vineyard is on the table.&lt;/p&gt;
&lt;h3 id=&#34;assumptions&#34;&gt;Assumptions&lt;/h3&gt;
&lt;ol&gt;
&lt;li&gt;We assume the swimmer is able to maintain a &lt;strong&gt;constant pace&lt;/strong&gt; of 1:40/100yd.  This is a reasonable pace for an average open water swimmer in good shape.&lt;/li&gt;
&lt;li&gt;We assume the swimmer departs from a &lt;strong&gt;fixed location&lt;/strong&gt; on Nobska Beach, MA, sometime between 2:00a and 11:00a on July 20th, 2022.&lt;/li&gt;
&lt;/ol&gt;
&lt;h3 id=&#34;parameters&#34;&gt;Parameters&lt;/h3&gt;
&lt;ol&gt;
&lt;li&gt;&lt;strong&gt;Heading Sequence&lt;/strong&gt;:  Each heading is pursued for a fixed time interval, $\Delta t;$ the total number of heading changes is not known in advance.&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Start Time&lt;/strong&gt;:  This parameter determines the state of the tidal current vector field during the simulated swim, and thus exerts a strong influence on the crossing time and optimal heading sequence.&lt;/li&gt;
&lt;/ol&gt;
&lt;h3 id=&#34;methodology&#34;&gt;Methodology&lt;/h3&gt;
&lt;ol&gt;
&lt;li&gt;
&lt;p&gt;Define &lt;strong&gt;swim constants&lt;/strong&gt;:&lt;/p&gt;
&lt;ul&gt;
&lt;li&gt;Departure location&lt;/li&gt;
&lt;li&gt;Start time&lt;/li&gt;
&lt;li&gt;Time step&lt;/li&gt;
&lt;/ul&gt;
&lt;/li&gt;
&lt;li&gt;
&lt;p&gt;Define &lt;strong&gt;optimization constants&lt;/strong&gt;:&lt;/p&gt;
&lt;ul&gt;
&lt;li&gt;Initial annealing temperature&lt;/li&gt;
&lt;li&gt;Initial Gaussian bump amplitude limit&lt;/li&gt;
&lt;li&gt;Number of iterations&lt;/li&gt;
&lt;/ul&gt;
&lt;/li&gt;
&lt;li&gt;
&lt;p&gt;Determine a reasonable &lt;strong&gt;initial guess&lt;/strong&gt;.&lt;/p&gt;
&lt;ol&gt;
&lt;li&gt;Simulate a series of &lt;em&gt;constant-heading&lt;/em&gt; swims.&lt;/li&gt;
&lt;li&gt;Store the solutions which produce valid trajectories (i.e., those that reach Martha&amp;rsquo;s Vineyard in under 3 hours).&lt;/li&gt;
&lt;li&gt;Pick the fastest valid &lt;em&gt;constant-heading&lt;/em&gt; solution as the initial guess.&lt;/li&gt;
&lt;li&gt;Compute the cost (swim duration) of the initial guess.&lt;/li&gt;
&lt;/ol&gt;
&lt;/li&gt;
&lt;li&gt;
&lt;p&gt;Randomly choose whether to modify, grow, or shrink the heading vector,  using the following logic:&lt;/p&gt;
&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-python&#34; data-lang=&#34;python&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pb&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.33&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pd&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.33&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pm&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.34&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;move_type&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;np&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;random&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;choice&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;([&lt;/span&gt;&lt;span class=&#34;s1&#34;&gt;&amp;#39;birth&amp;#39;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;s1&#34;&gt;&amp;#39;death&amp;#39;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;s1&#34;&gt;&amp;#39;modify&amp;#39;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pb&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;pd&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;pm&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;if&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;move_type&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;==&lt;/span&gt; &lt;span class=&#34;s1&#34;&gt;&amp;#39;modify&amp;#39;&lt;/span&gt; &lt;span class=&#34;ow&#34;&gt;and&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;len&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;new_h&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;&amp;gt;&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;:&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;perturbation&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;generate_bump&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;new_h&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;bump_size&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;new_h&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;+=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;perturbation&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;elif&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;move_type&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;==&lt;/span&gt; &lt;span class=&#34;s1&#34;&gt;&amp;#39;birth&amp;#39;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;:&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;new_heading&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;np&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;random&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;normal&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;birth_bearing&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;20.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;insert_idx&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;len&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;new_h&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;new_h&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;np&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;insert&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;new_h&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;insert_idx&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;new_heading&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt; &lt;span class=&#34;k&#34;&gt;elif&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;move_type&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;==&lt;/span&gt; &lt;span class=&#34;s1&#34;&gt;&amp;#39;death&amp;#39;&lt;/span&gt; &lt;span class=&#34;ow&#34;&gt;and&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;len&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;new_h&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;&amp;gt;&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;:&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;     &lt;span class=&#34;n&#34;&gt;idx&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;len&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;new_h&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;     &lt;span class=&#34;n&#34;&gt;removed_heading&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;new_h&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;idx&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;     &lt;span class=&#34;n&#34;&gt;new_h&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;np&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;delete&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;new_h&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;idx&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;The &lt;code&gt;bump_size&lt;/code&gt; parameter is typically set between 5 and 15 degrees at the outset, depending on the user&amp;rsquo;s confidence in their first guess, and reduced $0.1\%$ after every iteration; this allows the optimization routine to conduct a thorough exploration of the parameter space early on, and then refine the solution using smaller perturbations.  The use of a larger initial &lt;code&gt;bump_size&lt;/code&gt; results in very robust performance, but can sometimes yield sub-optimal solutions; conversely, the use of a smaller initial &lt;code&gt;bump_size&lt;/code&gt; works better if the initial guess is already pretty good, such that the odds of getting trapped in a local minimum are small.  This function is described in greater detail in the &lt;a href=&#34;https://www.blakecole.com/post/2-mcmc-mv-swim/#gaussian-bump&#34;&gt;Flourishes&lt;/a&gt; section.&lt;/p&gt;
&lt;/li&gt;
&lt;li&gt;
&lt;p&gt;Compute the cost of the candidate solution.&lt;/p&gt;
&lt;/li&gt;
&lt;li&gt;
&lt;p&gt;Compute the acceptance probability: $\alpha_k = \exp{\left( \frac{\Delta_k}{T_k} \right)}$, where $\Delta_k = C(x_k) - C(x_{k-1})$ is the cost difference between the candidate and previous solutions, and $T_k$ is the annealing temperature parameter.  We reduce $T_k$ gradually over time using a &lt;em&gt;linear cooling schedule&lt;/em&gt; to discourage premature convergence.&lt;/p&gt;
&lt;/li&gt;
&lt;li&gt;
&lt;p&gt;Keep the candidate solution if &lt;em&gt;&lt;strong&gt;BOTH&lt;/strong&gt;&lt;/em&gt; the following conditions are met:&lt;/p&gt;
&lt;ol&gt;
&lt;li&gt;&lt;em&gt;&lt;strong&gt;EITHER&lt;/strong&gt;&lt;/em&gt; the candidate cost is lower than the previous cost, $C(x_k) &lt; C(x_{k-1})$, &lt;em&gt;&lt;strong&gt;OR&lt;/strong&gt;&lt;/em&gt; the acceptance probability, $\alpha_k$, is greater than a random number drawn from a uniform distribution, $\mathcal{U}(0,1)$.&lt;/li&gt;
&lt;li&gt;The swim trajectory terminates inside the boundary of Martha&amp;rsquo;s Vineyard.&lt;/li&gt;
&lt;/ol&gt;
&lt;/li&gt;
&lt;li&gt;
&lt;p&gt;If the candidate solution is the lowest-cost solution yet, store it.&lt;/p&gt;
&lt;/li&gt;
&lt;li&gt;
&lt;p&gt;If the iteration limit has not been reached, return to &lt;strong&gt;Step 4&lt;/strong&gt;.&lt;/p&gt;
&lt;/li&gt;
&lt;/ol&gt;
&lt;h3 id=&#34;useful-functions&#34;&gt;Useful Functions&lt;/h3&gt;
&lt;ol&gt;
&lt;li&gt;
&lt;p&gt;&lt;strong&gt;Flowfield Interpolation&lt;/strong&gt;:  Extract a tidal current vector from the discrete computational domain using SciPy&amp;rsquo;s &lt;code&gt;NearestNDInterpolator&lt;/code&gt; function.  I also experimented with the &lt;code&gt;LinearNDInterpolator&lt;/code&gt; function, but the minor fidelity improvement did not justify the enormous (roughly $1000x$) increase* in computational cost.&lt;/p&gt;
&lt;/li&gt;
&lt;li&gt;
&lt;p&gt;&lt;strong&gt;Dead Reckon&lt;/strong&gt;: Given a starting point (lon, lat), initial bearing (in degrees), and a distance (in meters), compute the destination point along a &lt;em&gt;great-circle path&lt;/em&gt; using a modified form of the &lt;a href=&#34;https://en.wikipedia.org/wiki/Haversine_formula&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Haversine formula&lt;/a&gt; (note: if you like geodesy, check out my &lt;a href=&#34;../../publication/jrnl-aor-2022/&#34;&gt;proof of this relationship&lt;/a&gt;).  This function permits the simulation of swim trajectories knowing only a starting point, a set of sequential headings, and the swimmer&amp;rsquo;s speed over ground.&lt;/p&gt;
&lt;/li&gt;
&lt;li&gt;
&lt;p&gt;&lt;strong&gt;Arrival Detection&lt;/strong&gt;: Given a point (lon, lat) and a list of points describing a closed region, determine whether the point is inside the closed region using the &lt;a href=&#34;https://en.wikipedia.org/wiki/Point_in_polygon#Ray_casting_algorithm&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;ray casting algorithm&lt;/a&gt;.  This function is used to determine whether a given sequence of headings produces a swim trajectory that reaches Martha&amp;rsquo;s Vineyard.&lt;/p&gt;
&lt;/li&gt;
&lt;li&gt;
&lt;p&gt;&lt;strong&gt;Closest Approach&lt;/strong&gt; (Optional): Compute the closest distance (in meters), and the bearing (in degrees), from a geodetic point (lon, lat) to a closed polygon defined as a (Nx2) NumPy ndarray of (lon, lat) points, using vectorized geodetic functions.  Although this function is not strictly necessary, it was useful when I was debugging my optimization routine.&lt;/p&gt;
&lt;/li&gt;
&lt;/ol&gt;
&lt;p&gt;*You may have found this surprising.  How could simple linear interpolation incur such an immense computational burden?  Here&amp;rsquo;s how: SciPy&amp;rsquo;s  &lt;code&gt;LinearNDInterpolator&lt;/code&gt; works by first creating a &lt;a href=&#34;https://en.wikipedia.org/wiki/Delaunay_triangulation&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Delaunay triangulation&lt;/a&gt; structure, and then iterating over the &lt;em&gt;entire structure&lt;/em&gt; for every evaluation.  Consequently, the lookup time scales poorly with the number of points: $\mathcal{O}(N~log(N))$ for construction, and $\mathcal{O}(N)$ per query.  Conversely, &lt;code&gt;NearestNDInterpolator&lt;/code&gt; uses a &lt;a href=&#34;https://en.wikipedia.org/wiki/K-d_tree&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;k-d tree&lt;/a&gt; to efficiently partition space for nearest-neighbor searches, reducing the lookup time to $\mathcal{O}(log(N))$ per query.&lt;/p&gt;
&lt;h2 id=&#34;results&#34;&gt;Results&lt;/h2&gt;
&lt;p&gt;Before I present my own solutions, I&amp;rsquo;d like to show Casey&amp;rsquo;s work.  His results aren&amp;rsquo;t quite as polished as mine, but he probably cranked them out in about the same time I spent formatting one of my plots (that being said, I wholeheartedly believe that data visualization is important, beauty is important, and making &lt;a href=&#34;https://www.datawrapper.de/blog/beautifulcolors#1&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;good plots&lt;/a&gt; is worth the time and effort; I will die on this hill).  Further, Casey&amp;rsquo;s findings were compelling enough to inspire me to write my own MCMC code, and publish this blog.&lt;/p&gt;
&lt;h3 id=&#34;caseys-solution&#34;&gt;Casey&amp;rsquo;s Solution&lt;/h3&gt;
&lt;p&gt;Shortly after I sent Casey my tide simulation data, I received the following email:&lt;/p&gt;
&lt;div style=&#34;background-color: #00131A; color: #F2F2F2; padding: 10px;&#34;&gt;
Hi Blake, 
&lt;p&gt;Finally getting around to shooting you this info.&lt;/p&gt;
&lt;p&gt;I relaxed the constraint that the headings be constant to see if it was possible to find a faster routing. I coded a function that would (quickly) attempt to find an optimal sequence of headings as a function of&lt;/p&gt;
&lt;ul&gt;
&lt;li&gt;departure location and time&lt;/li&gt;
&lt;li&gt;arrival location&lt;/li&gt;
&lt;li&gt;minimizing time in the ferry zone&lt;/li&gt;
&lt;li&gt;pace together with an interpolated background dataset of known (simulated) current flows.&lt;/li&gt;
&lt;/ul&gt;
&lt;p&gt;This graph shows an example output, with headings in degrees as a function of time (units in multiples of 3 minutes). For a particular combination of start time and pace, close to 140 degrees the whole way is adequate, as shown with the orange line. Leaving earlier or later, favors different headings to take advantage of the water speed.&lt;/p&gt;






&lt;figure style=&#34;text-align: center; margin-top: 50px; margin-bottom: 50px;&#34;&gt;

  &lt;a data-fancybox=&#34;&#34; href=&#34;casey_heading_vs_time.png&#34; style=&#34;display:inline-block; margin:0; padding:0; line-height:0;&#34; data-caption=&#34;Optimal swim heading sequences for assorted start times. [source: Casey Handmer]&#34;&gt;

&lt;img src=&#34;casey_heading_vs_time.png&#34; style=&#34;display:block; margin:0; padding:0;&#34; alt=&#34;&amp;lt;center&amp;gt;Optimal swim heading sequences for assorted start times. [source: Casey Handmer]&amp;lt;/center&amp;gt;&#34; width=&#34;600&#34; &gt;
&lt;/a&gt;


&lt;figcaption style=&#34;text-align: left;&#34; data-pre=&#34;Figure &#34; data-post=&#34;:&#34; &gt;
  
  &lt;p&gt;
    &lt;center&gt;Optimal swim heading sequences for assorted start times. [source: Casey Handmer]&lt;/center&gt;
    
    
    
  &lt;/p&gt; 
&lt;/figcaption&gt;

&lt;/figure&gt;

&lt;p&gt;This ground track shows, as a function of latitude and longitude in degrees, the ground tracks corresponding to the curves above. The bright green and red are baselines premised on 140 degree headings the whole time, while the colored red and green dots demarcate the boundaries of the ferry zone. Blue dots show the grid on which tide current values are known with certainty, and hence an inference of the shoreline location. Not every start time converged to the desired landing point but most of them did.&lt;/p&gt;






&lt;figure style=&#34;text-align: center; margin-top: 50px; margin-bottom: 50px;&#34;&gt;

  &lt;a data-fancybox=&#34;&#34; href=&#34;casey_trajectories.png&#34; style=&#34;display:inline-block; margin:0; padding:0; line-height:0;&#34; data-caption=&#34;Optimal swim trajectories for assorted start times. [source: Casey Handmer]&#34;&gt;

&lt;img src=&#34;casey_trajectories.png&#34; style=&#34;display:block; margin:0; padding:0;&#34; alt=&#34;&amp;lt;center&amp;gt;Optimal swim trajectories for assorted start times. [source: Casey Handmer]&amp;lt;/center&amp;gt;&#34; width=&#34;600&#34; &gt;
&lt;/a&gt;


&lt;figcaption style=&#34;text-align: left;&#34; data-pre=&#34;Figure &#34; data-post=&#34;:&#34; &gt;
  
  &lt;p&gt;
    &lt;center&gt;Optimal swim trajectories for assorted start times. [source: Casey Handmer]&lt;/center&gt;
    
    
    
  &lt;/p&gt; 
&lt;/figcaption&gt;

&lt;/figure&gt;

