From b11ddf5f9517f14fdcd0121f20dbc5aa5883134d Mon Sep 17 00:00:00 2001 From: lmoresi Date: Fri, 14 Aug 2026 16:26:36 +1000 Subject: [PATCH 01/15] Draft UWTN 2026-014: Setting Up Full Multigrid The G1 note from the writing plan, and the pair to UWTN 2026-011. Four things that decide whether multigrid helps on the meshes we actually build, none of which are solver options: - turning it on, and what "auto" will and will not override; - a level is a coarsening ratio, not an engine pass -- with the box-fault timings, 93.7s at nine levels against 23.3s at five, same mesh and same answer; - how transfers are built when the levels were not made from each other, why the geometric fallback works despite agreeing with the real cells only 17% of the time in 3D, and the pullback that keeps the nested construction conditioned; - the outer Krylov has to be flexible, with the notch table: gmres diverging at 983 velocity iterations a step against fgmres at 58, and the iteration cap inert in every combination. Closes by saying what the hierarchy is worth, which is the claim UWTN 2026-011 defers to this note for. ID 2026-014: 2026-012 is gthyagi's note on PR #11 and 2026-013 the news post, neither of which the allocator can see from this branch. Underworld development team with AI support from Claude Code --- .../setting-up-full-multigrid/metadata.yml | 34 +++ .../setting-up-full-multigrid.md | 241 ++++++++++++++++++ classification.yml | 5 + 3 files changed, 280 insertions(+) create mode 100644 articles/setting-up-full-multigrid/metadata.yml create mode 100644 articles/setting-up-full-multigrid/setting-up-full-multigrid.md diff --git a/articles/setting-up-full-multigrid/metadata.yml b/articles/setting-up-full-multigrid/metadata.yml new file mode 100644 index 0000000..dd904f9 --- /dev/null +++ b/articles/setting-up-full-multigrid/metadata.yml @@ -0,0 +1,34 @@ +# Validated in CI against schemas/article-metadata.schema.json. +# `pixi run validate` checks this and the cross-file invariants a schema cannot +# express -- that the article file is named .md, that canonical_path +# matches the slug, and that no legacy DOI is ever paired with a new registrant. +id: UWTN 2026-014 +slug: setting-up-full-multigrid +title: Setting Up Full Multigrid +article_type: technical-note +status: review +authors: + - name: Louis Moresi + orcid: 0000-0003-3685-174X + affiliation: Australian National University +publication_date: null +version: 1.0.0 +# The deposit writes archive_doi and repository_record_id when the note is +# published; leave them out until then. `doi` and `doi_registrant` were here +# once and are not fields the schema knows -- every note made from this +# template failed `pixi run validate` on all three of them. +license: CC-BY-4.0 +canonical_path: /setting-up-full-multigrid/ +legacy_paths: [] +# Facets, from vocabulary.yml. Both keys must be present even when empty: a +# note with no subject is normal -- many are purely about method. +subjects: + - mantle-convection +methods: + - solvers + - meshing + - parallel-hpc +ghost_tags: + - Underworld Code +figures: 0 +source: native diff --git a/articles/setting-up-full-multigrid/setting-up-full-multigrid.md b/articles/setting-up-full-multigrid/setting-up-full-multigrid.md new file mode 100644 index 0000000..82bb77c --- /dev/null +++ b/articles/setting-up-full-multigrid/setting-up-full-multigrid.md @@ -0,0 +1,241 @@ +--- +title: Setting Up Full Multigrid +description: >- + Geometric multigrid is the fastest preconditioner we have for Stokes on a + refined mesh, and most of what decides whether it works is not in the solver + options. What counts as a level, how the transfers are built when the levels + were not made from each other, and why the outer Krylov has to be flexible. +date: 2026-08-14 +authors: + - name: Louis Moresi + orcid: 0000-0003-3685-174X + affiliations: + - Australian National University +license: CC-BY-4.0 +keywords: + - Underworld Code + - Tricks of the Trade + - development +exports: + - format: typst + logo: ../../static/uwtn-logo.png + series: "Underworld Technical Notes" + origin_url: https://www.underworldcode.org/setting-up-full-multigrid/ + template: ../../templates/pdf + output: setting-up-full-multigrid.pdf + article_id: UWTN 2026-014 + article_version: 1.0.0 + software_version: underworld3 0.0.0 +--- +A Stokes solve is where the time goes, and multigrid is how we stop it going +there. The idea is old and well described: solve the problem on a sequence of +grids, let each grid remove the part of the error it can see cheaply, and the +work per unknown stops growing with the size of the