Variance Reduction Techniques in Monte Carlo: Sampling Pi via Circle-in-a-Square

Variance Reduction Techniques (Engines):

• Random
  Plain IID draws $(x,y) \sim U([0,1]^2)$, $f = 4 \cdot I[x^2+y^2≤1]$.

• Stratified
  Partition $[0,1]^2$ into an $(m \times m)$ grid (where $m^2 \approx N$), draw one random point per cell, and average.

• Exponential
  Importance sampling via an exponential function: $p(x, y) = λ^2 e^{-\lambda(x+y)} / (1 - e^{-\lambda})^2$. $\lambda = 0.6$ is hard-coded.

• ControlVariate
  Uniform draws + control variate $g=x^2+y^2$. Adjusts each $f_i$ by $\beta (2/3 − g_i)$ to reduce variance.

• Antithetic
  Forms antithetic pairs $[(x,y),(1−x,1−y)]$. Returns floor($N/2$) samples, each sample’s value = $(f_1+f_2) / 2$.

• ControlAntithetic
  Antithetic pairs + control variate. For each pair, calculate $f_{avg} = (f_1 + f_2) / 2$ and $g_{avg} = (g_1 + g_2) / 2$, compute $\beta$, and adjust $f_{avg}$ by $\beta (2/3 − g_{avg})$.

Variance Comparisons:

Using $N = 25 \times 10^6$ samples:

Method	Variance
Random	2.6966
Stratified	2.6968
Importance (Exponential)	2.1597
Control Variate	1.1548
Antithetic Variate	0.9799
Control + Antithetic	0.7665

For this specific problem, the Control Variate technique, combined with Antithetic pair samples, appears to yield the best variance. The Stratified method did not result in any resolvable improvement in variance.

To Build (C++14, CMake 3.10 required):

mkdir build
cd build
cmake ..
make

To Run:

cd build
cp ../examples/input.in . # Adjust the input as needed
./monte_carlo_pi

Output

results.log will include running statistics for the number of samples, $(x, y)$ sample coordinates, sample value, and running values for mean, variance, and standard error.