&lt;p&gt;The time saved compared to 140 degrees baseline varied between 5 and 20 minutes, some of which could be attributed to reducing the overall distance needed to be covered. Fortunately Martha&amp;rsquo;s Vineyard has a big beachy coast which is hard to miss!&lt;/p&gt;
&lt;p&gt;Compared to Blake and friends&amp;rsquo; actual swim, the solutions are about 20-30 minutes faster. I think this is probably because the simulated swimmers routed close to the shore on departure, waiting until they were in strong current before zooming across the ferry zone. Blake&amp;rsquo;s strategy to swim directly across the zone in still water meant their group took longer to get into the faster water.&lt;/p&gt;
&lt;p&gt;This optimization effort was very hacky, and I&amp;rsquo;m sure there&amp;rsquo;s another 3-5 minutes of performance to be squeezed out with concerted effort.&lt;/p&gt;
&lt;p&gt;Casey&lt;/p&gt;
&lt;/div&gt;
&lt;p&gt;I have a few thoughts on this.&lt;/p&gt;
&lt;p&gt;First, &lt;em&gt;nice&lt;/em&gt;.&lt;/p&gt;
&lt;p&gt;I think the coolest part of Casey&amp;rsquo;s work is actually hiding behind the figures: it is his &lt;a href=&#34;https://www.blakecole.com/post/2-mcmc-mv-swim/#gaussian-bump&#34;&gt;Gaussian bump&lt;/a&gt; feature.  Extremely clever.  I initially tried to run my optimization algorithm without it, and the results were nightmarish: the heading vector was perturbed every which way, yielding a chaotic, jagged swim trajectory.  Casey realized that such drastic changes in sequential headings were counterproductive, and developed a &lt;em&gt;representation&lt;/em&gt; that encoded this knowledge.  The result?  Gorgeously smooth solutions, and much faster convergence.  Bravo!&lt;/p&gt;
&lt;p&gt;Casey&amp;rsquo;s results are &lt;em&gt;quick-and-dirty&lt;/em&gt; but credible.  It makes sense that earlier departures &amp;ndash; which subject the swimmer to stronger westward currents &amp;ndash; would necessitate a course adjustment to the east.  Plus, given that my &lt;a href=&#34;https://www.blakecole.com/post/1-adcirc-mv-swim/#600a-departure--140100yd-pace&#34;&gt;initial analysis&lt;/a&gt; suggested that a constant $140^{\circ}$ heading was a reasonably efficient solution (i.e., the dot product of current velocity and swimmer velocity is nonnegative at nearly every timestep), we wouldn&amp;rsquo;t expect the optimal solution to be dramatically different.&lt;/p&gt;
&lt;p&gt;I also thought it was cool that he was able to modify the cost function to penalize solutions that spent too much time in the ferry channel, and arrived too far from Lake Tashmoo.&lt;/p&gt;
&lt;p&gt;Tinkering with the cost function sometimes leads to unexpected behavior.  Early on, I was using a composite cost function that combined &lt;code&gt;swim_time&lt;/code&gt; and &lt;code&gt;distance_from_shore&lt;/code&gt; as a weighted sum, hoping this would guide the optimization routine toward solutions that were both fast &lt;em&gt;and&lt;/em&gt; reached the shoreline.  However, I soon found that the routine often prioritized a smaller value of &lt;code&gt;swim_time&lt;/code&gt; over the necessary (but as-of-yet unenforced) condition that &lt;code&gt;distance_from_shore = 0&lt;/code&gt;.  I initially rectified this issue by tuning the cost function weights, but I hated this solution &amp;ndash; it felt flimsy.  So, I eventually figured out a way to reject invalid solutions, and remove &lt;code&gt;distance_from_shore&lt;/code&gt; from the cost function.&lt;/p&gt;
&lt;p&gt;I think Casey would be the first to admit that there is room for improvement here (from the horse&amp;rsquo;s mouth: &amp;ldquo;This optimization effort was very hacky&amp;rdquo;).&lt;/p&gt;
&lt;p&gt;He didn&amp;rsquo;t digitize the shoreline of Martha&amp;rsquo;s Vineyard, so I&amp;rsquo;m guessing his arrival detection function might have introduced errors on the order of a few minutes.  Practically speaking, this isn&amp;rsquo;t a big deal; however, I found that optimization generally only improves the initial guess by a few minutes, so an error of this magnitude is problematic.&lt;/p&gt;
&lt;p&gt;Finally, I think a good chunk of the &amp;ldquo;20-30 minute improvement&amp;rdquo; Casey referenced could be explained by the fact that we took a 10-minute break in the middle of Vineyard Sound to chat, hydrate, and eat snacks.  But, I don&amp;rsquo;t think he knew about this, and thus cannot be blamed for pointing his hypotheses elsewhere.  I agree with his general contention that swimming south into stronger currents before crossing the ferry channel may have reduced our total swim time &amp;ndash; but not by 30 minutes.&lt;/p&gt;
&lt;h3 id=&#34;my-solution&#34;&gt;My Solution&lt;/h3&gt;
&lt;p&gt;My approach improves on Casey&amp;rsquo;s in a number of ways:&lt;/p&gt;
&lt;ul&gt;
&lt;li&gt;
&lt;p&gt;&lt;strong&gt;Reversible Jump MCMC&lt;/strong&gt;: Rather than assuming a fixed number of headings, my approach allows the heading vector to shrink and grow as necessary.  You can imagine the tip of the swim trajectory rapidly bouncing around the shoreline, probing different solutions like some sort of disgusting proboscis.&lt;/p&gt;
&lt;/li&gt;
&lt;li&gt;
&lt;p&gt;&lt;strong&gt;Pre-optimization&lt;/strong&gt;: Prior to firing up the main optimization engine, I conduct an automated sweep of &lt;em&gt;constant-heading&lt;/em&gt; simulations, keeping track of the ones that reach Martha&amp;rsquo;s Vineyard, and pick the fastest solution as my first guess.  Surprisingly, this very dumb technique produced solutions that were only a few minutes slower than the final optimized solutions.&lt;/p&gt;
&lt;/li&gt;
&lt;li&gt;
&lt;p&gt;&lt;strong&gt;Enhanced arrival detection&lt;/strong&gt;: I use &lt;a href=&#34;https://en.wikipedia.org/wiki/Point_in_polygon#Ray_casting_algorithm&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;ray casting&lt;/a&gt; to determine if swim trajectories terminate inside the digitized boundary of Martha&amp;rsquo;s Vineyard.&lt;/p&gt;
&lt;/li&gt;
&lt;li&gt;
&lt;p&gt;&lt;strong&gt;Better temporal resolution&lt;/strong&gt;: I reduced the simulation timestep from $\Delta t = 3 min$ to $\Delta t = 18 s$.  Without the use of a smaller timestep, the optimizer will reject candidate solutions unless they yield at least a $2.99 min$ reduction in swim time.&lt;/p&gt;
&lt;/li&gt;
&lt;li&gt;
&lt;p&gt;&lt;strong&gt;Exponentially-cooled Gaussian bump&lt;/strong&gt;: Rather than selecting the amplitude of each &lt;em&gt;Gaussian bump&lt;/em&gt; from a uniform distribution with fixed bounds, I allow the bounds to shrink over time.  This enhancement yielded a dramatic improvement in both solver robustness and solution quality.&lt;/p&gt;
&lt;/li&gt;
&lt;li&gt;
&lt;p&gt;&lt;strong&gt;Linearly-cooled annealing temperature&lt;/strong&gt;: The use of a linear cooling schedule resulted in a reasonable reduction in the acceptance probability over time.  This feature worked in conjunction with the &lt;em&gt;exponentially-cooled Gaussian bump&lt;/em&gt; to improve convergence.&lt;/p&gt;
&lt;/li&gt;
&lt;li&gt;
&lt;p&gt;&lt;strong&gt;Pretty dark plots&lt;/strong&gt;: Who doesn&amp;rsquo;t like dark plots?&lt;/p&gt;
&lt;/li&gt;
&lt;/ul&gt;
&lt;h4 id=&#34;optimal-trajectories&#34;&gt;Optimal Trajectories&lt;/h4&gt;
&lt;p&gt;In the following plots, the pre-optimization (i.e., &lt;em&gt;constant-heading&lt;/em&gt;) solutions are shown as grayscale lines, with a corresponding colorbar showing the total swim time in minutes.  The final, optimized swim trajectory is shown in yellow, and the textbox in the upper righthand corner shows the optimized swim distance and time.&lt;/p&gt;
&lt;p&gt;Without further ado, here are the results you presumably came here for.








  
  


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&lt;/p&gt;
&lt;p&gt;Assuming a constant swim pace of 1:40/100yd, and no concerns about getting maimed by a ferry&amp;hellip;&lt;/p&gt;








  


&lt;div class=&#34;px-4 py-3 mb-6 rounded-md bg-primary-100 dark:bg-primary-900&#34;&gt;
  
    
    &lt;div class=&#34;dark:text-neutral-300&#34;&gt;&lt;em&gt;&lt;strong&gt;On July 30th, 2022, it would have been possible to depart Nobska Beach between 7:30a and 8:00a, and cross Vineyard Sound in 1 hour, 32 minutes&lt;/strong&gt;&lt;/em&gt;.&lt;/div&gt;
  
&lt;/div&gt;


&lt;p&gt;These plots reveal another interesting fact: &lt;em&gt;constant-heading solutions are actually pretty good&lt;/em&gt;.  Most of the time, the optimal trajectory sticks pretty close to at least one &lt;em&gt;constant-heading&lt;/em&gt; trajectory.  So, unless you&amp;rsquo;re really determined to shave a few minutes off your swim, using a &lt;em&gt;constant-heading&lt;/em&gt; strategy is totally fine.&lt;/p&gt;
&lt;h4 id=&#34;optimal-heading-sequences&#34;&gt;Optimal Heading Sequences&lt;/h4&gt;






&lt;figure style=&#34;text-align: center; margin-top: 50px; margin-bottom: 50px;&#34;&gt;

  &lt;a data-fancybox=&#34;&#34; href=&#34;mv_swim_heading_vs_time.png&#34; style=&#34;display:inline-block; margin:0; padding:0; line-height:0;&#34; &gt;

&lt;img src=&#34;mv_swim_heading_vs_time.png&#34; style=&#34;display:block; margin:0; padding:0;&#34; &gt;
&lt;/a&gt;

&lt;/figure&gt;

&lt;p&gt;Despite the fact that both 7:30a and 8:00a starts enable an optimal crossing, the required corresponding heading sequences couldn&amp;rsquo;t be more different!  While a 7:30a start requires a sequence of headings that varies between $150^{\circ}-155^{\circ}$, an 8:00a start requires a sequence of headings that hovers around $170^{\circ}$.  What&amp;rsquo;s going on?&lt;/p&gt;
&lt;p&gt;By this time in the morning, the tidal current is absolutely ripping from west to east (check out the &lt;a href=&#34;https://www.blakecole.com/post/2-mcmc-mv-swim/#recapitulation&#34;&gt;tidal current gif&lt;/a&gt;).  By starting later, you&amp;rsquo;re effectively getting all the longitudinal velocity you could possible hope for, and more, for free &amp;ndash; your only option is to swim due south, and hope you make it to the beach before you get ripped around the tip of West Chop.  Good luck!&lt;/p&gt;
&lt;p&gt;Probably a good time to reiterate my initial caveat: this would be an insanely high-risk strategy in practice, and is not recommended.  By the time the tidal current is this strong, swimming across Middle Ground would be dangerous, if not impossible.&lt;/p&gt;
&lt;h4 id=&#34;convergence&#34;&gt;Convergence&lt;/h4&gt;
&lt;p&gt;I wanted to display a few figures that highlight the robust convergence properties of this optimization routine.  In the following two figures, the grayscale lines represent the in-progress solution every 100 iterations, with a corresponding colorbar displaying the total swim time in seconds.  The optimized solution is shown in yellow, and the textbox in the upper righthand corner shows the optimized swim distance and time.&lt;/p&gt;






&lt;figure style=&#34;text-align: center; margin-top: 50px; margin-bottom: 50px;&#34;&gt;

  &lt;a data-fancybox=&#34;&#34; href=&#34;mv_swim_convergence_h180_0630.png&#34; style=&#34;display:inline-block; margin:0; padding:0; line-height:0;&#34; data-caption=&#34;Illustration of solver robustness: convergence is achieved despite poorly chosen initial conditions.&#34;&gt;

&lt;img src=&#34;mv_swim_convergence_h180_0630.png&#34; style=&#34;display:block; margin:0; padding:0;&#34; alt=&#34;Illustration of solver robustness: convergence is achieved despite poorly chosen initial conditions.&#34; &gt;
&lt;/a&gt;


&lt;figcaption style=&#34;text-align: left;&#34; data-pre=&#34;Figure &#34; data-post=&#34;:&#34; &gt;
  
  &lt;p&gt;
    Illustration of solver robustness: convergence is achieved despite poorly chosen initial conditions.
    
    
    
  &lt;/p&gt; 
&lt;/figcaption&gt;

&lt;/figure&gt;

&lt;p&gt;I ran this optimization shortly after I applied an exponential cooling schedule to the &lt;code&gt;bump_size&lt;/code&gt; parameter, and was shocked by how well it worked &amp;ndash; the resulting solutions were almost entirely insensitive to initial conditions.  In other words, the optimization procedure was &lt;em&gt;robust&lt;/em&gt;.  Knowing in advance that the optimal heading vector hovered around $135^{\circ}$ for this start time, I decided to push my luck and use a $180^{\circ}$ &lt;em&gt;constant-heading&lt;/em&gt; solution as my initial guess.&lt;/p&gt;
&lt;p&gt;Uncalled for.  Pathological.&lt;/p&gt;
&lt;p&gt;Nevertheless, the optimization routine quickly honed in on the correct solution.  I was so proud that it didn&amp;rsquo;t get mired in the morass of bullshit to which it was intentionally subjected &amp;ndash; now I understand why the &lt;a href=&#34;https://www.youtube.com/watch?v=zkv-_LqTeQA&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Boston Dynamics engineers are so mean to their robots&lt;/a&gt;.&lt;/p&gt;
&lt;p&gt;On to the next!&lt;/p&gt;






&lt;figure style=&#34;text-align: center; margin-top: 50px; margin-bottom: 50px;&#34;&gt;

  &lt;a data-fancybox=&#34;&#34; href=&#34;mv_swim_convergence_2_0630.png&#34; style=&#34;display:inline-block; margin:0; padding:0; line-height:0;&#34; data-caption=&#34;Illustration of solver ability to escape local minima.&#34;&gt;

&lt;img src=&#34;mv_swim_convergence_2_0630.png&#34; style=&#34;display:block; margin:0; padding:0;&#34; alt=&#34;Illustration of solver ability to escape local minima.&#34; &gt;
&lt;/a&gt;


&lt;figcaption style=&#34;text-align: left;&#34; data-pre=&#34;Figure &#34; data-post=&#34;:&#34; &gt;
  
  &lt;p&gt;
    Illustration of solver ability to escape local minima.
    
    
    
  &lt;/p&gt; 
&lt;/figcaption&gt;

&lt;/figure&gt;

&lt;p&gt;This particular optimization attempt did not yield the best solution &amp;ndash; note that the simulated swim lasted 1.5 minutes longer than the optimal solution for a 6:30a start &amp;ndash; but I liked how it again demonstrated &lt;em&gt;robustness&lt;/em&gt;, and an ability to escape local minima.  Look at how the solution periodically becomes stuck in local minima, before jumping to better solutions.&lt;/p&gt;
&lt;p&gt;Incredible!&lt;/p&gt;
&lt;p&gt;We love it.&lt;/p&gt;
&lt;h4 id=&#34;reflections&#34;&gt;Reflections&lt;/h4&gt;
&lt;p&gt;Ultimately, this was a highly academic exercise.  Maybe this sort of thing makes sense if you&amp;rsquo;re a professional athlete or a Navy SEAL with a &lt;a href=&#34;https://en.wikipedia.org/wiki/Head-up_display&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;head-up display&lt;/a&gt; in your goggles, but otherwise you&amp;rsquo;re better off sticking with a &lt;em&gt;constant-heading&lt;/em&gt; strategy.  During the actual swim, it was challenging enough to follow a single fixed heading with any appreciable degree of confidence &amp;ndash; the notion that a swimmer might be able to make sub-degree alterations to their bearing every eighteen seconds is laughable.&lt;/p&gt;
&lt;p&gt;But, on the other hand&amp;hellip; pretty cool, right?&lt;/p&gt;
&lt;p&gt;It is often said (by lunatics, mostly) that there are many ways to skin a cat; likewise, I am confident that there are other, perhaps better, ways to approach this problem.  But, then again, there are so many cats, and so little time.&lt;/p&gt;
&lt;p&gt;Feel free to &lt;a href=&#34;../../#contact&#34;&gt;reach out&lt;/a&gt; if you have any questions, comments, or want to collaborate on a cool project.&lt;/p&gt;
&lt;p&gt;Thanks for reading.&lt;/p&gt;
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&lt;h2 id=&#34;flourishes&#34;&gt;Flourishes&lt;/h2&gt;
&lt;h3 id=&#34;gaussian-bump&#34;&gt;Gaussian Bump&lt;/h3&gt;
&lt;p&gt;A common and straightforward method for proposing new candidate solutions in MCMC optimization involves directly perturbing the parameter vector with random noise.  For instance, one typically selects a random index from the parameter vector and adds a small perturbation drawn from a Gaussian distribution with zero mean and a chosen standard deviation.  A Python implementation of this approach might look like this:&lt;/p&gt;
&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-python&#34; data-lang=&#34;python&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;proposal_std&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;1.0&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;idx&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;np&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;random&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;randint&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;len&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;old_soln&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;perturbation&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;np&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;random&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;normal&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;proposal_std&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;new_soln&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;idx&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;old_soln&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;+&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;perturbation&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;Although easy to implement, this method tends to produce isolated, incremental changes, potentially limiting the ability of the algorithm to explore broader variations in the solution space efficiently.  We can often leverage domain-specific expertise to modify the &lt;em&gt;proposal step&lt;/em&gt;, and drastically improve both the convergence rate and quality of the resulting solutions.  For instance, when optimizing a swim route, we can be fairly confident that swimmer heading will not change abruptly, and can modify our proposals accordingly.&lt;/p&gt;
&lt;p&gt;Rather than simply adding random noise to the swimmer heading vector, we can preserve smoothness by adding a &lt;em&gt;Gaussian bump&lt;/em&gt;.  The &lt;em&gt;Gaussian bump&lt;/em&gt; is characterized by three randomly selected parameters: the &lt;strong&gt;center&lt;/strong&gt; (index in the heading vector), the &lt;strong&gt;width&lt;/strong&gt; (extent of the perturbation along the heading vector), and the &lt;strong&gt;amplitude&lt;/strong&gt; (magnitude and direction of the perturbation).&lt;/p&gt;






&lt;figure style=&#34;text-align: center; margin-top: 50px; margin-bottom: 50px;&#34;&gt;

  &lt;a data-fancybox=&#34;&#34; href=&#34;gaussian_bumps_invert.png&#34; style=&#34;display:inline-block; margin:0; padding:0; line-height:0;&#34; data-caption=&#34;An set of Gaussian functions with varying center (i.e. mean), width (i.e. variance), and amplitude [source: Wikimedia Commons].&#34;&gt;

&lt;img src=&#34;gaussian_bumps_invert.png&#34; style=&#34;display:block; margin:0; padding:0;&#34; alt=&#34;An set of Gaussian functions with varying center (i.e. mean), width (i.e. variance), and amplitude [source: [Wikimedia Commons](https://en.wikipedia.org/wiki/File:Normal_Distribution_PDF.svg)].&#34; &gt;
&lt;/a&gt;


&lt;figcaption style=&#34;text-align: left;&#34; data-pre=&#34;Figure &#34; data-post=&#34;:&#34; &gt;
  
  &lt;p&gt;
    An set of Gaussian functions with varying center (i.e. mean), width (i.e. variance), and amplitude [source: &lt;a href=&#34;https://en.wikipedia.org/wiki/File:Normal_Distribution_PDF.svg&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Wikimedia Commons&lt;/a&gt;].
    