problem. + +What is much less well described is what to do when the hierarchy you have is +not the tidy one in the textbook. Ours rarely is. A mesh gets refined where a +fault or a boundary layer needs it, so the levels differ by a factor that is +not two. A mesh gets cut, or moved, or adapted between timesteps, so the levels +were not all built from one another. This note is about the parts that decide +whether multigrid actually helps on a mesh like that, and none of them are the +options you type into PETSc. + +## Turning it on + +Build the mesh with a refinement hierarchy and it is already on: + +```python +mesh = uw.meshing.UnstructuredSimplexBox(cellSize=0.05, refinement=2) +stokes = uw.systems.Stokes(mesh) +stokes.solve() +``` + +`refinement=2` builds the mesh twice-refined from a coarse base and keeps the +coarse levels. The solver's `preconditioner` property defaults to `"auto"`, +which uses geometric full multigrid when the mesh carries a genuine hierarchy — +`len(mesh.dm_hierarchy) > 1` — and algebraic multigrid (GAMG) when it does not. +You can ask for either explicitly: + +```python +stokes.preconditioner = "fmg" # geometric; warns and falls back if no hierarchy +stokes.preconditioner = "gamg" # algebraic +stokes.preconditioner_settings # what actually got applied +``` + +`preconditioner_settings` is worth knowing about. It reports the option keys UW3 +has put in the database for this solver, so what was applied can be asserted on +rather than inferred from how long the solve took. + +Two things about `"auto"` that are deliberate. It only ever *adds* geometric +multigrid to an untouched default: if you have configured the preconditioner +yourself, it leaves your configuration alone. And on a single-field scalar or +vector solver it stays with GAMG even when a hierarchy exists, because the +native geometric path there is unreliable; asking for `"fmg"` explicitly routes +it through custom transfers instead. Where a request is declined, the reason is +recorded in `pc_fallbacks` rather than being silent. + +## A level is a coarsening ratio + +This is the one that costs the most time when it is wrong. + +A multigrid level earns its place by seeing error that no other level can see +cheaply. That is a statement about *resolution*: each level should differ from +its neighbour by something near a factor of two in cell size. It is not a +statement about how the mesh was made. + +Refinement engines do not work that way. A pass is how an engine *reaches* a +target size — independence conditions cap how many edges may be split at once, +and a conforming closure cascades — so the number of passes reflects the +engine's constraints, not the geometry. Recording one level per pass produces +levels that coarsen almost nothing. We measured ratios of 1.06, 1.02 and 1.007 +sitting in a hierarchy, each costing a full Galerkin triple product and a +smoother sweep on every cycle to remove error that its neighbour had already +removed. + +Cutting a mesh has the same problem in a purer form. A cut re-represents the +same grid with a surface conformed to it; it adds no resolution at all. Two +cuts once produced two levels coarsening by 1.11 and 1.17 against a threshold +of 1.8. + +The cost is not subtle. On a box-fault shear solve, dropping the levels that +were not genuine coarsenings: + +| levels | how | wall clock | +|---|---|---| +| 9 | one per pass | 93.7 s | +| 7 | `mg_coarsening_ratio=2.0` | 47.5 s | +| 5 | `mg_coarsening_ratio=3.0` | **23.3 s** | + +Same mesh, same answer, four times the speed. + +:::{warning} Measure resolution, not cell count +Under adapt-on-top the mesh only grows where the feature is, so a genuine +halving of $h$ can show as a global cell-count ratio near 1. Counting cells to +decide whether a level is real will tell you that every level is redundant on +exactly the meshes where the levels matter most. +::: + +## Levels that were not built from each other + +The textbook hierarchy is nested: every coarse node is a fine node, every +coarse cell is a union of fine cells, and the transfer between them follows +from the refinement relation. When that holds you can write the prolongation +down exactly, for any element degree, from the parent-cell map alone. With the +Lagrange basis dual to its nodal points, + +$$ +W = B\,M^{-1}, +\qquad +M_{im} = \mu_m(x_i), +\qquad +B_{tm} = \mu_m(x_t), +$$ + +where $\mu_m$ are the parent cell's basis functions, $x_i$ its own nodes and +$x_t$ the fine degrees of freedom inside it. No reference element, no +per-degree formulae, no point location. One detail is not optional: pull the +coordinates back through the parent's own vertices before evaluating. Raw +monomials give a conditioning that grows like $h^{-k}$ — around $10^{6}$ at +$P_3$ — while under the pullback the conditioning depends only on the degree +and the dimension. + +Often, though, the levels were *not* built from each other. A mesh that has +been cut, moved or adapted has coarse levels that are related to the fine one +geometrically but not combinatorially. Then the transfer has to be constructed +rather than derived, and UW3 falls back to locating fine degrees of freedom +inside a triangulation of the coarse degree-of-freedom cloud. + +That fallback works, and it is worth being clear about *why* it works, because +it is not the reason one might assume. The triangulation is of the coarse +**degree-of-freedom points**, and it never consults the mesh cells. How often +its simplices agree with the actual cells: + +| mesh | agreement with real cells | +|---|---| +| 2D uniform, $P_1$ | 100% | +| 2D adapt child | 90.8% | +| 3D uniform, $P_1$ | 58.8% | +| 3D adapt child | **17.1%** | + +At 17% agreement the construction is not reproducing the mesh in any meaningful +sense. It works because it reproduces *linear functions* exactly, and that is +the property multigrid transfers actually need. Knowing this changes what you +check when a hierarchy underperforms in 3D: the question is not whether the +transfer found the right cells, because mostly it did not. + +## The outer solver has to be flexible + +This one is not about the hierarchy at all, and it is the single largest effect +we have measured on a hard nonlinear problem. + +In the Schur factorisation used for Stokes, the velocity block is solved with +its own Krylov method. That solve is not a passive preconditioner: it computes +the search direction. Being Krylov, it is inexact, and being inexact, the +operator effectively applied *differs between outer iterations*. A non-flexible +GMRES assumes a fixed preconditioner and its residual recurrence has no +guarantee against an operator that drifts. + +On a notch problem at refinement 3, varying only this: + +| outer | inner rtol | velocity its/step | result | +|---|---|---|---| +| `gmres` | 3.3e-08 | 983 | DIVERGED_LINEAR_SOLVE | +| `gmres` | 1e-03 | 915 | DIVERGED_LINEAR_SOLVE | +| **`fgmres`** | 3.3e-08 | **58** | no linear failure | +| **`fgmres`** | 1e-03 | **23** | no linear failure | + +Raising the velocity iteration cap from 200 to 2000 changed nothing at all — +the rows were byte-identical. Loosening the inner tolerance by five orders of +magnitude while the outer stayed non-flexible was worth 7%. The flexible outer +Krylov is the entire effect, and once it is in place the same loosening is +worth a further factor of 2.6. + +The inner solves being deliberately inexact is an old design, inherited from +Citcom. Two halves of one decision: inexact inner solves require a flexible +outer method, and *how* inexact is bounded by requiring the inner solves to +converge well below the tolerance demanded of the outer one. The factors UW3 +uses — 0.033 for velocity, 0.1 for pressure — are that margin. Their existence +is principled; their particular sizes are inherited convention. + +## What it is worth + +The comparison that matters is against algebraic multigrid, which is what you +get without a hierarchy and which needs nothing from the mesh. + +On a refined mesh with a genuine hierarchy, geometric multigrid is +substantially better conditioned, because it is built from the refinement +relation rather than inferred from the operator's connectivity — which is +exactly what mesh anisotropy makes misleading. In the runs behind the +[mesh-mover note](/moving-the-mesh-without-remaking-it/), a four-level +`pc=mg` converged the velocity block in three to four iterations; where the +hierarchy was lost and the solve fell back to algebraic multigrid, the same +block took 23. + +The corollary is the reason those two notes are a pair. A hierarchy is a +capital asset: it is built once and it is what makes the solve cheap. Anything +that destroys it — remeshing in particular — is not only paying for the +remesh, it is paying for every solve afterwards until a new hierarchy is built. +That is what moving the mesh instead of remaking it is protecting. + +## Using it + +```python +# A hierarchy: two refinements from a coarse base, coarse levels kept. +mesh = uw.meshing.UnstructuredSimplexBox(cellSize=0.05, refinement=2) + +stokes = uw.systems.Stokes(mesh) +stokes.preconditioner = "auto" # the default: FMG if there is a hierarchy +stokes.solve() + +print(stokes.preconditioner_settings) # what was applied +print(stokes.pc_fallbacks) # anything declined, and why +``` + +When adapting, ask for levels by coarsening ratio rather than taking one per +engine pass: + +```python +child = mesh.adapt(metric, max_levels=3, mg_coarsening_ratio=2.0) +``` + +