    
    
  &lt;/p&gt; 
&lt;/figcaption&gt;

&lt;/figure&gt;

&lt;p&gt;By adjusting these parameters randomly within carefully chosen ranges, the function introduces controlled yet diverse variability, allowing the optimization routine to explore different route possibilities without creating unrealistic or overly abrupt path changes.  A Python implementation of the Gaussian bump might look like this:&lt;/p&gt;
&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-python&#34; data-lang=&#34;python&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;def&lt;/span&gt; &lt;span class=&#34;nf&#34;&gt;generate_bump&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;headings&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;bump_size&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;):&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;c1&#34;&gt;# Random center in [-0.2, 1.2]&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;center&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;np&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;random&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;uniform&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;1.2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;c1&#34;&gt;# Random width: generate log-width uniformly between log(0.05) and log(1), then exponentiate&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;log_width&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;np&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;random&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;uniform&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;np&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;log&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.05&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;np&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;log&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;width&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;np&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;exp&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;log_width&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;c1&#34;&gt;# Random amplitude in [-bump_size, bump_size]&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;amplitude&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;np&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;random&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;uniform&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;bump_size&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;bump_size&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;c1&#34;&gt;# Create sample points: the number of points equals the length of headings&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;n_points&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;len&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;headings&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;x_vals&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;np&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;linspace&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;n_points&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;c1&#34;&gt;# Evaluate the Gaussian bump at each sample point&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;bump&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;amplitude&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;*&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;np&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;exp&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;((&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x_vals&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;center&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;**&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;/&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;width&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;**&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;bump&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;To enhance convergence toward an optimal solution, the amplitude range of these perturbations &amp;ndash; controlled by the parameter &lt;code&gt;bump_size&lt;/code&gt; &amp;ndash; is reduced after each iteration (e.g., using &lt;code&gt;bump_size *= 0.999&lt;/code&gt; in the optimization loop).  Initially, larger perturbations encourage exploration across a broad set of candidate solutions, preventing premature convergence on local minima. As the optimization progresses, decreasing &lt;code&gt;bump_size&lt;/code&gt; narrows the search area, transitioning from broad exploration toward refined, localized adjustments.  This adaptive strategy balances efficient global exploration with precise local optimization, resulting in a systematically improved swim trajectory.&lt;/p&gt;
&lt;p&gt;I was honestly shocked at how well this worked.  Initially, I was using a fixed value of &lt;code&gt;bump_size = 3.0&lt;/code&gt; (units: degrees), and found that the final solution was highly dependent on the initial guess.  The Gaussian bump technique, with exponential cooling, yielded a much more robust solver capable of converging to the global optimum, almost regardless of the initial guess.&lt;/p&gt;
&lt;h3 id=&#34;simulated-annealing&#34;&gt;Simulated Annealing&lt;/h3&gt;
&lt;p&gt;Generally speaking, simulated annealing is a technique used to gradually reduce the rate of change experienced by a system over time; indeed, its name was inspired by the physical process of slowly cooling molten metal to achieve a stable crystal structure.  In the context of optimization, it involves slowly decreasing a &lt;em&gt;temperature parameter&lt;/em&gt; to control the acceptance probability of proposed solutions.  In the Metropolis-Hastings algorithm, the acceptance probability is typically computed in the following way:&lt;/p&gt;
$$
\alpha_k = \exp{\left( \frac{\Delta_k}{T_k} \right)}
$$&lt;p&gt;where $T_k$ is the temperature parameter, and $\Delta_k = C(x_k) - C(x_{k-1})$ is the difference between the new and old solution costs. For simple problems (as in the &lt;a href=&#34;https://www.blakecole.com/post/2-mcmc-mv-swim/#python-demo&#34;&gt;Python Demo&lt;/a&gt; provided above), it is often sufficient to simply set $T=1$; however, by allowing the temperature parameter to gradually decrease over time, we can achieve both broad &lt;em&gt;exploration&lt;/em&gt; early in the optimization process, and focused &lt;em&gt;exploitation&lt;/em&gt; and refinement as convergence progresses.&lt;/p&gt;
&lt;p&gt;The temperature parameter is often reduced according to a predefined &lt;em&gt;cooling schedule&lt;/em&gt;, a few of which will be outlined here:&lt;/p&gt;
&lt;h4 id=&#34;exponential&#34;&gt;Exponential&lt;/h4&gt;
&lt;p&gt;This is likely the most widely-used cooling method, due to its simplicity and effectiveness:&lt;/p&gt;
$$
T_k = \beta T_{k-1}
$$&lt;p&gt;
where $T_k$ is the temperature at iteration $k$, and $\beta$ is a constant slightly less than $1$ (typically $0.80 \leq \beta \leq 0.99$), controlling how rapidly temperature decreases.  However, it should be noted that exponential decay reduces the temperature proportionally each step, leading to a rapid reduction in temperature early on, quickly suppressing exploration.  I found that this cooling technique resulted in an overly-aggressive reduction of the acceptance probability, even when I set $\beta$ very close to $1$; however, this approach was perfectly suited to the &lt;code&gt;bump_size&lt;/code&gt; parameter, which controls the maximum amplitude of &lt;em&gt;Gaussian bump&lt;/em&gt; perturbations.&lt;/p&gt;
&lt;h4 id=&#34;linear&#34;&gt;Linear&lt;/h4&gt;
&lt;p&gt;This is straightforward method that is useful when the number of iterations, $N$, is known in advance.  Linear cooling reduces temperature at a constant rate rather than proportionally:&lt;/p&gt;
$$
T_k = T_{k-1} - \Delta T, \qquad \Delta T = \frac{T_0 - T_{final}}{N}
$$&lt;p&gt;
This method is arguably simpler to implement, but less flexible since it requires knowing the final iteration count in advance.  It also results in much slower cooling, which was more appropriate for my application.&lt;/p&gt;
</description>
    </item>
    
    <item>
      <title>Blog</title>
      <link>https://www.blakecole.com/projects/</link>
      <pubDate>Sun, 19 May 2024 00:00:00 +0000</pubDate>
      <guid>https://www.blakecole.com/projects/</guid>
      <description></description>
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      <title>Pandas</title>
      <link>https://www.blakecole.com/project/pandas/</link>
      <pubDate>Thu, 26 Oct 2023 00:00:00 +0000</pubDate>
      <guid>https://www.blakecole.com/project/pandas/</guid>
      <description>&lt;p&gt;Flexible and powerful data analysis / manipulation library for Python, providing labeled data structures.&lt;/p&gt;</description>
    </item>
    
    <item>
      <title>PyTorch</title>
      <link>https://www.blakecole.com/project/pytorch/</link>
      <pubDate>Thu, 26 Oct 2023 00:00:00 +0000</pubDate>
      <guid>https://www.blakecole.com/project/pytorch/</guid>
      <description>&lt;p&gt;PyTorch is a Python package that provides tensor computation (like NumPy) with strong GPU acceleration.&lt;/p&gt;</description>
    </item>
    
    <item>
      <title>scikit-learn</title>
      <link>https://www.blakecole.com/project/scikit/</link>
      <pubDate>Thu, 26 Oct 2023 00:00:00 +0000</pubDate>
      <guid>https://www.blakecole.com/project/scikit/</guid>
      <description>&lt;p&gt;scikit-learn is a Python module for machine learning built on top of SciPy and is distributed under the 3-Clause BSD license.&lt;/p&gt;</description>
    </item>
    
    <item>
      <title>Experience</title>
      <link>https://www.blakecole.com/experience/</link>
      <pubDate>Tue, 24 Oct 2023 00:00:00 +0000</pubDate>
      <guid>https://www.blakecole.com/experience/</guid>
      <description></description>
    </item>
    
    <item>
      <title>Learn JavaScript</title>
      <link>https://www.blakecole.com/teaching/js/</link>
      <pubDate>Tue, 24 Oct 2023 00:00:00 +0000</pubDate>
      <guid>https://www.blakecole.com/teaching/js/</guid>
      <description>&lt;p&gt;&lt;a href=&#34;https://hugoblox.com&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Hugo Blox Builder&lt;/a&gt; is designed to give technical content creators a seamless experience. You can focus on the content and the Hugo Blox Builder which this template is built upon handles the rest.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Embed videos, podcasts, code, LaTeX math, and even test students!&lt;/strong&gt;&lt;/p&gt;
&lt;p&gt;On this page, you&amp;rsquo;ll find some examples of the types of technical content that can be rendered with Hugo Blox.&lt;/p&gt;
&lt;h2 id=&#34;video&#34;&gt;Video&lt;/h2&gt;
&lt;p&gt;Teach your course by sharing videos with your students. Choose from one of the following approaches:&lt;/p&gt;

    &lt;div style=&#34;position: relative; padding-bottom: 56.25%; height: 0; overflow: hidden;&#34;&gt;
      &lt;iframe allow=&#34;accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share&#34; allowfullscreen=&#34;allowfullscreen&#34; loading=&#34;eager&#34; referrerpolicy=&#34;strict-origin-when-cross-origin&#34; src=&#34;https://www.youtube.com/embed/D2vj0WcvH5c?autoplay=0&amp;amp;controls=1&amp;amp;end=0&amp;amp;loop=0&amp;amp;mute=0&amp;amp;start=0&#34; style=&#34;position: absolute; top: 0; left: 0; width: 100%; height: 100%; border:0;&#34; title=&#34;YouTube video&#34;&gt;&lt;/iframe&gt;
    &lt;/div&gt;

&lt;p&gt;&lt;strong&gt;Youtube&lt;/strong&gt;:&lt;/p&gt;
&lt;pre&gt;&lt;code&gt;{{&amp;lt; youtube w7Ft2ymGmfc &amp;gt;}}
&lt;/code&gt;&lt;/pre&gt;
&lt;p&gt;&lt;strong&gt;Bilibili&lt;/strong&gt;:&lt;/p&gt;
&lt;pre&gt;&lt;code&gt;{{&amp;lt; bilibili id=&amp;quot;BV1WV4y1r7DF&amp;quot; &amp;gt;}}
&lt;/code&gt;&lt;/pre&gt;
&lt;p&gt;&lt;strong&gt;Video file&lt;/strong&gt;&lt;/p&gt;
&lt;p&gt;Videos may be added to a page by either placing them in your &lt;code&gt;assets/media/&lt;/code&gt; media library or in your &lt;a href=&#34;https://gohugo.io/content-management/page-bundles/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;page&amp;rsquo;s folder&lt;/a&gt;, and then embedding them with the &lt;em&gt;video&lt;/em&gt; shortcode:&lt;/p&gt;
&lt;pre&gt;&lt;code&gt;{{&amp;lt; video src=&amp;quot;my_video.mp4&amp;quot; controls=&amp;quot;yes&amp;quot; &amp;gt;}}
&lt;/code&gt;&lt;/pre&gt;
&lt;h2 id=&#34;podcast&#34;&gt;Podcast&lt;/h2&gt;
&lt;p&gt;You can add a podcast or music to a page by placing the MP3 file in the page&amp;rsquo;s folder or the media library folder and then embedding the audio on your page with the &lt;em&gt;audio&lt;/em&gt; shortcode:&lt;/p&gt;
&lt;pre&gt;&lt;code&gt;{{&amp;lt; audio src=&amp;quot;ambient-piano.mp3&amp;quot; &amp;gt;}}
&lt;/code&gt;&lt;/pre&gt;
&lt;p&gt;Try it out:&lt;/p&gt;








  








&lt;audio controls &gt;
  &lt;source src=&#34;https://www.blakecole.com/teaching/js/ambient-piano.mp3&#34; type=&#34;audio/mpeg&#34;&gt;
&lt;/audio&gt;

&lt;h2 id=&#34;test-students&#34;&gt;Test students&lt;/h2&gt;
&lt;p&gt;Provide a simple yet fun self-assessment by revealing the solutions to challenges with the &lt;code&gt;spoiler&lt;/code&gt; shortcode:&lt;/p&gt;
&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-markdown&#34; data-lang=&#34;markdown&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;{{&lt;span class=&#34;p&#34;&gt;&amp;lt;&lt;/span&gt; &lt;span class=&#34;nt&#34;&gt;spoiler&lt;/span&gt; &lt;span class=&#34;na&#34;&gt;text&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;👉 Click to view the solution&amp;#34;&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;&amp;gt;&lt;/span&gt;}}
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;You found me!
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;{{&lt;span class=&#34;p&#34;&gt;&amp;lt;&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;/&lt;/span&gt;&lt;span class=&#34;nt&#34;&gt;spoiler&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;&amp;gt;&lt;/span&gt;}}
&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;renders as&lt;/p&gt;
&lt;details class=&#34;spoiler &#34;  id=&#34;spoiler-2&#34;&gt;
  &lt;summary class=&#34;cursor-pointer&#34;&gt;👉 Click to view the solution&lt;/summary&gt;
  &lt;div class=&#34;rounded-lg bg-neutral-50 dark:bg-neutral-800 p-2&#34;&gt;
    You found me 🎉
  &lt;/div&gt;
&lt;/details&gt;
&lt;h2 id=&#34;math&#34;&gt;Math&lt;/h2&gt;
&lt;p&gt;Hugo Blox Builder supports a Markdown extension for $\LaTeX$ math. You can enable this feature by toggling the &lt;code&gt;math&lt;/code&gt; option in your &lt;code&gt;config/_default/params.yaml&lt;/code&gt; file.&lt;/p&gt;
&lt;p&gt;To render &lt;em&gt;inline&lt;/em&gt; or &lt;em&gt;block&lt;/em&gt; math, wrap your LaTeX math with &lt;code&gt;{{&amp;lt; math &amp;gt;}}$...${{&amp;lt; /math &amp;gt;}}&lt;/code&gt; or &lt;code&gt;{{&amp;lt; math &amp;gt;}}$$...$${{&amp;lt; /math &amp;gt;}}&lt;/code&gt;, respectively.&lt;/p&gt;
&lt;div class=&#34;px-4 py-3 mb-6 rounded-md bg-primary-100 dark:bg-primary-900&#34;&gt;
&lt;pre&gt;&lt;code&gt;&amp;lt;div class=&amp;quot;flex&amp;quot;&amp;gt;
  &amp;lt;span class=&amp;quot;pr-3 pt-1 text-primary-600 dark:text-primary-300&amp;quot;&amp;gt;
    &amp;lt;svg height=&amp;quot;24&amp;quot; xmlns=&amp;quot;http://www.w3.org/2000/svg&amp;quot; viewBox=&amp;quot;0 0 24 24&amp;quot;&amp;gt;&amp;lt;path fill=&amp;quot;none&amp;quot; stroke=&amp;quot;currentColor&amp;quot; stroke-linecap=&amp;quot;round&amp;quot; stroke-linejoin=&amp;quot;round&amp;quot; stroke-width=&amp;quot;1.5&amp;quot; d=&amp;quot;m11.25 11.25l.041-.02a.75.75 0 0 1 1.063.852l-.708 2.836a.75.75 0 0 0 1.063.853l.041-.021M21 12a9 9 0 1 1-18 0a9 9 0 0 1 18 0m-9-3.75h.008v.008H12z&amp;quot;/&amp;gt;&amp;lt;/svg&amp;gt;
  &amp;lt;/span&amp;gt;
  &amp;lt;span class=&amp;quot;dark:text-neutral-300&amp;quot;&amp;gt;We wrap the LaTeX math in the Hugo Blox &amp;lt;em&amp;gt;math&amp;lt;/em&amp;gt; shortcode to prevent Hugo rendering our math as Markdown.&amp;lt;/span&amp;gt;
&amp;lt;/div&amp;gt;
&lt;/code&gt;&lt;/pre&gt;
&lt;/div&gt;
&lt;p&gt;Example &lt;strong&gt;math block&lt;/strong&gt;:&lt;/p&gt;
&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-latex&#34; data-lang=&#34;latex&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nb&#34;&gt;{{&lt;/span&gt;&amp;lt; math &amp;gt;&lt;span class=&#34;nb&#34;&gt;}}&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;sb&#34;&gt;$$&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;
&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nb&#34;&gt;&lt;/span&gt;&lt;span class=&#34;nv&#34;&gt;\gamma&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;_{n} &lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt; &lt;/span&gt;&lt;span class=&#34;nv&#34;&gt;\frac&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;{ &lt;/span&gt;&lt;span class=&#34;nv&#34;&gt;\left&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt; | &lt;/span&gt;&lt;span class=&#34;nv&#34;&gt;\left&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt; &lt;/span&gt;&lt;span class=&#34;o&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;nv&#34;&gt;\mathbf&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt; x_{n} &lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt; &lt;/span&gt;&lt;span class=&#34;nv&#34;&gt;\mathbf&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt; x_{n&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;m&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;} &lt;/span&gt;&lt;span class=&#34;nv&#34;&gt;\right&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt; &lt;/span&gt;&lt;span class=&#34;o&#34;&gt;)&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;^T &lt;/span&gt;&lt;span class=&#34;nv&#34;&gt;\left&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt; &lt;/span&gt;&lt;span class=&#34;o&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;nv&#34;&gt;\nabla&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt; F &lt;/span&gt;&lt;span class=&#34;o&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;nv&#34;&gt;\mathbf&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt; x_{n}&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;)&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt; &lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt; &lt;/span&gt;&lt;span class=&#34;nv&#34;&gt;\nabla&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt; F &lt;/span&gt;&lt;span class=&#34;o&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;nv&#34;&gt;\mathbf&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt; x_{n&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;m&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;}&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;)&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt; &lt;/span&gt;&lt;span class=&#34;nv&#34;&gt;\right&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt; &lt;/span&gt;&lt;span class=&#34;o&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt; &lt;/span&gt;&lt;span class=&#34;nv&#34;&gt;\right&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt; |}{&lt;/span&gt;&lt;span class=&#34;nv&#34;&gt;\left&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt; &lt;/span&gt;&lt;span class=&#34;nv&#34;&gt;\|\nabla&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt; F&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;nv&#34;&gt;\mathbf&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;{x}_{n}&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;)&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt; &lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt; &lt;/span&gt;&lt;span class=&#34;nv&#34;&gt;\nabla&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt; F&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;nv&#34;&gt;\mathbf&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;{x}_{n&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;m&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;}&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;)&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt; &lt;/span&gt;&lt;span class=&#34;nv&#34;&gt;\right&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt; &lt;/span&gt;&lt;span class=&#34;nv&#34;&gt;\|&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;^&lt;/span&gt;&lt;span class=&#34;m&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;}
&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nb&#34;&gt;&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;$$&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nb&#34;&gt;{{&lt;/span&gt;&amp;lt; /math &amp;gt;&lt;span class=&#34;nb&#34;&gt;}}&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;renders as&lt;/p&gt;

$$\gamma_{n} = \frac{ \left | \left (\mathbf x_{n} - \mathbf x_{n-1} \right )^T \left [\nabla F (\mathbf x_{n}) - \nabla F (\mathbf x_{n-1}) \right ] \right |}{\left \|\nabla F(\mathbf{x}_{n}) - \nabla F(\mathbf{x}_{n-1}) \right \|^2}$$