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diff --git a/classification.yml b/classification.yml index 68d9c8e..65201be 100644 --- a/classification.yml +++ b/classification.yml @@ -317,3 +317,8 @@ introducing-the-technical-notes: article_type: news subjects: [] methods: [] + +setting-up-full-multigrid: + article_type: technical-note + subjects: [mantle-convection] + methods: [solvers, meshing, parallel-hpc] From bd3ed92292570a5e5bbcddc80182712edd6d53bf Mon Sep 17 00:00:00 2001 From: lmoresi Date: Fri, 14 Aug 2026 16:47:05 +1000 Subject: [PATCH 02/15] Rewrite as an introduction to geometric multigrid in Underworld The first draft opened on non-nested hierarchies and transfer construction, which assumes a reader who already knows what FMG is in this context. Nothing has introduced it. It now runs: what multigrid does, the two ways of building coarse levels and what each needs, how to build a mesh that carries a hierarchy, why curved boundaries need the snap callback the annulus already has, what the coarse mesh's triangulation leaves in the fine one and how relax() loosens it, and finally when the choice between GAMG and FMG is worth making. The comparison is measured rather than asserted: same box, same discretisation, twice refined. At constant viscosity both converge and the difference is seconds. Across a 1e4 viscous layer GAMG does not converge the velocity block at all -- capped at 200 and again at 2000, DIVERGED_ITS, 78 seconds to fail -- against three iterations and three seconds for FMG. The note says the GAMG configuration is the untuned default and that supplying the near-null-space would improve it. The material the first draft led with -- non-nested transfers, the Delaunay fallback, the flexible outer Krylov -- is not here. It belongs after this. Example script shipped with the note. Underworld development team with AI support from Claude Code --- .../examples/fmg-vs-gamg.py | 40 +++ .../setting-up-full-multigrid.md | 330 ++++++++---------- 2 files changed, 183 insertions(+), 187 deletions(-) create mode 100644 articles/setting-up-full-multigrid/examples/fmg-vs-gamg.py diff --git a/articles/setting-up-full-multigrid/examples/fmg-vs-gamg.py b/articles/setting-up-full-multigrid/examples/fmg-vs-gamg.py new file mode 100644 index 0000000..e52b779 --- /dev/null +++ b/articles/setting-up-full-multigrid/examples/fmg-vs-gamg.py @@ -0,0 +1,40 @@ +"""FMG against GAMG: when the choice does not matter, and when it does. + +Same mesh and same discretisation each time; the only differences are the +viscosity structure and which preconditioner the velocity block uses. +""" +import time, numpy as np, sympy, underworld3 as uw + +def run(pc, contrast, refinement=2, cell=1/8): + mesh = uw.meshing.UnstructuredSimplexBox( + minCoords=(0.,0.), maxCoords=(1.,1.), cellSize=cell, + qdegree=3, refinement=refinement) + x, y = mesh.X.coords_sym if hasattr(mesh.X,"coords_sym") else mesh.X + v = uw.discretisation.MeshVariable("U", mesh, mesh.dim, degree=2) + p = uw.discretisation.MeshVariable("P", mesh, 1, degree=1) + stokes = uw.systems.Stokes(mesh, velocityField=v, pressureField=p) + stokes.constitutive_model = uw.constitutive_models.ViscousFlowModel + # A viscous layer across the middle: contrast=1 is the easy case. + eta = 1.0 + (contrast - 1.0) * sympy.exp(-(((y - 0.5) / 0.08) ** 2)) + stokes.constitutive_model.Parameters.shear_viscosity_0 = eta + stokes.bodyforce = sympy.Matrix([0, -sympy.sin(sympy.pi * x) * sympy.cos(sympy.pi * y)]) + for wall, comps in (("Left",(0,)),("Right",(0,)),("Bottom",(1,)),("Top",(1,))): + stokes.add_dirichlet_bc((0.,)*mesh.dim, wall, comps) + stokes.preconditioner = pc + stokes.tolerance = 1e-6 + t0 = time.time() + stokes.solve() + dt = time.time() - t0 + ksp = stokes.snes.getKSP() + try: + its = ksp.getPC().getFieldSplitSubKSP()[0].getIterationNumber() + except Exception: + its = -1 + print(" %-5s contrast %-7g %6.2f s velocity its %s levels %d" + % (pc, contrast, dt, its, len(mesh.dm_hierarchy)), flush=True) + return dt + +for contrast in (1.0, 1.0e4): + print("viscosity contrast %g" % contrast, flush=True) + for pc in ("gamg", "fmg"): + run(pc, contrast) diff --git a/articles/setting-up-full-multigrid/setting-up-full-multigrid.md b/articles/setting-up-full-multigrid/setting-up-full-multigrid.md index 82bb77c..ba8f8e9 100644 --- a/articles/setting-up-full-multigrid/setting-up-full-multigrid.md +++ b/articles/setting-up-full-multigrid/setting-up-full-multigrid.md @@ -1,10 +1,10 @@ --- title: Setting Up Full Multigrid description: >- - Geometric multigrid is the fastest preconditioner we have for Stokes on a - refined mesh, and most of what decides whether it works is not in the solver - options. What counts as a level, how the transfers are built when the levels - were not made from each other, and why the outer Krylov has to be flexible. + Underworld can precondition a Stokes solve with geometric multigrid built + from a real hierarchy of meshes, rather than with the algebraic multigrid it + falls back to. What the difference is, how to build a mesh that has a + hierarchy, and when the choice is worth making. date: 2026-08-14 authors: - name: Louis Moresi @@ -27,215 +27,171 @@ exports: article_version: 1.0.0 software_version: underworld3 0.0.0 --- -A Stokes solve is where the time goes, and multigrid is how we stop it going -there. The idea is old and well described: solve the problem on a sequence of -grids, let each grid remove the part of the error it can see cheaply, and the -work per unknown stops growing with the size of the problem. +Most of the time in a geodynamics model goes into solving the Stokes equations, +and most of that goes into the velocity block. How that block is preconditioned +decides how long a model takes to run, and it is one of the few settings where +the right choice is worth an order of magnitude. + +## What multigrid does + +An iterative solver reduces error unevenly. Simple relaxation methods — the +smoothers — are good at removing error that varies rapidly from cell to cell, +and poor at removing error that varies smoothly across the whole domain. The +smooth part is the expensive part: on a fine mesh it takes many sweeps to move +information from one side of the model to the other. + +Multigrid removes that part somewhere else. Error that is smooth on a fine mesh +is *not* smooth on a mesh twice as coarse, where it spans half as many cells, so +a few sweeps on the coarse mesh remove what the fine mesh could not. The method +is a recursion: smooth on the fine mesh, transfer what is left to a coarser one, +solve there (by recursing again), transfer the correction back, smooth again. +Each level handles the band of error it can see cheaply, and the work per +unknown stops growing as the model gets bigger. + +Two things have to be supplied: the coarse levels, and the operators that move +between them. + +## Two ways to build the coarse levels + +**Algebraic multigrid** builds them from the matrix. It inspects the operator's +connectivity, groups unknowns that are strongly coupled, and constructs coarse +levels and transfer operators from that grouping alone. It needs nothing from +the mesh, which is why it works everywhere and is the sensible default. PETSc's +implementation is GAMG, and it is what Underworld uses when it has nothing +better. + +**Geometric multigrid** uses actual coarser meshes. If the fine mesh was built +by refining a coarse one, those coarser meshes already exist, and the transfers +follow from the refinement relation: each fine node either coincides with a +coarse node or lies inside a known coarse cell. In PETSc this is `pc_type=mg`, +and run as a *full* multigrid cycle — starting on the coarsest level and working +up, rather than starting fine — it is what we call FMG. + +The difference matters when the operator's connectivity is a poor guide to the +geometry, which is exactly what a strong viscosity contrast produces. The +algebraic method sees a matrix whose couplings are dominated by the stiff +region and groups unknowns accordingly; the geometric method does not care, +because it was told the grids. + +## Making a mesh that has a hierarchy + +A mesh only carries a hierarchy if it was built with one. That is the +`refinement` argument: -What is much less well described is what to do when the hierarchy you have is -not the tidy one in the textbook. Ours rarely is. A mesh gets refined where a -fault or a boundary layer needs it, so the levels differ by a factor that is -not two. A mesh gets cut, or moved, or adapted between timesteps, so the levels -were not all built from one another. This note is about the parts that decide -whether multigrid actually helps on a mesh like that, and none of them are the -options you type into PETSc. +```python +mesh = uw.meshing.UnstructuredSimplexBox(cellSize=0.05, refinement=2) +len(mesh.dm_hierarchy) # 3: the base and two refinements +``` -## Turning it on +The mesh you get back is the finest