&lt;p&gt;Example &lt;strong&gt;inline math&lt;/strong&gt; &lt;code&gt;{{&amp;lt; math &amp;gt;}}$\nabla F(\mathbf{x}_{n})${{&amp;lt; /math &amp;gt;}}&lt;/code&gt; renders as $\nabla F(\mathbf{x}_{n})$
.&lt;/p&gt;
&lt;p&gt;Example &lt;strong&gt;multi-line math&lt;/strong&gt; using the math linebreak (&lt;code&gt;\\&lt;/code&gt;):&lt;/p&gt;
&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-latex&#34; data-lang=&#34;latex&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nb&#34;&gt;{{&lt;/span&gt;&amp;lt; math &amp;gt;&lt;span class=&#34;nb&#34;&gt;}}&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;sb&#34;&gt;$$&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;k;p_{&lt;/span&gt;&lt;span class=&#34;m&#34;&gt;0&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;}^{&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;}&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;)&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt; &lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt; &lt;/span&gt;&lt;span class=&#34;nv&#34;&gt;\begin&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;{cases}p_{&lt;/span&gt;&lt;span class=&#34;m&#34;&gt;0&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;}^{&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;} &amp;amp; &lt;/span&gt;&lt;span class=&#34;nv&#34;&gt;\text&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;{if }k&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;m&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;, &lt;/span&gt;&lt;span class=&#34;nv&#34;&gt;\\&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;
&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nb&#34;&gt;&lt;/span&gt;&lt;span class=&#34;m&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;p_{&lt;/span&gt;&lt;span class=&#34;m&#34;&gt;0&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;}^{&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;} &amp;amp; &lt;/span&gt;&lt;span class=&#34;nv&#34;&gt;\text&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;{if }k&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;m&#34;&gt;0&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;nv&#34;&gt;\end&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;{cases}&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;$$&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nb&#34;&gt;{{&lt;/span&gt;&amp;lt; /math &amp;gt;&lt;span class=&#34;nb&#34;&gt;}}&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;renders as&lt;/p&gt;


$$
f(k;p_{0}^{*}) = \begin{cases}p_{0}^{*} &amp; \text{if }k=1, \\
1-p_{0}^{*} &amp; \text{if }k=0.\end{cases}
$$



&lt;h2 id=&#34;code&#34;&gt;Code&lt;/h2&gt;
&lt;p&gt;Hugo Blox Builder utilises Hugo&amp;rsquo;s Markdown extension for highlighting code syntax. The code theme can be selected in the &lt;code&gt;config/_default/params.yaml&lt;/code&gt; file.&lt;/p&gt;
&lt;pre&gt;&lt;code&gt;```python
import pandas as pd
data = pd.read_csv(&amp;quot;data.csv&amp;quot;)
data.head()
```
&lt;/code&gt;&lt;/pre&gt;
&lt;p&gt;renders as&lt;/p&gt;
&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-python&#34; data-lang=&#34;python&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;kn&#34;&gt;import&lt;/span&gt; &lt;span class=&#34;nn&#34;&gt;pandas&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;as&lt;/span&gt; &lt;span class=&#34;nn&#34;&gt;pd&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;data&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;pd&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;read_csv&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;s2&#34;&gt;&amp;#34;data.csv&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;data&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;head&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;()&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;h2 id=&#34;inline-images&#34;&gt;Inline Images&lt;/h2&gt;
&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-go&#34; data-lang=&#34;go&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;p&#34;&gt;{{&amp;lt;&lt;/span&gt; &lt;span class=&#34;nx&#34;&gt;icon&lt;/span&gt; &lt;span class=&#34;nx&#34;&gt;name&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;python&amp;#34;&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;&amp;gt;}}&lt;/span&gt; &lt;span class=&#34;nx&#34;&gt;Python&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;renders as&lt;/p&gt;
&lt;p&gt;
  &lt;span class=&#34;inline-block  pr-1&#34;&gt;
    &lt;svg style=&#34;height: 1em; transform: translateY(0.1em);&#34; xmlns=&#34;http://www.w3.org/2000/svg&#34; height=&#34;1em&#34; viewBox=&#34;0 0 448 512&#34; fill=&#34;currentColor&#34;&gt;&lt;path d=&#34;M439.8 200.5c-7.7-30.9-22.3-54.2-53.4-54.2h-40.1v47.4c0 36.8-31.2 67.8-66.8 67.8H172.7c-29.2 0-53.4 25-53.4 54.3v101.8c0 29 25.2 46 53.4 54.3 33.8 9.9 66.3 11.7 106.8 0 26.9-7.8 53.4-23.5 53.4-54.3v-40.7H226.2v-13.6h160.2c31.1 0 42.6-21.7 53.4-54.2 11.2-33.5 10.7-65.7 0-108.6zM286.2 404c11.1 0 20.1 9.1 20.1 20.3 0 11.3-9 20.4-20.1 20.4-11 0-20.1-9.2-20.1-20.4.1-11.3 9.1-20.3 20.1-20.3zM167.8 248.1h106.8c29.7 0 53.4-24.5 53.4-54.3V91.9c0-29-24.4-50.7-53.4-55.6-35.8-5.9-74.7-5.6-106.8.1-45.2 8-53.4 24.7-53.4 55.6v40.7h106.9v13.6h-147c-31.1 0-58.3 18.7-66.8 54.2-9.8 40.7-10.2 66.1 0 108.6 7.6 31.6 25.7 54.2 56.8 54.2H101v-48.8c0-35.3 30.5-66.4 66.8-66.4zm-6.7-142.6c-11.1 0-20.1-9.1-20.1-20.3.1-11.3 9-20.4 20.1-20.4 11 0 20.1 9.2 20.1 20.4s-9 20.3-20.1 20.3z&#34;/&gt;&lt;/svg&gt;
  &lt;/span&gt; Python&lt;/p&gt;
&lt;h2 id=&#34;did-you-find-this-page-helpful-consider-sharing-it-&#34;&gt;Did you find this page helpful? Consider sharing it 🙌&lt;/h2&gt;
</description>
    </item>
    
    <item>
      <title>Learn Python</title>
      <link>https://www.blakecole.com/teaching/python/</link>
      <pubDate>Tue, 24 Oct 2023 00:00:00 +0000</pubDate>
      <guid>https://www.blakecole.com/teaching/python/</guid>
      <description>&lt;p&gt;&lt;a href=&#34;https://hugoblox.com&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Hugo Blox Builder&lt;/a&gt; is designed to give technical content creators a seamless experience. You can focus on the content and the Hugo Blox Builder which this template is built upon handles the rest.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Embed videos, podcasts, code, LaTeX math, and even test students!&lt;/strong&gt;&lt;/p&gt;
&lt;p&gt;On this page, you&amp;rsquo;ll find some examples of the types of technical content that can be rendered with Hugo Blox.&lt;/p&gt;
&lt;h2 id=&#34;video&#34;&gt;Video&lt;/h2&gt;
&lt;p&gt;Teach your course by sharing videos with your students. Choose from one of the following approaches:&lt;/p&gt;

    &lt;div style=&#34;position: relative; padding-bottom: 56.25%; height: 0; overflow: hidden;&#34;&gt;
      &lt;iframe allow=&#34;accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share&#34; allowfullscreen=&#34;allowfullscreen&#34; loading=&#34;eager&#34; referrerpolicy=&#34;strict-origin-when-cross-origin&#34; src=&#34;https://www.youtube.com/embed/D2vj0WcvH5c?autoplay=0&amp;amp;controls=1&amp;amp;end=0&amp;amp;loop=0&amp;amp;mute=0&amp;amp;start=0&#34; style=&#34;position: absolute; top: 0; left: 0; width: 100%; height: 100%; border:0;&#34; title=&#34;YouTube video&#34;&gt;&lt;/iframe&gt;
    &lt;/div&gt;

&lt;p&gt;&lt;strong&gt;Youtube&lt;/strong&gt;:&lt;/p&gt;
&lt;pre&gt;&lt;code&gt;{{&amp;lt; youtube w7Ft2ymGmfc &amp;gt;}}
&lt;/code&gt;&lt;/pre&gt;
&lt;p&gt;&lt;strong&gt;Bilibili&lt;/strong&gt;:&lt;/p&gt;
&lt;pre&gt;&lt;code&gt;{{&amp;lt; bilibili id=&amp;quot;BV1WV4y1r7DF&amp;quot; &amp;gt;}}
&lt;/code&gt;&lt;/pre&gt;
&lt;p&gt;&lt;strong&gt;Video file&lt;/strong&gt;&lt;/p&gt;
&lt;p&gt;Videos may be added to a page by either placing them in your &lt;code&gt;assets/media/&lt;/code&gt; media library or in your &lt;a href=&#34;https://gohugo.io/content-management/page-bundles/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;page&amp;rsquo;s folder&lt;/a&gt;, and then embedding them with the &lt;em&gt;video&lt;/em&gt; shortcode:&lt;/p&gt;
&lt;pre&gt;&lt;code&gt;{{&amp;lt; video src=&amp;quot;my_video.mp4&amp;quot; controls=&amp;quot;yes&amp;quot; &amp;gt;}}
&lt;/code&gt;&lt;/pre&gt;
&lt;h2 id=&#34;podcast&#34;&gt;Podcast&lt;/h2&gt;
&lt;p&gt;You can add a podcast or music to a page by placing the MP3 file in the page&amp;rsquo;s folder or the media library folder and then embedding the audio on your page with the &lt;em&gt;audio&lt;/em&gt; shortcode:&lt;/p&gt;
&lt;pre&gt;&lt;code&gt;{{&amp;lt; audio src=&amp;quot;ambient-piano.mp3&amp;quot; &amp;gt;}}
&lt;/code&gt;&lt;/pre&gt;
&lt;p&gt;Try it out:&lt;/p&gt;








  








&lt;audio controls &gt;
  &lt;source src=&#34;https://www.blakecole.com/teaching/python/ambient-piano.mp3&#34; type=&#34;audio/mpeg&#34;&gt;
&lt;/audio&gt;

&lt;h2 id=&#34;test-students&#34;&gt;Test students&lt;/h2&gt;
&lt;p&gt;Provide a simple yet fun self-assessment by revealing the solutions to challenges with the &lt;code&gt;spoiler&lt;/code&gt; shortcode:&lt;/p&gt;
&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-markdown&#34; data-lang=&#34;markdown&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;{{&lt;span class=&#34;p&#34;&gt;&amp;lt;&lt;/span&gt; &lt;span class=&#34;nt&#34;&gt;spoiler&lt;/span&gt; &lt;span class=&#34;na&#34;&gt;text&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;👉 Click to view the solution&amp;#34;&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;&amp;gt;&lt;/span&gt;}}
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;You found me!
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;{{&lt;span class=&#34;p&#34;&gt;&amp;lt;&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;/&lt;/span&gt;&lt;span class=&#34;nt&#34;&gt;spoiler&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;&amp;gt;&lt;/span&gt;}}
&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;renders as&lt;/p&gt;
&lt;details class=&#34;spoiler &#34;  id=&#34;spoiler-2&#34;&gt;
  &lt;summary class=&#34;cursor-pointer&#34;&gt;👉 Click to view the solution&lt;/summary&gt;
  &lt;div class=&#34;rounded-lg bg-neutral-50 dark:bg-neutral-800 p-2&#34;&gt;
    You found me 🎉
  &lt;/div&gt;
&lt;/details&gt;
&lt;h2 id=&#34;math&#34;&gt;Math&lt;/h2&gt;
&lt;p&gt;Hugo Blox Builder supports a Markdown extension for $\LaTeX$ math. You can enable this feature by toggling the &lt;code&gt;math&lt;/code&gt; option in your &lt;code&gt;config/_default/params.yaml&lt;/code&gt; file.&lt;/p&gt;
&lt;p&gt;To render &lt;em&gt;inline&lt;/em&gt; or &lt;em&gt;block&lt;/em&gt; math, wrap your LaTeX math with &lt;code&gt;{{&amp;lt; math &amp;gt;}}$...${{&amp;lt; /math &amp;gt;}}&lt;/code&gt; or &lt;code&gt;{{&amp;lt; math &amp;gt;}}$$...$${{&amp;lt; /math &amp;gt;}}&lt;/code&gt;, respectively.&lt;/p&gt;
&lt;div class=&#34;px-4 py-3 mb-6 rounded-md bg-primary-100 dark:bg-primary-900&#34;&gt;
&lt;pre&gt;&lt;code&gt;&amp;lt;div class=&amp;quot;flex&amp;quot;&amp;gt;
  &amp;lt;span class=&amp;quot;pr-3 pt-1 text-primary-600 dark:text-primary-300&amp;quot;&amp;gt;
    &amp;lt;svg height=&amp;quot;24&amp;quot; xmlns=&amp;quot;http://www.w3.org/2000/svg&amp;quot; viewBox=&amp;quot;0 0 24 24&amp;quot;&amp;gt;&amp;lt;path fill=&amp;quot;none&amp;quot; stroke=&amp;quot;currentColor&amp;quot; stroke-linecap=&amp;quot;round&amp;quot; stroke-linejoin=&amp;quot;round&amp;quot; stroke-width=&amp;quot;1.5&amp;quot; d=&amp;quot;m11.25 11.25l.041-.02a.75.75 0 0 1 1.063.852l-.708 2.836a.75.75 0 0 0 1.063.853l.041-.021M21 12a9 9 0 1 1-18 0a9 9 0 0 1 18 0m-9-3.75h.008v.008H12z&amp;quot;/&amp;gt;&amp;lt;/svg&amp;gt;
  &amp;lt;/span&amp;gt;
  &amp;lt;span class=&amp;quot;dark:text-neutral-300&amp;quot;&amp;gt;We wrap the LaTeX math in the Hugo Blox &amp;lt;em&amp;gt;math&amp;lt;/em&amp;gt; shortcode to prevent Hugo rendering our math as Markdown.&amp;lt;/span&amp;gt;
&amp;lt;/div&amp;gt;
&lt;/code&gt;&lt;/pre&gt;
&lt;/div&gt;
&lt;p&gt;Example &lt;strong&gt;math block&lt;/strong&gt;:&lt;/p&gt;
&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-latex&#34; data-lang=&#34;latex&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nb&#34;&gt;{{&lt;/span&gt;&amp;lt; math &amp;gt;&lt;span class=&#34;nb&#34;&gt;}}&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;sb&#34;&gt;$$&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;
&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nb&#34;&gt;&lt;/span&gt;&lt;span class=&#34;nv&#34;&gt;\gamma&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;_{n} &lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt; &lt;/span&gt;&lt;span class=&#34;nv&#34;&gt;\frac&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;{ &lt;/span&gt;&lt;span class=&#34;nv&#34;&gt;\left&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt; | &lt;/span&gt;&lt;span class=&#34;nv&#34;&gt;\left&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt; &lt;/span&gt;&lt;span class=&#34;o&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;nv&#34;&gt;\mathbf&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt; x_{n} &lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt; &lt;/span&gt;&lt;span class=&#34;nv&#34;&gt;\mathbf&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt; x_{n&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;m&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;} &lt;/span&gt;&lt;span class=&#34;nv&#34;&gt;\right&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt; &lt;/span&gt;&lt;span class=&#34;o&#34;&gt;)&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;^T &lt;/span&gt;&lt;span class=&#34;nv&#34;&gt;\left&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt; &lt;/span&gt;&lt;span class=&#34;o&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;nv&#34;&gt;\nabla&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt; F &lt;/span&gt;&lt;span class=&#34;o&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;nv&#34;&gt;\mathbf&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt; x_{n}&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;)&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt; &lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt; &lt;/span&gt;&lt;span class=&#34;nv&#34;&gt;\nabla&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt; F &lt;/span&gt;&lt;span class=&#34;o&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;nv&#34;&gt;\mathbf&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt; x_{n&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;m&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;}&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;)&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt; &lt;/span&gt;&lt;span class=&#34;nv&#34;&gt;\right&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt; &lt;/span&gt;&lt;span class=&#34;o&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt; &lt;/span&gt;&lt;span class=&#34;nv&#34;&gt;\right&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt; |}{&lt;/span&gt;&lt;span class=&#34;nv&#34;&gt;\left&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt; &lt;/span&gt;&lt;span class=&#34;nv&#34;&gt;\|\nabla&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt; F&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;nv&#34;&gt;\mathbf&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;{x}_{n}&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;)&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt; &lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt; &lt;/span&gt;&lt;span class=&#34;nv&#34;&gt;\nabla&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt; F&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;nv&#34;&gt;\mathbf&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;{x}_{n&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;m&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;}&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;)&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt; &lt;/span&gt;&lt;span class=&#34;nv&#34;&gt;\right&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt; &lt;/span&gt;&lt;span class=&#34;nv&#34;&gt;\|&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;^&lt;/span&gt;&lt;span class=&#34;m&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;}
&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nb&#34;&gt;&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;$$&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nb&#34;&gt;{{&lt;/span&gt;&amp;lt; /math &amp;gt;&lt;span class=&#34;nb&#34;&gt;}}&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;renders as&lt;/p&gt;

$$\gamma_{n} = \frac{ \left | \left (\mathbf x_{n} - \mathbf x_{n-1} \right )^T \left [\nabla F (\mathbf x_{n}) - \nabla F (\mathbf x_{n-1}) \right ] \right |}{\left \|\nabla F(\mathbf{x}_{n}) - \nabla F(\mathbf{x}_{n-1}) \right \|^2}$$


&lt;p&gt;Example &lt;strong&gt;inline math&lt;/strong&gt; &lt;code&gt;{{&amp;lt; math &amp;gt;}}$\nabla F(\mathbf{x}_{n})${{&amp;lt; /math &amp;gt;}}&lt;/code&gt; renders as $\nabla F(\mathbf{x}_{n})$
.&lt;/p&gt;
&lt;p&gt;Example &lt;strong&gt;multi-line math&lt;/strong&gt; using the math linebreak (&lt;code&gt;\\&lt;/code&gt;):&lt;/p&gt;
&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-latex&#34; data-lang=&#34;latex&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nb&#34;&gt;{{&lt;/span&gt;&amp;lt; math &amp;gt;&lt;span class=&#34;nb&#34;&gt;}}&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;sb&#34;&gt;$$&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;k;p_{&lt;/span&gt;&lt;span class=&#34;m&#34;&gt;0&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;}^{&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;}&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;)&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt; &lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt; &lt;/span&gt;&lt;span class=&#34;nv&#34;&gt;\begin&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;{cases}p_{&lt;/span&gt;&lt;span class=&#34;m&#34;&gt;0&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;}^{&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;} &amp;amp; &lt;/span&gt;&lt;span class=&#34;nv&#34;&gt;\text&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;{if }k&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;m&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;, &lt;/span&gt;&lt;span class=&#34;nv&#34;&gt;\\&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;
&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nb&#34;&gt;&lt;/span&gt;&lt;span class=&#34;m&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;p_{&lt;/span&gt;&lt;span class=&#34;m&#34;&gt;0&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;}^{&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;} &amp;amp; &lt;/span&gt;&lt;span class=&#34;nv&#34;&gt;\text&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;{if }k&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;m&#34;&gt;0&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;nv&#34;&gt;\end&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;{cases}&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;$$&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nb&#34;&gt;{{&lt;/span&gt;&amp;lt; /math &amp;gt;&lt;span class=&#34;nb&#34;&gt;}}&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;renders as&lt;/p&gt;


$$
f(k;p_{0}^{*}) = \begin{cases}p_{0}^{*} &amp; \text{if }k=1, \\
1-p_{0}^{*} &amp; \text{if }k=0.\end{cases}
$$



&lt;h2 id=&#34;code&#34;&gt;Code&lt;/h2&gt;
&lt;p&gt;Hugo Blox Builder utilises Hugo&amp;rsquo;s Markdown extension for highlighting code syntax. The code theme can be selected in the &lt;code&gt;config/_default/params.yaml&lt;/code&gt; file.&lt;/p&gt;
&lt;pre&gt;&lt;code&gt;```python
import pandas as pd
data = pd.read_csv(&amp;quot;data.csv&amp;quot;)
data.head()
```
&lt;/code&gt;&lt;/pre&gt;
&lt;p&gt;renders as&lt;/p&gt;
&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-python&#34; data-lang=&#34;python&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;kn&#34;&gt;import&lt;/span&gt; &lt;span class=&#34;nn&#34;&gt;pandas&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;as&lt;/span&gt; &lt;span class=&#34;nn&#34;&gt;pd&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;data&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;pd&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;read_csv&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;s2&#34;&gt;&amp;#34;data.csv&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;data&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;head&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;()&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;h2 id=&#34;inline-images&#34;&gt;Inline Images&lt;/h2&gt;
&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-go&#34; data-lang=&#34;go&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;p&#34;&gt;{{&amp;lt;&lt;/span&gt; &lt;span class=&#34;nx&#34;&gt;icon&lt;/span&gt; &lt;span class=&#34;nx&#34;&gt;name&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;python&amp;#34;&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;&amp;gt;}}&lt;/span&gt; &lt;span class=&#34;nx&#34;&gt;Python&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;renders as&lt;/p&gt;
&lt;p&gt;
  &lt;span class=&#34;inline-block  pr-1&#34;&gt;
    &lt;svg style=&#34;height: 1em; transform: translateY(0.1em);&#34; xmlns=&#34;http://www.w3.org/2000/svg&#34; height=&#34;1em&#34; viewBox=&#34;0 0 448 512&#34; fill=&#34;currentColor&#34;&gt;&lt;path d=&#34;M439.8 200.5c-7.7-30.9-22.3-54.2-53.4-54.2h-40.1v47.4c0 36.8-31.2 67.8-66.8 67.8H172.7c-29.2 0-53.4 25-53.4 54.3v101.8c0 29 25.2 46 53.4 54.3 33.8 9.9 66.3 11.7 106.8 0 26.9-7.8 53.4-23.5 53.4-54.3v-40.7H226.2v-13.6h160.2c31.1 0 42.6-21.7 53.4-54.2 11.2-33.5 10.7-65.7 0-108.6zM286.2 404c11.1 0 20.1 9.1 20.1 20.3 0 11.3-9 20.4-20.1 20.4-11 0-20.1-9.2-20.1-20.4.1-11.3 9.1-20.3 20.1-20.3zM167.8 248.1h106.8c29.7 0 53.4-24.5 53.4-54.3V91.9c0-29-24.4-50.7-53.4-55.6-35.8-5.9-74.7-5.6-106.8.1-45.2 8-53.4 24.7-53.4 55.6v40.7h106.9v13.6h-147c-31.1 0-58.3 18.7-66.8 54.2-9.8 40.7-10.2 66.1 0 108.6 7.6 31.6 25.7 54.2 56.8 54.2H101v-48.8c0-35.3 30.5-66.4 66.8-66.4zm-6.7-142.6c-11.1 0-20.1-9.1-20.1-20.3.1-11.3 9-20.4 20.1-20.4 11 0 20.1 9.2 20.1 20.4s-9 20.3-20.1 20.3z&#34;/&gt;&lt;/svg&gt;
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</description>
    </item>
    
    <item>
      <title>Leveraging the ADCIRC Model to Plan an Open-Water Swim from Woods Hole to Martha&#39;s Vineyard</title>
      <link>https://www.blakecole.com/post/1-adcirc-mv-swim/</link>
      <pubDate>Mon, 08 Aug 2022 13:09:10 -0400</pubDate>
      <guid>https://www.blakecole.com/post/1-adcirc-mv-swim/</guid>
      <description>


&lt;details class=&#34;print:hidden xl:hidden&#34; open&gt;
  &lt;summary&gt;Table of Contents&lt;/summary&gt;
  &lt;div class=&#34;text-sm&#34;&gt;
  &lt;nav id=&#34;TableOfContents&#34;&gt;
  &lt;ul&gt;
    &lt;li&gt;&lt;a href=&#34;#preamble&#34;&gt;Preamble&lt;/a&gt;&lt;/li&gt;
    &lt;li&gt;&lt;a href=&#34;#the-chappy-swimmers&#34;&gt;The Chappy Swimmers&lt;/a&gt;&lt;/li&gt;
    &lt;li&gt;&lt;a href=&#34;#the-plan&#34;&gt;The Plan&lt;/a&gt;&lt;/li&gt;
    &lt;li&gt;&lt;a href=&#34;#the-adcirc-hydrodynamic-model&#34;&gt;The ADCIRC Hydrodynamic Model&lt;/a&gt;
      &lt;ul&gt;
        &lt;li&gt;&lt;a href=&#34;#background&#34;&gt;Background&lt;/a&gt;&lt;/li&gt;
        &lt;li&gt;&lt;a href=&#34;#tidal-constituent-corrections&#34;&gt;Tidal Constituent Corrections&lt;/a&gt;&lt;/li&gt;
        &lt;li&gt;&lt;a href=&#34;#swimmer-simulation&#34;&gt;Swimmer Simulation&lt;/a&gt;&lt;/li&gt;
        &lt;li&gt;&lt;a href=&#34;#assumptions--caveats&#34;&gt;Assumptions &amp;amp; Caveats&lt;/a&gt;&lt;/li&gt;
        &lt;li&gt;&lt;a href=&#34;#results&#34;&gt;Results:&lt;/a&gt;&lt;/li&gt;
        &lt;li&gt;&lt;a href=&#34;#key-findings&#34;&gt;Key Findings&lt;/a&gt;&lt;/li&gt;
      &lt;/ul&gt;
    &lt;/li&gt;
    &lt;li&gt;&lt;a href=&#34;#the-swim&#34;&gt;The Swim&lt;/a&gt;&lt;/li&gt;
    &lt;li&gt;&lt;a href=&#34;#conclusion&#34;&gt;Conclusion&lt;/a&gt;
      &lt;ul&gt;
        &lt;li&gt;&lt;/li&gt;
      &lt;/ul&gt;
    &lt;/li&gt;
    &lt;li&gt;&lt;a href=&#34;#conclusion-ii&#34;&gt;Conclusion II&lt;/a&gt;&lt;/li&gt;
  &lt;/ul&gt;
&lt;/nav&gt;
  &lt;/div&gt;
&lt;/details&gt;

&lt;h2 id=&#34;preamble&#34;&gt;Preamble&lt;/h2&gt;
&lt;p&gt;At 5:58am on the morning of July 30th 2022, a small group of swimmers dove into the shallow waters off Woods Hole, Massachusetts, and began swimming steadily southwest toward Martha&amp;rsquo;s Vineyard.  The island was already visible in the early dawn light, though its features were obscured by mist and atmospheric scattering.  Lacking terrestrial landmarks, the swimmers would need to orient themselves relative to nearby guide boats in order to make it safely to their intended destination near Lake Tashmoo.&lt;/p&gt;
&lt;p&gt;As the crow flies, Martha&amp;rsquo;s Vineyard and Woods Hole are separated by only 3.3 miles; however, the sea separating these two landmasses is infamously tempestuous, with currents routinely exceeding 5 knots.  These strong currents are driven in large part by tidal resonance in the Gulf of Maine, which produces an isotropic pressure gradient between Cape Cod Bay and Buzzards Bay.&lt;/p&gt;
&lt;p&gt;A strong ocean swimmer can maintain a speed of 1.4-1.8 knots in ideal conditions, and sprint at 3 knots for a brief time.  Thus, it is vitally important that one seeking to swim across Vineyard Sound make a careful accounting of its powerful tidal currents.&lt;/p&gt;
&lt;h2 id=&#34;the-chappy-swimmers&#34;&gt;The Chappy Swimmers&lt;/h2&gt;
&lt;p&gt;The genesis of this swim can be traced back to an eclectic cadre of goggle-clad humans who call themselves the Chappy Swimmers.  This Falmouth-based swim club is a tight-knit group of long-time friends who range in age from 30 to 70, and simply enjoy swimming together.  &lt;a href=&#34;https://www.usgs.gov/staff-profiles/rob-thieler&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Rob Thieler&lt;/a&gt;, the Center Director of the USGS Woods Hole Coastal and Marine Science Center, manages the Chappy Swimmers group text, and is responsible for informing members where they will be swimming each weekend.  Rob is one of the eldest members (by seniority, not age), and has been swimming with the group since its founding in the early 1990&amp;rsquo;s.  Anyone can become a Chappy Swimmer &amp;ndash; the group is exceedingly friendly and inclusive.  One of the regulars named John McCadam started training with the group a few years ago, despite having no formal swim background.  He now swims every day, even in the icy winter months when water temperatures routinely dip into the low-30s.&lt;/p&gt;






&lt;figure style=&#34;text-align: center; margin-top: 50px; margin-bottom: 50px;&#34;&gt;

  &lt;a data-fancybox=&#34;&#34; href=&#34;chappy-swimmers.jpg&#34; style=&#34;display:inline-block; margin:0; padding:0; line-height:0;&#34; data-caption=&#34;A small subset of the Chappy Swimmers, featuring: Rick Rheinhardt, Blake Cole, Kalina Grabb, Meri Gilson, Mary Kaminski, and Moira McCollough. [Photo: Chuck Redington]&#34;&gt;

&lt;img src=&#34;chappy-swimmers.jpg&#34; style=&#34;display:block; margin:0; padding:0;&#34; alt=&#34;A small subset of the Chappy Swimmers, featuring: Rick Rheinhardt, Blake Cole, Kalina Grabb, Meri Gilson, Mary Kaminski, and Moira McCollough. [Photo: Chuck Redington]&#34; &gt;
&lt;/a&gt;


&lt;figcaption style=&#34;text-align: left;&#34; data-pre=&#34;Figure &#34; data-post=&#34;:&#34; &gt;
  
  &lt;p&gt;
    A small subset of the Chappy Swimmers, featuring: Rick Rheinhardt, Blake Cole, Kalina Grabb, Meri Gilson, Mary Kaminski, and Moira McCollough. [Photo: Chuck Redington]
    
    
    
  &lt;/p&gt; 
&lt;/figcaption&gt;

&lt;/figure&gt;

&lt;p&gt;Shockingly, John isn&amp;rsquo;t the only blue-lipped masochist in the group.  Of the thirty or so active summer members, approximately five are gnarly enough to swim through the winter.  Sometimes, on cold, dark January mornings, veterans of the group would speak in hushed tones about the time &amp;ndash; nearly a decade ago &amp;ndash; when they successfully swam across Vineyard Sound.  These stories piqued the interest of recent MIT-WHOI Joint Program graduate Dr. Kalina Grabb.&lt;/p&gt;
&lt;p&gt;Kalina is an intrepid human.  She played collegiate waterpolo, and still carries herself with a subtle alertness that surely grew out of the need to remain calm while grim-faced aquatic gladiators tried to drown her in various chlorinated cerulean coliseums nationwide.  However, this quiet intensity is often obscured by her overwhelming geniality.  Kalina is one of the nicest people you will ever meet, and has an infectious laugh.  She seems capable of weathering life&amp;rsquo;s most brutal tempests with a smile and a wave.  So, in Dr. Grabb, we have a highly-competent swimmer who is both endlessly game for adventure, and affable enough to coax the fence-sitters (including yours truly) into action.&lt;/p&gt;
&lt;h2 id=&#34;the-plan&#34;&gt;The Plan&lt;/h2&gt;
&lt;p&gt;In March 2022, a small group of the winter Chappy Swimmers, including John McCadam, Meri Gilson, Mary Kaminski, and Kalina Grabb were out freezing their asses off somewhere in Buzzards Bay when Kalina popped the question: &amp;ldquo;So who wants to swim to Martha&amp;rsquo;s Vineyard this year?&amp;rdquo;  A salvo of enthusiastic replies filled the chilly Spring air, echoing off the rows of vacant summer mansions which line the waterfront.  A plan had been hatched.&lt;/p&gt;
&lt;p&gt;However, a plan once hatched, must be nurtured until it is able to sustain itself, and take flight.  Months passed, and the grand plan remained anaemic.  In early July 2022, Kalina texted the Chappy Swimmers to alert them that she was serious about the idea, and would be starting a new group chat with anyone interested in participating.  One man was particularly interested, and would prove himself invaluable in the planning process: Rick Rheinhardt.&lt;/p&gt;
&lt;p&gt;Rick grabbed the ball from Kalina and ran with it.&lt;/p&gt;
&lt;p&gt;On July 14th, 2022, Rick sent out a detailed email covering swim logistics, safety concerns, ferry schedules, support boat logistics, and swim date proposals.  Long before most people had even seriously considered how messy and complicated this swim could get, Rick had basically already hashed the whole thing out.  An excerpt from his email read:&lt;/p&gt;
&lt;blockquote&gt;
&lt;p&gt;I have been looking at tides and currents in the 2022 Eldridge Tide and Pilot Book&amp;hellip; Current information is critical to us because we cannot outswim them, but we can use them to our advantage&amp;hellip; There is a countercurrent off MV from Norton Pt to Vineyard Haven that floods NE while the main part of the Vineyard [Sound] ebbs southwest. If we catch [the current] off Norton Point, it will assist us the last 1/3rd of our swim.  If we catch it too far North, it could pull us into Vineyard Haven Bay and into the ferry route&amp;hellip;. &lt;strong&gt;I think we should plan to leave on an ebb tide in Vineyard Sound before slack water at dead low tide.  Riding an ebb tide will take us southward and away from most boat and ferry traffic.&lt;/strong&gt;&lt;/p&gt;
&lt;/blockquote&gt;






&lt;figure style=&#34;text-align: center; margin-top: 50px; margin-bottom: 50px;&#34;&gt;

  &lt;a data-fancybox=&#34;&#34; href=&#34;eldridge.jpg&#34; style=&#34;display:inline-block; margin:0; padding:0; line-height:0;&#34; data-caption=&#34;An excerpt from the 2022 Eldridge Tide and Pilot Book, which was used to determine an ideal time to start the swim: a neap ebb tide, just before slack. [Photo: Rick Rheinhardt]&#34;&gt;

&lt;img src=&#34;eldridge.jpg&#34; style=&#34;display:block; margin:0; padding:0;&#34; alt=&#34;An excerpt from the 2022 Eldridge Tide and Pilot Book, which was used to determine an ideal time to start the swim: a neap ebb tide, just before slack. [Photo: Rick Rheinhardt]&#34; &gt;
&lt;/a&gt;


&lt;figcaption style=&#34;text-align: left;&#34; data-pre=&#34;Figure &#34; data-post=&#34;:&#34; &gt;
  
  &lt;p&gt;
    An excerpt from the 2022 Eldridge Tide and Pilot Book, which was used to determine an ideal time to start the swim: a neap ebb tide, just before slack. [Photo: Rick Rheinhardt]
    
    
    
  &lt;/p&gt; 
&lt;/figcaption&gt;

&lt;/figure&gt;

&lt;p&gt;Rick&amp;rsquo;s analysis tipped the scales: for the first time, it felt real, and everyone started to believe that the swim was actually going to happen.  A target date had been selected: July 30th, 2022.  This gave the swimmers two weeks to start conditioning, and to recruit additional support vessels.&lt;/p&gt;
&lt;h2 id=&#34;the-adcirc-hydrodynamic-model&#34;&gt;The ADCIRC Hydrodynamic Model&lt;/h2&gt;
&lt;h3 id=&#34;background&#34;&gt;Background&lt;/h3&gt;
&lt;p&gt;On July 20th, I mentioned  to my advisor, &lt;a href=&#34;https://www2.whoi.edu/staff/ptraykovski/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Dr. Peter Traykovski&lt;/a&gt;, that I intended to swim to Martha&amp;rsquo;s Vineyard in a few week&amp;rsquo;s time.  Peter is an avid waterman himself, and immediately took a keen interest in the endeavor.  Understanding the severity of the tidal currents in Vineyard Sound, he inquired as to whether we had developed a detailed plan to contend with these currents.  I told Peter that we had narrowed down a set of prospective dates using the 2022 Eldgride Tide and Pilot Book, but still needed a specific swim plan for the day (i.e. a set of waypoints or headings to follow).&lt;/p&gt;
&lt;p&gt;Mostly for fun, he suggested I pursue  a technical approach, using the output of a tidally-driven, depth-averaged hydrodynamic model of Vineyard Sound to assess the effect of current action on a simulated swimmer.  So, I got to work!&lt;/p&gt;
&lt;p&gt;In 2012, the National Oceanic and Atmospheric Administration (NOAA) used a shallow-water equation solver called the &lt;a href=&#34;https://adcirc.org/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;ADvanced CIRCulation model (ADCIRC)&lt;/a&gt; to create a vertical datum transformation tool called &lt;a href=&#34;https://vdatum.noaa.gov/download/publications/TM_NOS_CS_31_FY13_04_Yang_VDatumGulfOfMaine.pdf&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;VDATUM&lt;/a&gt;.  The purpose of this tool was to facilitate the comparison of water level measurements made with respect to different vertical references.  For instance, some measurements might be made with respect to a tidal datum, like Mean Lower Low Water (MLLW), while others might be made with respect to an orthometric datum, like &lt;a href=&#34;https://en.wikipedia.org/wiki/North_American_Vertical_Datum_of_1988&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;NAVD88&lt;/a&gt;.  In order to create this tool, NOAA needed to simulate sea surface levels under tidal action, from the Gulf of Maine south to Rhode Island, over a span of 55 days.  For the purposes of this project, NOAA was not particularly interested in tidal currents.&lt;/p&gt;
&lt;p&gt;However, the shallow-water equations require the simultaneous computation of water level &lt;em&gt;and&lt;/em&gt; flow velocity at each computational grid point.  It is useful to look at a simplified 1D form of the shallow-water equations (sometimes called the &lt;em&gt;Saint-Venant equations&lt;/em&gt;) to illustrate this point:&lt;/p&gt;
$$\frac{\partial A}{\partial t} + \frac{\partial \left(Au \right)}{\partial x} = 0$$$$\frac{\partial u}{\partial t} + u\frac{\partial u}{\partial x} + g\frac{\partial \zeta}{\partial x} = - \frac{P}{A} \frac{\tau}{\rho}$$&lt;p&gt;The first of these equations can be recognized as the continuity equation.  Together, these two equations describe one-dimensional flow of an incompressible fluid in an open channel, where $x$ is the spatial coordinate running along the length of the channel; $t$ represents time; $A(x,t)$ is the cross-sectional area of the flow; $\tau(x,t)$ is the shear stress along the wetted perimeter $P(x,t)$; $u(x,t)$ is the &lt;strong&gt;flow velocity&lt;/strong&gt; along the channel axis; and $\zeta(x,t)$ is the &lt;strong&gt;free surface elevation&lt;/strong&gt;.&lt;/p&gt;
&lt;p&gt;This is probably a good time to mention that the good folks at NOAA and the USGS really did all the heavy lifting here (specifically, Zizang Yang, Edward P. Myers, Erin Twomey, and Rich Signell): they generated the computational grid; synthesized bathymetric data from multiple sources; interpolated this data onto the computational grid; and used nine tidal constituents to drive the free surface elevation at the model boundaries.  ADCIRC handled the rest, predicting the free surface elevation and flow velocity at each grid point over a 55 day span in 2012.  The model output was then compared to field measurements to assess its accuracy, and determine appropriate empirical corrections.  These corrections were then incorporated back into the ADCIRC model, further enhancing the accuracy of its predictions.&lt;/p&gt;
&lt;h3 id=&#34;tidal-constituent-corrections&#34;&gt;Tidal Constituent Corrections&lt;/h3&gt;
&lt;p&gt;Due to the fact that tidally-driven currents are periodic in nature, I was able to adapt these ADCIRC model results from 2012 to my application in 2022.&lt;/p&gt;
&lt;p&gt;To do this, I needed to complete the following steps:&lt;/p&gt;
&lt;ol&gt;
&lt;li&gt;Apply a phase offset (&lt;em&gt;nodal phase modulation&lt;/em&gt;) and amplitude correction to each tidal constituent.  These corrections are dependent on date and latitude.&lt;/li&gt;
&lt;li&gt;Determine the tidal constituent phases corresponding to the morning of July 30th, 2022.&lt;/li&gt;
&lt;li&gt;At each timestep, sum the contribution of each tidal constituent at each spatial grid point.&lt;/li&gt;
&lt;/ol&gt;
&lt;p&gt;Fortunately, there is a MATLAB toolbox called &lt;a href=&#34;https://www.eoas.ubc.ca/~rich/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;T_TIDE&lt;/a&gt;, written by Rich Pawlowicz, which facilitates the determination of the relevant tidal constituent corrections.  The implementation details of this toolbox are covered in a paper by Rich Pawlowicz, Bob Beardsley, and Steve Lentz, titled &lt;a href=&#34;https://doi.org/10.1016/S0098-3004%2802%2900013-4&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Classical tidal harmonic analysis including error estimates in MATLAB using T_TIDE&lt;/a&gt;.&lt;/p&gt;
&lt;h3 id=&#34;swimmer-simulation&#34;&gt;Swimmer Simulation&lt;/h3&gt;
&lt;p&gt;Once I was able to determine the time-varying vector field of current velocities in Vineyard Sound, I set about simulating the motion of a swimmer.  The basic procedure is as follows:&lt;/p&gt;
&lt;ol&gt;
&lt;li&gt;Specify an initial swimmer position in geodetic coordinates (i.e. latitude and longitude).&lt;/li&gt;
&lt;li&gt;Specify swimmer heading and speed-through-water.  These values can be held constant, or dynamically updated over the course of the simulation.  This permits the simulation of a swimmer slowing down due to fatigue, or following a non-constant heading.&lt;/li&gt;
&lt;li&gt;Linearly interpolate the ADCIRC current prediction to the swimmer&amp;rsquo;s position.&lt;/li&gt;
&lt;li&gt;Compute the vector sum of the swimmer velocity and interpolated current velocity to estimate the swimmer&amp;rsquo;s speed over ground (SOG) and course over ground (COG).&lt;/li&gt;
&lt;li&gt;Using the swimmer&amp;rsquo;s SOG and COG, solve &lt;a href=&#34;https://en.wikipedia.org/wiki/Vincenty%27s_formulae#Direct_problem&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Vincenty&amp;rsquo;s Direct Problem&lt;/a&gt; to propagate the swimmer position along the WGS84 ellipsoid, and repeat steps 3-5 at each timestep.&lt;/li&gt;
&lt;/ol&gt;
&lt;h3 id=&#34;assumptions--caveats&#34;&gt;Assumptions &amp;amp; Caveats&lt;/h3&gt;
&lt;ul&gt;
&lt;li&gt;
&lt;p&gt;The model timestep was chosen to be $\Delta t = 0.1~hr = 6~min$.  This seemed like the shortest time over which a swimmer and guide boat could realistically perceive trajectory errors, and make changes to their course or speed.  Additional refinement of temporal resolution would not have been useful.  Because I was not actually solving the shallow-water equations, it was not necessary to consider the &lt;a href=&#34;https://en.wikipedia.org/wiki/Courant-Friedrichs-Lewy_condition&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Courant–Friedrichs–Lewy condition&lt;/a&gt;.&lt;/p&gt;
&lt;/li&gt;
&lt;li&gt;
&lt;p&gt;The spatial resolution of the bathymetric data is not fine enough to fully capture the shallow ridge at Middle Ground.  This will likely lead to an underestimation of the tidal current velocity in this region.&lt;/p&gt;
&lt;/li&gt;
&lt;li&gt;
&lt;p&gt;The ADCIRC model is depth-averaged.  This assumption is only valid in circumstances when the horizontal flow velocity is much faster than the vertical flow velocity.  This assumption may not hold true in parts of the computational domain where there are rapid changes in bathymetry (like Middle Ground).&lt;/p&gt;
&lt;/li&gt;
&lt;li&gt;
&lt;p&gt;The Vineyard Sound specific ADCIRC model is driven by barotropic pressure gradients only (i.e. surface set up resulting from tidal action).  It does not account for wind shear, nor wave-induced radiation stresses.&lt;/p&gt;
&lt;/li&gt;
&lt;/ul&gt;
&lt;h3 id=&#34;results&#34;&gt;Results:&lt;/h3&gt;
&lt;p&gt;Well, if you made it this far, congratulations!  Your reward is a handful of colorful plots and animations.&lt;/p&gt;
&lt;p&gt;Recall that the point of all of this was to determine a feasible, effective swim strategy.  The three key components of a simple swim strategy include:&lt;/p&gt;
&lt;ul&gt;
&lt;li&gt;Departure time&lt;/li&gt;
&lt;li&gt;Swimmer speed&lt;/li&gt;
&lt;li&gt;Swimmer heading&lt;/li&gt;
&lt;/ul&gt;
&lt;p&gt;Ideally, the modified ADCIRC model would tell us something about how a change in any of these variables changes the fundamental characteristics of the swim, which can be described by:&lt;/p&gt;
&lt;ul&gt;
&lt;li&gt;Swim duration&lt;/li&gt;
&lt;li&gt;Swim distance&lt;/li&gt;
&lt;li&gt;Arrival destination&lt;/li&gt;
&lt;/ul&gt;
&lt;p&gt;Of these three characteristics, &lt;em&gt;swim duration&lt;/em&gt; is most important.  Assuming the swimmer is providing an approximately constant level of effort, swim duration serves as a good heuristic for &lt;em&gt;fatigue&lt;/em&gt;.  Conversely, &lt;em&gt;swim distance&lt;/em&gt; is less informative.  Consider the boundary cases:&lt;/p&gt;
&lt;ul&gt;
&lt;li&gt;A swimmer opposing a strong current will move nowhere, and nevertheless become completely exhausted after some time.&lt;/li&gt;
&lt;li&gt;A swimmer aided by a strong current could exert no effort, and nevertheless arrive at their destination some arbitrary distance downstream.&lt;/li&gt;
&lt;/ul&gt;
&lt;p&gt;For this particular swim, &lt;em&gt;arrival destination&lt;/em&gt; is also quite important, as it serves as a heuristic for risk of encountering ferry traffic.  Finishing too far to the northeast, near West Chop Point, places one in a strong northeasterly current, directly adjacent to ferry traffic entering and exiting Vineyard Haven Bay.  In this region, a small miscalculation could have catastrophic effects.&lt;/p&gt;
&lt;p&gt;One final caveat before I dive into the simulation results:&lt;/p&gt;
&lt;blockquote&gt;
&lt;p&gt;&lt;strong&gt;Nothing about what I have done here is &amp;ldquo;optimal.&amp;rdquo;&lt;/strong&gt;&lt;/p&gt;
&lt;/blockquote&gt;
&lt;p&gt;I simply poked around the parameter space (departure time, swimmer speed, and swimmer heading) until I found trajectories that seemed &amp;ldquo;reasonable.&amp;rdquo;  Specifically, I looked for paths that were:&lt;/p&gt;
&lt;ul&gt;
&lt;li&gt;&lt;strong&gt;Safe.&lt;/strong&gt;  The route must avoid ferry traffic.&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Efficient.&lt;/strong&gt;  The swimmer should not fight the current (mathematically speaking, the dot product of swimmer velocity and current velocity should never be negative).&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Timely.&lt;/strong&gt;  The swim must be completed in less than 2.5 hours to avoid the strongest currents.&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Practical.&lt;/strong&gt;  The swimmer must arrive near the intended destination.&lt;/li&gt;
&lt;/ul&gt;
&lt;p&gt;This struck me as a justifiable approach, given that the parameter search space was actually quite small; indeed, Rick had already identified a relatively compact range of feasible departure dates and times.  Further, we already knew where we wanted to start (Nobska Beach), where we wanted to finish (Lake Tashmoo), and how fast we could swim.  The only remaining unknowns were the exact departure time, swim pace, and where we should point our bodies!&lt;/p&gt;
&lt;p&gt;Here is a quick summary of the simulations I ran:&lt;/p&gt;
&lt;ul&gt;
&lt;li&gt;Constant $145^{\circ}$ Heading&lt;/li&gt;
&lt;li&gt;Forced Transit Through Predefined Waypoints&lt;/li&gt;
&lt;li&gt;Hybrid Strategy: Transit 1st Waypoint + Constant $140^{\circ}$ Heading
&lt;ul&gt;
&lt;li&gt;Departure Times
&lt;ul&gt;
&lt;li&gt;6:00a&lt;/li&gt;
&lt;li&gt;6:18a&lt;/li&gt;
&lt;li&gt;6:30a&lt;/li&gt;
&lt;/ul&gt;
&lt;/li&gt;
&lt;li&gt;Swim Pace
&lt;ul&gt;
&lt;li&gt;1:40/100yd&lt;/li&gt;
&lt;li&gt;2:00/100yd&lt;/li&gt;
&lt;/ul&gt;
&lt;/li&gt;
&lt;/ul&gt;
&lt;/li&gt;
&lt;/ul&gt;
&lt;p&gt;Finally, here are a few notes on the animations that might be useful:&lt;/p&gt;
&lt;ul&gt;
&lt;li&gt;The &lt;strong&gt;colormap&lt;/strong&gt; corresponds to the magnitude of current velocity.&lt;/li&gt;
&lt;li&gt;The &lt;strong&gt;black arrows&lt;/strong&gt; indicate the direction of the current velocity.&lt;/li&gt;
&lt;li&gt;The &lt;strong&gt;black lines&lt;/strong&gt; are rhumb lines between predefined waypoints, represented by &lt;strong&gt;black circles&lt;/strong&gt;.&lt;/li&gt;
&lt;li&gt;The &lt;strong&gt;dashed magenta line&lt;/strong&gt; represents the swimmer trajectory at each time step.&lt;/li&gt;
&lt;li&gt;The &lt;strong&gt;magenta arrow&lt;/strong&gt; indicates the swimmer Course Over Ground (COG): this is the net direction of swimmer movement, including current action.&lt;/li&gt;
&lt;li&gt;The &lt;strong&gt;dark purple arrow&lt;/strong&gt; indicates swimmer heading: this is the direction in which the swimmer is producing thrust.  Note that when the current goes completely slack, the &lt;strong&gt;magenta arrow&lt;/strong&gt; and &lt;strong&gt;dark purple arrow&lt;/strong&gt; become perfectly aligned.&lt;/li&gt;
&lt;/ul&gt;
&lt;p&gt;Enjoy!&lt;/p&gt;
&lt;h4 id=&#34;constant-145circ-heading&#34;&gt;Constant $145^{\circ}$ Heading&lt;/h4&gt;
&lt;p&gt;


&lt;div style=&#34;padding:79.41% 0 0 0;position:relative;&#34;&gt;
  &lt;iframe src=&#34;https://player.vimeo.com/video/737636662?h=5d956f822a&amp;portrait=0#t=0m0s&#34; style=&#34;position:absolute;top:0;left:0;width:100%;height:100%;&#34; frameborder=&#34;0&#34; allow=&#34;autoplay; fullscreen; picture-in-picture&#34; allowfullscreen&gt;
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&lt;br&gt;
Initially, I was curious to see whether there existed a &lt;strong&gt;simple, constant-heading solution&lt;/strong&gt; that would deliver a swimmer moving at a constant pace to a desired destination (e.g. Lake Tashmoo).  So I ran a few simulations, and found that a swimmer departing Nobska Beach at 6:18a on the morning of July 30th, 2022, who was able to hold an average 2:00/100yd pace, and a constant heading of 145 degrees, would have quite a pleasant, efficient swim.  At nearly all points throughout the journey, the swimmer&amp;rsquo;s thrust vector is either perpendicular to, or somewhat aligned with, the tidal current vector (non-negative scalar product).  This is ideal, as it means the swimmer is not &amp;ldquo;fighting&amp;rdquo; the current.&lt;/p&gt;
&lt;p&gt;Unfortunately, this strategy places the swimmer directly in the Steamship Authority&amp;rsquo;s ferry lane for much of the journey across Vineyard Sound.  Efficiency is a delightful thing, but truly nothing compares to the joy and satisfaction one feels when not being run over by a 154 ft ferry capable of producing 9,400 HP thrust, moving at 35 knots.  We needed a different strategy.&lt;/p&gt;
&lt;h4 id=&#34;forced-transit-through-predefined-waypoints&#34;&gt;Forced Transit Through Predefined Waypoints&lt;/h4&gt;
&lt;p&gt;


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&lt;br&gt;
Realizing that we needed to first cut southwest across the ferry lane (again, to avoid being run over), before pivoting southeast toward Martha&amp;rsquo;s Vineyard, we selected a set of waypoints that seemed like they would deliver us to our destination.  Unfortunately, this resulted in a much longer swim, both in terms of distance and time.  Following this strategy would also force the swimmers to fight hard against the current over Middle Ground, toward the end of the swim.  Back to the drawing board!&lt;/p&gt;
&lt;h4 id=&#34;hybrid-strategy-transit-1st-waypoint--constant-140circ-heading&#34;&gt;Hybrid Strategy: Transit 1st Waypoint + Constant $140^{\circ}$ Heading&lt;/h4&gt;
&lt;p&gt;At this point, we had a fairly good idea of what we needed to do: we would transit across the ferry channel to the first waypoint as quickly as possible, and then pursue a constant-heading trajectory thereafter.  So, I ran a 2D matrix of experiments, varying both &lt;em&gt;departure time&lt;/em&gt; and &lt;em&gt;swim pace.&lt;/em&gt;&lt;/p&gt;
&lt;p&gt;I was quite concerned about the impact of fatigue.  If a swimmer started slowing down somewhere in Vineyard Haven, I wanted to know exactly what might happen to them.  This is why I ran both &amp;ldquo;fast pace&amp;rdquo; simulations (1:40/100yd) and &amp;ldquo;slow pace&amp;rdquo; (2:00/100yd) simulations.&lt;/p&gt;
&lt;p&gt;As for the departure times, well&amp;hellip; the Chappy Swimmers are a lovely group, but also a loquacious one!  We like to chat before swims.  Accordingly, I thought it might also be useful to assess the impact of punctuality (Incidentally, I think everyone was so nervous, we ended up departing Nobska Beach a minute or two early).&lt;/p&gt;
&lt;p&gt;Here are the resulting six simulations:&lt;/p&gt;
&lt;h5 id=&#34;600a-departure--140100yd-pace&#34;&gt;6:00a Departure | 1:40/100yd Pace&lt;/h5&gt;
&lt;p&gt;


&lt;div style=&#34;padding:79.41% 0 0 0;position:relative;&#34;&gt;
  &lt;iframe src=&#34;https://player.vimeo.com/video/737626385?h=5d956f822a&amp;portrait=0#t=0&#34; style=&#34;position:absolute;top:0;left:0;width:100%;height:100%;&#34; frameborder=&#34;0&#34; allow=&#34;autoplay; fullscreen; picture-in-picture&#34; allowfullscreen&gt;
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&lt;script src=&#34;https://player.vimeo.com/api/player.js&#34;&gt;&lt;/script&gt;







&lt;figure style=&#34;text-align: center; margin-top: 50px; margin-bottom: 50px;&#34;&gt;

  &lt;a data-fancybox=&#34;&#34; href=&#34;dpt_600_hyb_140_fast_speed.png&#34; style=&#34;display:inline-block; margin:0; padding:0; line-height:0;&#34; data-caption=&#34;Swimmer speed over ground (SOG) and speed-through-water.  Ideally, SOG should always be greater than or equal to speed-through-water, indicating that the swimmer is never fighting the current.  In math terms, this efficiency condition is achieved by ensuring that that the dot product of swimmer velocity and current velocity is non-negative at each timestep.  For this particular simulation, the efficiency condition is met at every time except $t=1.4~hr$.&#34;&gt;

&lt;img src=&#34;dpt_600_hyb_140_fast_speed.png&#34; style=&#34;display:block; margin:0; padding:0;&#34; alt=&#34;Swimmer speed over ground (SOG) and speed-through-water.  Ideally, SOG should always be greater than or equal to speed-through-water, indicating that the swimmer is never fighting the current.  In math terms, this *efficiency condition* is achieved by ensuring that that the dot product of swimmer velocity and current velocity is non-negative at each timestep.  For this particular simulation, the efficiency condition is met at every time except $t=1.4~hr$.&#34; &gt;
&lt;/a&gt;


&lt;figcaption style=&#34;text-align: left;&#34; data-pre=&#34;Figure &#34; data-post=&#34;:&#34; &gt;
  
  &lt;p&gt;
    Swimmer speed over ground (SOG) and speed-through-water.  Ideally, SOG should always be greater than or equal to speed-through-water, indicating that the swimmer is never fighting the current.  In math terms, this &lt;em&gt;efficiency condition&lt;/em&gt; is achieved by ensuring that that the dot product of swimmer velocity and current velocity is non-negative at each timestep.  For this particular simulation, the efficiency condition is met at every time except $t=1.4~hr$.
    
    
    
  &lt;/p&gt; 
&lt;/figcaption&gt;

&lt;/figure&gt;
&lt;/p&gt;
&lt;h5 id=&#34;600a-departure--200100yd-pace&#34;&gt;6:00a Departure | 2:00/100yd Pace&lt;/h5&gt;



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  &lt;iframe src=&#34;https://player.vimeo.com/video/737624155?h=5d956f822a&amp;portrait=0#t=0&#34; style=&#34;position:absolute;top:0;left:0;width:100%;height:100%;&#34; frameborder=&#34;0&#34; allow=&#34;autoplay; fullscreen; picture-in-picture&#34; allowfullscreen&gt;
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&lt;h5 id=&#34;618a-departure--140100yd-pace&#34;&gt;6:18a Departure | 1:40/100yd Pace&lt;/h5&gt;



&lt;div style=&#34;padding:79.41% 0 0 0;position:relative;&#34;&gt;
  &lt;iframe src=&#34;https://player.vimeo.com/video/737634275?h=5d956f822a&amp;portrait=0#t=0&#34; style=&#34;position:absolute;top:0;left:0;width:100%;height:100%;&#34; frameborder=&#34;0&#34; allow=&#34;autoplay; fullscreen; picture-in-picture&#34; allowfullscreen&gt;
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&lt;h5 id=&#34;618a-departure--200100yd-pace&#34;&gt;6:18a Departure | 2:00/100yd Pace&lt;/h5&gt;



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  &lt;iframe src=&#34;https://player.vimeo.com/video/737629259?h=5d956f822a&amp;portrait=0#t=0&#34; style=&#34;position:absolute;top:0;left:0;width:100%;height:100%;&#34; frameborder=&#34;0&#34; allow=&#34;autoplay; fullscreen; picture-in-picture&#34; allowfullscreen&gt;
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&lt;h5 id=&#34;630a-departure--140100yd-pace&#34;&gt;6:30a Departure | 1:40/100yd Pace&lt;/h5&gt;



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  &lt;iframe src=&#34;https://player.vimeo.com/video/737635428?h=5d956f822a&amp;portrait=0#t=0&#34; style=&#34;position:absolute;top:0;left:0;width:100%;height:100%;&#34; frameborder=&#34;0&#34; allow=&#34;autoplay; fullscreen; picture-in-picture&#34; allowfullscreen&gt;
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&lt;h5 id=&#34;630a-departure--200100yd-pace&#34;&gt;6:30a Departure | 2:00/100yd Pace&lt;/h5&gt;



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  &lt;iframe src=&#34;https://player.vimeo.com/video/737634948?h=5d956f822a&amp;portrait=0#t=0&#34; style=&#34;position:absolute;top:0;left:0;width:100%;height:100%;&#34; frameborder=&#34;0&#34; allow=&#34;autoplay; fullscreen; picture-in-picture&#34; allowfullscreen&gt;
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&lt;h3 id=&#34;key-findings&#34;&gt;Key Findings&lt;/h3&gt;
&lt;p&gt;I think more than anything, these simulations really underscore the extent to which &lt;strong&gt;timing is crucial&lt;/strong&gt; for this swim.&lt;/p&gt;
&lt;p&gt;While the tide is ebbing, a strong southwesterly current develops in Vineyard Sound, reaching maximum speeds of 3-5 knots, depending on the lunar cycle.  The current then goes slack for a mere twenty minutes, reverses, and begins ripping back toward the northeast.  In shallow regions of Vineyard Sound, like Middle Ground, the large volume of water being funneled between Woods Hole and Martha&amp;rsquo;s Vineyard can produce extremely fast local current velocities, and even hydraulic jumps.  Obviously, these are not ideal conditions for swimming!&lt;/p&gt;
&lt;p&gt;These simulations point to a useful conclusion: it is far safer and more conservative to err on the side of departing too early, as opposed to departing too late.  Departing at 6:00a, instead of 6:30a, may result in a slightly longer swim, but also gives swimmers more wiggle room to contend with exhaustion, cramps, heading misalignment, and various other unforeseen circumstances.&lt;/p&gt;
&lt;p&gt;I think simply visualizing how strong the currents can become (2-3 times our sustainable swim speed) gave the Chappy Swimmers a healthy dose of humility.  These simulations helped us understand that we might have be a bit flexible if things didn’t go exactly as planned.&lt;/p&gt;
&lt;h2 id=&#34;the-swim&#34;&gt;The Swim&lt;/h2&gt;
&lt;p&gt;The morning of the swim arrived, and the damp July air was buzzing with electricity.  Amplified laughter betrayed jittery nerves.  Beaming smiles were interspersed with heavy sighs and idle stretching.  The mood felt vaguely schizophrenic and unstable.  Each of the ten swimmers would occasionally gaze off into the distance, somehow both focused and vacant, contemplating the crucible that lay ahead.&lt;/p&gt;






&lt;figure style=&#34;text-align: center; margin-top: 50px; margin-bottom: 50px;&#34;&gt;

  &lt;a data-fancybox=&#34;&#34; href=&#34;blake-friends.jpg&#34; style=&#34;display:inline-block; margin:0; padding:0; line-height:0;&#34; data-caption=&#34;Chappy Swimmers Blake Cole, Kalina Grabb, Karen Mareb, and Dina Pandya; along with support boat crew Kelsey Chenoweth and Captain Eve Cinquino. [Photo: Wan Grabb]&#34;&gt;

&lt;img src=&#34;blake-friends.jpg&#34; style=&#34;display:block; margin:0; padding:0;&#34; alt=&#34;Chappy Swimmers Blake Cole, Kalina Grabb, Karen Mareb, and Dina Pandya; along with support boat crew Kelsey Chenoweth and Captain Eve Cinquino. [Photo: Wan Grabb]&#34; &gt;
&lt;/a&gt;


&lt;figcaption style=&#34;text-align: left;&#34; data-pre=&#34;Figure &#34; data-post=&#34;:&#34; &gt;
  
  &lt;p&gt;
    Chappy Swimmers Blake Cole, Kalina Grabb, Karen Mareb, and Dina Pandya; along with support boat crew Kelsey Chenoweth and Captain Eve Cinquino. [Photo: Wan Grabb]
    
    
    
  &lt;/p&gt; 
&lt;/figcaption&gt;

&lt;/figure&gt;

&lt;p&gt;Fortunately, the conditions looked ideal: there was very little wind, and no waves.  The sea surface was smooth and glassy.  The water visibility was at least 15-20 feet, and the water temperature was in the neighborhood of $70^{\circ}F$.  It seemed as though the sea gods had smiled upon our venture, and might see fit to deliver us safely across Vineyard Sound.  Only time would tell.&lt;/p&gt;
&lt;p&gt;At 5:58a, it was time to go: each swimmer took a final deep breath, their body weight still supported by the silty sands off Nobska Beach, and dove into the water.  They would touch neither solid ground nor boat for the following two hours.&lt;/p&gt;
&lt;p&gt;


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&lt;br&gt;
The swim itself was fortunately quite uneventful for seven out of the ten swimmers.  We settled into a rhythm, and enjoyed the strange solace of gliding through an unfamiliar aquamarine world.  Some of us tripped out a little.  John mentioned that he felt like he was flying through the Cosmos, and my inner dialogue was dominated by a scientifically apocryphal mantra about humans being the descendants of whales.&lt;/p&gt;






&lt;figure style=&#34;text-align: center; margin-top: 50px; margin-bottom: 50px;&#34;&gt;

  &lt;a data-fancybox=&#34;&#34; href=&#34;placental-mammal.png&#34; style=&#34;display:inline-block; margin:0; padding:0; line-height:0;&#34; data-caption=&#34;The ancestral placental mammal. [Credit: Carl Buell]&#34;&gt;

&lt;img src=&#34;placental-mammal.png&#34; style=&#34;display:block; margin:0; padding:0;&#34; alt=&#34;The ancestral placental mammal. [Credit: Carl Buell]&#34; &gt;
&lt;/a&gt;


&lt;figcaption style=&#34;text-align: left;&#34; data-pre=&#34;Figure &#34; data-post=&#34;:&#34; &gt;
  
  &lt;p&gt;
    The ancestral placental mammal. [Credit: Carl Buell]
    
    
    
  &lt;/p&gt; 
&lt;/figcaption&gt;

&lt;/figure&gt;

&lt;p&gt;[Incidentally, I looked this up afterwards, an apparently we &lt;a href=&#34;https://www.nationalgeographic.com/science/article/meet-the-ancestor-of-every-human-bat-cat-whale-and-mouse&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;share a common ancestor&lt;/a&gt;, but are in no way the descendants of whales.  Disappointing.]&lt;/p&gt;
&lt;p&gt;The conditions remained favorable throughout the swim, and just after 8:00a, John McCadam arrived on the shore of Martha&amp;rsquo;s Vineyard, just north of Lake Tashmoo.  Blake Cole, Meri Gilson, and Kalina Grabb made landfall a few minutes later, followed by Megan Titas, Dina Pandya, and Karen Mareb.  Karen had endured a painful leg cramp during the last half of the swim. Nevertheless, she managed to persevere through the discomfort, and finished the swim using only her arms.&lt;/p&gt;






&lt;figure style=&#34;text-align: center; margin-top: 50px; margin-bottom: 50px;&#34;&gt;

  &lt;a data-fancybox=&#34;&#34; href=&#34;garmin-path.jpg&#34; style=&#34;display:inline-block; margin:0; padding:0; line-height:0;&#34; data-caption=&#34;Swim path traversed by Blake Cole, recording using a Garmin Fenix 7S smartwatch, with multi-band GNSS mode enabled.&#34;&gt;

&lt;img src=&#34;garmin-path.jpg&#34; style=&#34;display:block; margin:0; padding:0;&#34; alt=&#34;Swim path traversed by Blake Cole, recording using a Garmin Fenix 7S smartwatch, with multi-band GNSS mode enabled.&#34; &gt;
&lt;/a&gt;


&lt;figcaption style=&#34;text-align: left;&#34; data-pre=&#34;Figure &#34; data-post=&#34;:&#34; &gt;
  
  &lt;p&gt;
    Swim path traversed by Blake Cole, recording using a Garmin Fenix 7S smartwatch, with multi-band GNSS mode enabled.
    
    
    
  &lt;/p&gt; 
&lt;/figcaption&gt;

&lt;/figure&gt;

&lt;p&gt;One of the most memorable aspects of the swim was the kindness bestowed upon us by the homeowners upon whose beach we landed.  Having various preconceptions about Martha&amp;rsquo;s Vineyard and its mostly-too-affluent inhabitants, I had been expecting a chilly reception, at best.  I couldn&amp;rsquo;t have been more wrong.  The husband and wife who greeted us were named Tom and Casey, and they were the sweetest people ever: not only did they congratulate us, but they offered us hot coffee, water, and snacks.  It was such a touching human moment, and meant so much to me!&lt;/p&gt;
&lt;p&gt;







  
  


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  &lt;img src=&#34;https://www.blakecole.com/post/1-adcirc-mv-swim/mv-arrival-photos/IMG_2866_hu3470360377756939317.jpg&#34; alt=&#34;&#34;&gt;
  &lt;/a&gt;
  

  
&lt;/div&gt;

&lt;br&gt;
But, as time drew on, the folks on the beach became increasingly concerned: three of the original ten swimmers remained unaccounted for, and could not be seen from the shore.  Gazing out toward Middle Ground, small ripples had begun forming on the sea surface, indicating that the dangerous northeasterly current we had hoped to avoid was now rapidly building in strength.&lt;/p&gt;
&lt;p&gt;At 8:30a, Rick Rheinhardt, Mary Kaminski, and Moira McCullough had still yet to arrive.  With each passing minute, we understood that the northeasterly current would continue accelerating, eventually reaching a top speed of over 5 knots.  Eventually, around 8:50a, reports came trickling in from support boats: the three swimmers had indeed become stuck in the current, and were still fighting hard to make it to shore.  Fortunately, over an hour after John McCadam&amp;rsquo;s toes touched sand on Martha&amp;rsquo;s Vineyard, we received word that the trio had made landfall near the tip of West Chop Point, after swimming over 5.3 miles.  This was an incredible feat of strength and determination &amp;ndash; kudos to Rick, Mary, and Moira!&lt;/p&gt;
&lt;h2 id=&#34;conclusion&#34;&gt;Conclusion&lt;/h2&gt;
&lt;p&gt;So, the question remains: &lt;strong&gt;how effectively did the ADCIRC model predict our swim trajectory?&lt;/strong&gt;&lt;/p&gt;
&lt;p&gt;Unfortunately, we will never know.&lt;/p&gt;
&lt;p&gt;This question cannot be answered, because there is no way to determine the &lt;em&gt;control effort&lt;/em&gt; exerted by each swimmer; in other words, while a GNSS track can tell us a swimmer&amp;rsquo;s speed over ground (SOG), course over ground (COG), and true position over time, these metrics do not distinguish the influence of the current from the effort exerted by the swimmer.  This disambiguation would require knowledge of the swimmer&amp;rsquo;s &lt;em&gt;true heading&lt;/em&gt; and &lt;em&gt;speed-through-water&lt;/em&gt; during the swim.&lt;/p&gt;
&lt;p&gt;Nevertheless, we can make a few observations, based on the data we were able to collect.&lt;/p&gt;






&lt;figure style=&#34;text-align: center; margin-top: 50px; margin-bottom: 50px;&#34;&gt;

  &lt;a data-fancybox=&#34;&#34; href=&#34;sim-vs-reality.png&#34; style=&#34;display:inline-block; margin:0; padding:0; line-height:0;&#34; data-caption=&#34;True swim path compared to simulations produced using the ADCRIC hydrodynamic model.&#34;&gt;

&lt;img src=&#34;sim-vs-reality.png&#34; style=&#34;display:block; margin:0; padding:0;&#34; alt=&#34;True swim path compared to simulations produced using the ADCRIC hydrodynamic model.&#34; &gt;
&lt;/a&gt;


&lt;figcaption style=&#34;text-align: left;&#34; data-pre=&#34;Figure &#34; data-post=&#34;:&#34; &gt;
  
  &lt;p&gt;
    True swim path compared to simulations produced using the ADCRIC hydrodynamic model.
    
    
    
  &lt;/p&gt; 
&lt;/figcaption&gt;

&lt;/figure&gt;

&lt;p&gt;Immediately, it is clear that the path predicted by the ADCIRC simulations is significantly different than the true path traversed by the first wave of swimmers on July 30th, 2022.  Why might this be?  What might have led to this inconsistency?  Allow me to propose a few hypotheses:&lt;/p&gt;
&lt;h4 id=&#34;hypothesis-1-insufficient-bathymetric-resolution&#34;&gt;Hypothesis #1: Insufficient Bathymetric Resolution&lt;/h4&gt;
&lt;p&gt;As mentioned earlier in the &lt;strong&gt;Assumptions &amp;amp; Caveats&lt;/strong&gt; section, it is possible that the coarse spatial resolution of the ADCIRC bathymetric data led to an underestimation of the northeasterly current that develops over Middle Ground during the initial stages of a flood tide.  If this were true, the swimmers would be swept substantially farther northeast than the model predicted.&lt;/p&gt;
&lt;p&gt;The data supports this hypothesis.  One need only to observe the radical easterly deviation in trajectory that occurs just as the swimmers are passing over Middle Ground.  Assuming the swimmers maintained a constant speed-through-water and heading, this deviation can almost certainly be attributed, at least in part, to stronger than expected tidal currents over Middle Ground.&lt;/p&gt;
&lt;h4 id=&#34;hypothesis-2-heading-error&#34;&gt;Hypothesis #2: Heading Error&lt;/h4&gt;
&lt;p&gt;It is possible that the swimmers were not, in fact, following a $140^{\circ}$ heading.  Lacking terrestrial landmarks, the &amp;ldquo;objective function&amp;rdquo; of each swimmer was simply to swim directly toward their guide vessel at all times.  Thus, in order for the swimmers to remain on a fixed $140^{\circ}$ heading, their respective guide boats needed to maintain a $320^{\circ}$ relative bearing to them (say, using a handheld compass).  This is a subtle, but crucial distinction: it is not the instantaneous heading of the guide vessel that matters, but rather, its &lt;strong&gt;relative bearing&lt;/strong&gt; to the swimmers, measured with respect to true north.  The figure below illustrates a situation in which a guide vessel is pursuing a $140^{\circ}$ heading, but is nevertheless inadvertently leading its swimmers in a more easterly direction.&lt;/p&gt;






&lt;figure style=&#34;text-align: center; margin-top: 50px; margin-bottom: 50px;&#34;&gt;

  &lt;a data-fancybox=&#34;&#34; href=&#34;relative-bearing-both.png&#34; style=&#34;display:inline-block; margin:0; padding:0; line-height:0;&#34; data-caption=&#34;Depiction of the importance of relative bearing between swimmer and support vessel.  In the left-hand figure, the support vessel is following a $140^{\circ}$ heading; however, the relative bearing from the swimmer to the support vessel is $120^{\circ}$ ($-20^{\circ}$ offset from the target heading of $140^{\circ}$).  In this scenario, the support vessel should observe that its relative bearing to the swimmer is $300^{\circ}$ instead of $320^{\circ}$, and begin clocking around to the right until the relative bearing reads $320^{\circ}$.  This corrected configuration is displayed in the right-hand figure.  Note that the instantaneous, absolute vessel heading is not relevant; only its bearing relative to the swimmers matters.&#34;&gt;

&lt;img src=&#34;relative-bearing-both.png&#34; style=&#34;display:block; margin:0; padding:0;&#34; alt=&#34;Depiction of the importance of **relative bearing** between swimmer and support vessel.  In the left-hand figure, the support vessel is following a $140^{\circ}$ heading; however, the relative bearing from the swimmer to the support vessel is $120^{\circ}$ ($-20^{\circ}$ offset from the target heading of $140^{\circ}$).  In this scenario, the support vessel should observe that its relative bearing to the swimmer is $300^{\circ}$ instead of $320^{\circ}$, and begin clocking around to the right until the relative bearing reads $320^{\circ}$.  This corrected configuration is displayed in the right-hand figure.  Note that the instantaneous, absolute vessel heading is not relevant; only its bearing relative to the swimmers matters.&#34; &gt;
&lt;/a&gt;


&lt;figcaption style=&#34;text-align: left;&#34; data-pre=&#34;Figure &#34; data-post=&#34;:&#34; &gt;
  
  &lt;p&gt;
    Depiction of the importance of &lt;strong&gt;relative bearing&lt;/strong&gt; between swimmer and support vessel.  In the left-hand figure, the support vessel is following a $140^{\circ}$ heading; however, the relative bearing from the swimmer to the support vessel is $120^{\circ}$ ($-20^{\circ}$ offset from the target heading of $140^{\circ}$).  In this scenario, the support vessel should observe that its relative bearing to the swimmer is $300^{\circ}$ instead of $320^{\circ}$, and begin clocking around to the right until the relative bearing reads $320^{\circ}$.  This corrected configuration is displayed in the right-hand figure.  Note that the instantaneous, absolute vessel heading is not relevant; only its bearing relative to the swimmers matters.
    
    
    
  &lt;/p&gt; 
&lt;/figcaption&gt;

&lt;/figure&gt;

&lt;p&gt;Rob Thieler suggested another potential source of heading error: magnetic declination.  Indeed, in the vicinity of Vineyard Sound, there is a $14^{\circ}~13&#39;~W$ separation between magnetic north (the direction a magnetized compass needle points) and true north (the direction along a meridian, towards the geographic North Pole).  Many marine GNSS systems account for magnetic declination by default, but some do not, and it isn&amp;rsquo;t always easy to determine the active compass mode.  Following a 140 degree bearing relative to magnetic north would place us on a 126 degree bearing relative to true north.&lt;/p&gt;
&lt;p&gt;It is also possible that heading errors were simply inevitable.  A propeller-driven ocean vessel moving at slow speeds (1-2 knots) has virtually no heading control authority.  Holding a fixed bearing, relative to a moving school of swimmers, while contending with currents, wind, and waves is no easy feat.  We were incredibly fortunate to have such experienced support vessel captains, and literally could not have completed this swim without them.  A massive thank you to our captains: Chuck Redington, Robert Catilano, Eve Cinquino, Christie Hegermiller, and Kevin Manganini.&lt;/p&gt;
&lt;p&gt;Regardless of the source of the heading error, the data seems to suggest that there was &lt;em&gt;some&lt;/em&gt; heading error, for some reason.  Roughly halfway through the swim, we took a brief break to hydrate.  This break can be seen in the data, when the black line jogs slightly southwest.  This small hop seems to suggest that the dominant tidal current in Vineyard Sound was still driving us southwest at this time, and had not yet gone slack.  Yet, our course over ground appears to be approximately 140 degrees; this would imply that our instantaneous heading must have been something less than 140 degrees (more easterly) at this time.&lt;/p&gt;
&lt;p&gt;In some ways, this is the most obvious explanation: pursuing a more easterly course will result in a path that is shifted in an easterly direction.&lt;/p&gt;
&lt;h4 id=&#34;hypothesis-3-systemic-model-error&#34;&gt;Hypothesis #3: Systemic Model Error&lt;/h4&gt;
&lt;p&gt;Finally, it is possible that the ADCIRC model is simply inaccurate, or lacks sufficient spatial or temporal resolution.  However, given the strong consistency of its predictions with the Eldridge Tide and Pilot Book, I think this is unlikely.&lt;/p&gt;
&lt;h2 id=&#34;conclusion-ii&#34;&gt;Conclusion II&lt;/h2&gt;
&lt;p&gt;Well, that just about wraps up this longer-than-expected blog post.  If you have any questions, concerns, or alternative hypotheses regarding the discrepancy between the true swimmer trajectory and the path predicted by ADCIRC, please reach out.  Thanks for reading.&lt;/p&gt;
</description>
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    <item>
      <title>Hello World &#43; Bio</title>
      <link>https://www.blakecole.com/post/0-hello-world/</link>
      <pubDate>Sun, 07 Aug 2022 08:18:13 -0400</pubDate>
      <guid>https://www.blakecole.com/post/0-hello-world/</guid>
      <description>


&lt;details class=&#34;print:hidden xl:hidden&#34; open&gt;
  &lt;summary&gt;Table of Contents&lt;/summary&gt;
  &lt;div class=&#34;text-sm&#34;&gt;
  &lt;nav id=&#34;TableOfContents&#34;&gt;
  &lt;ul&gt;
    &lt;li&gt;&lt;a href=&#34;#hi-everyone&#34;&gt;Hi everyone!&lt;/a&gt;&lt;/li&gt;
    &lt;li&gt;&lt;a href=&#34;#autobiographical-miscellany&#34;&gt;Autobiographical Miscellany&lt;/a&gt;&lt;/li&gt;
  &lt;/ul&gt;
&lt;/nav&gt;
  &lt;/div&gt;
&lt;/details&gt;

&lt;h2 id=&#34;hi-everyone&#34;&gt;Hi everyone!&lt;/h2&gt;
&lt;p&gt;Welcome to my blog.  In case you skipped over the &lt;a href=&#34;https://www.blakecole.com/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;About&lt;/a&gt; section, my name is Blake, and I am currently a doctoral candidate in the &lt;a href=&#34;https://mit.whoi.edu/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;MIT-WHOI Joint Program&lt;/a&gt;.  My high-level research objective is to &lt;strong&gt;&lt;em&gt;improve the performance, reliability, and useability of autonomous sailing vessels&lt;/em&gt;&lt;/strong&gt;.  I am incredibly excited about this project, and will try to post incremental updates over the course of the next year.&lt;/p&gt;
&lt;p&gt;However, the primary purpose of this blog is not to disseminate my thesis research; rather, I hope it will serve as a repository for discussions about my non-academic interests and side-projects.  I will focus mostly on the projects and experiences I think other people might find useful or illuminating.  Though I will do my best to make sure the information contained herein is as accurate as possible, I might make mistakes, and am counting on you (yes, &lt;em&gt;you!&lt;/em&gt;) to bring them to my attention.  I will also try to keep these posts consise, but basically anyone who has ever emailed or texted me knows that brevity is not exactly my strong suit.  There&amp;rsquo;s a lot of words out there, and I intend to use them all.&lt;/p&gt;
&lt;h2 id=&#34;autobiographical-miscellany&#34;&gt;Autobiographical Miscellany&lt;/h2&gt;
&lt;p&gt;In 2013, long before I began learning how to build ocean robots, I wanted to learn how to build ocean &lt;em&gt;models&lt;/em&gt;.  As a lifelong surfer, I had relied extensively upon wave forecasts to determine where and when I should try to hop in the ocean, but I had no clue how these forecasts worked.  I found the prospect of using math to predict the movement of seemingly chaotic ocean waves and currents completely fascinating.  At some point, I watched the video &lt;a href=&#34;https://www.youtube.com/watch?v=MX5cKoOm6Pk&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Waves Across The Pacific&lt;/a&gt;, a short documentary describing the pioneering ocean wave research conducted by Walter Munk in the 1950s, and I was hooked!&lt;/p&gt;

    &lt;div style=&#34;position: relative; padding-bottom: 56.25%; height: 0; overflow: hidden;&#34;&gt;
      &lt;iframe allow=&#34;accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share&#34; allowfullscreen=&#34;allowfullscreen&#34; loading=&#34;eager&#34; referrerpolicy=&#34;strict-origin-when-cross-origin&#34; src=&#34;https://www.youtube.com/embed/MX5cKoOm6Pk?autoplay=0&amp;amp;controls=1&amp;amp;end=0&amp;amp;loop=0&amp;amp;mute=0&amp;amp;start=0&#34; style=&#34;position: absolute; top: 0; left: 0; width: 100%; height: 100%; border:0;&#34; title=&#34;YouTube video&#34;&gt;&lt;/iframe&gt;
    &lt;/div&gt;

&lt;p&gt;&lt;br&gt;
Soon thereafter, I began working toward a masters degree at the &lt;a href=&#34;https://cee.stanford.edu/research-impact/labs-centers/bob-and-norma-street-environmental-fluid-mechanics-laboratory-efml&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Stanford Environental Fluid Mechanics Laboratory&lt;/a&gt;, under the tutelage of professors &lt;a href=&#34;https://cee.stanford.edu/people/stephen-monismith&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Stephen Monismith&lt;/a&gt; and &lt;a href=&#34;https://cee.stanford.edu/people/oliver-fringer&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Oliver Fringer&lt;/a&gt;.  These two teachers really changed my life, and taught me a useful framework for approaching numerical integration problems, and &amp;ldquo;life problems&amp;rdquo; more broadly.  I owe them both a debt of gratitude.&lt;/p&gt;
&lt;p&gt;Reaching back even further into the distant past, long before I learned how to build ocean models, I was basically just a kid who loved the ocean.  I grew up in Redondo Beach, CA, and spent most of my teenage years at the beach surfing, swimming, and playing volleyball.  It wasn&amp;rsquo;t until my 3rd year at UCSD that I began seriously considering the possibility of pursuing ocean science as a carreer.&lt;/p&gt;
&lt;p&gt;Fortunately, the good folks at the Scripps Institution of Oceanography&amp;rsquo;s (SIO) &lt;a href=&#34;https://coralreefecology.ucsd.edu/about/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Coral Reef Ecology Lab&lt;/a&gt; gave me a shot, and let me volunteer a few hours per week in the lab.  My PI, &lt;a href=&#34;https://coralreefecology.ucsd.edu/people/#Smith&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Jen Smith&lt;/a&gt;, sponsored my AAUS Scientific Diving training, which was one of the most fun, memorable experiences of my entire life.  I fell in love with the culture, people, and mission at SIO, and my experience there set me off on an ocean science career trajectory upon which I remain comfortably situated today!&lt;/p&gt;
&lt;p&gt;Anyway, I hope you enjoy the blog.  Please feel free to contact me with any feedback or questions.&lt;/p&gt;






&lt;figure style=&#34;text-align: center; margin-top: 50px; margin-bottom: 50px;&#34;&gt;

  &lt;a data-fancybox=&#34;&#34; href=&#34;jump.jpg&#34; style=&#34;display:inline-block; margin:0; padding:0; line-height:0;&#34; data-caption=&#34;Longtime friends Mark H. and Blake C. jumping off rocks, somewhere between San Francisco, CA and Portland, OR.&#34;&gt;

&lt;img src=&#34;jump.jpg&#34; style=&#34;display:block; margin:0; padding:0;&#34; alt=&#34;Longtime friends Mark H. and Blake C. jumping off rocks, somewhere between San Francisco, CA and Portland, OR.&#34; &gt;
&lt;/a&gt;


&lt;figcaption style=&#34;text-align: left;&#34; data-pre=&#34;Figure &#34; data-post=&#34;:&#34; &gt;
  
  &lt;p&gt;
    Longtime friends Mark H. and Blake C. jumping off rocks, somewhere between San Francisco, CA and Portland, OR.
    
    
    
  &lt;/p&gt; 
&lt;/figcaption&gt;

&lt;/figure&gt;

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      <title>Unscented Kalman filter for long-distance vessel tracking in geodetic coordinates</title>
      <link>https://www.blakecole.com/publication/jrnl-aor-2022/</link>
      <pubDate>Fri, 01 Jul 2022 00:00:00 +0000</pubDate>
      <guid>https://www.blakecole.com/publication/jrnl-aor-2022/</guid>
      <description></description>
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      <title>Surface vessel collision avoidance in MOOS-IvP using a geodetic unscented Kalman filter</title>
      <link>https://www.blakecole.com/publication/conf-ieee-oceans-2021/</link>
      <pubDate>Mon, 20 Sep 2021 00:00:00 +0000</pubDate>
      <guid>https://www.blakecole.com/publication/conf-ieee-oceans-2021/</guid>
      <description></description>
    </item>
    
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      <title>A model integrating longshore and cross-shore processes for predicting long-term shoreline response to climate change</title>
      <link>https://www.blakecole.com/publication/jrnl-jgr-earth-surface-2017/</link>
      <pubDate>Wed, 15 Mar 2017 00:00:00 +0000</pubDate>
      <guid>https://www.blakecole.com/publication/jrnl-jgr-earth-surface-2017/</guid>
      <description></description>
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