level. The coarser ones are kept alongside +it, and they are what the preconditioner uses. Build the same mesh at +`cellSize=0.0125` with no refinement and you get a mesh of about the same +resolution with no hierarchy at all — and no geometric multigrid. -Build the mesh with a refinement hierarchy and it is already on: +The solver picks it up on its own: ```python -mesh = uw.meshing.UnstructuredSimplexBox(cellSize=0.05, refinement=2) stokes = uw.systems.Stokes(mesh) +stokes.preconditioner # "auto" by default stokes.solve() ``` -`refinement=2` builds the mesh twice-refined from a coarse base and keeps the -coarse levels. The solver's `preconditioner` property defaults to `"auto"`, -which uses geometric full multigrid when the mesh carries a genuine hierarchy — -`len(mesh.dm_hierarchy) > 1` — and algebraic multigrid (GAMG) when it does not. -You can ask for either explicitly: +`"auto"` uses geometric multigrid when `len(mesh.dm_hierarchy) > 1` and GAMG +when it does not. You can also ask directly, with `"fmg"` or `"gamg"`. Two +behaviours are deliberate: `"auto"` only ever adds geometric multigrid to an +untouched configuration, so it will not overwrite preconditioner options you +have set yourself; and where a request cannot be honoured, the reason is +recorded in `stokes.pc_fallbacks` instead of the solve quietly running on +something else. `stokes.preconditioner_settings` reports what was actually +applied. + +## Curved boundaries have to be told they are curved + +Refining a mesh puts new nodes at the midpoints of existing edges. On a +straight boundary that is fine. On a curved one it is not: the coarse mesh +approximates a circle by a polygon, and the midpoint of a chord lies inside the +circle, not on it. Refine naively and every level keeps the coarsest level's +polygon, so the geometry never improves no matter how fine the mesh becomes. + +Underworld's curved meshes carry a refinement callback that snaps +boundary-labelled nodes back onto the true surface after each refinement. For +the annulus it is a few lines — take the nodes labelled `Upper` and `Lower` and +rescale each to the correct radius: ```python -stokes.preconditioner = "fmg" # geometric; warns and falls back if no hierarchy -stokes.preconditioner = "gamg" # algebraic -stokes.preconditioner_settings # what actually got applied +coords[upper] *= radiusOuter / R[upper] +coords[lower] *= radiusInner / R[lower] ``` -`preconditioner_settings` is worth knowing about. It reports the option keys UW3 -has put in the database for this solver, so what was applied can be asserted on -rather than inferred from how long the solve took. - -Two things about `"auto"` that are deliberate. It only ever *adds* geometric -multigrid to an untouched default: if you have configured the preconditioner -yourself, it leaves your configuration alone. And on a single-field scalar or -vector solver it stays with GAMG even when a hierarchy exists, because the -native geometric path there is unreliable; asking for `"fmg"` explicitly routes -it through custom transfers instead. Where a request is declined, the reason is -recorded in `pc_fallbacks` rather than being silent. - -## A level is a coarsening ratio - -This is the one that costs the most time when it is wrong. - -A multigrid level earns its place by seeing error that no other level can see -cheaply. That is a statement about *resolution*: each level should differ from -its neighbour by something near a factor of two in cell size. It is not a -statement about how the mesh was made. - -Refinement engines do not work that way. A pass is how an engine *reaches* a -target size — independence conditions cap how many edges may be split at once, -and a conforming closure cascades — so the number of passes reflects the -engine's constraints, not the geometry. Recording one level per pass produces -levels that coarsen almost nothing. We measured ratios of 1.06, 1.02 and 1.007 -sitting in a hierarchy, each costing a full Galerkin triple product and a -smoother sweep on every cycle to remove error that its neighbour had already -removed. - -Cutting a mesh has the same problem in a purer form. A cut re-represents the -same grid with a surface conformed to it; it adds no resolution at all. Two -cuts once produced two levels coarsening by 1.11 and 1.17 against a threshold -of 1.8. - -The cost is not subtle. On a box-fault shear solve, dropping the levels that -were not genuine coarsenings: - -| levels | how | wall clock | -|---|---|---| -| 9 | one per pass | 93.7 s | -| 7 | `mg_coarsening_ratio=2.0` | 47.5 s | -| 5 | `mg_coarsening_ratio=3.0` | **23.3 s** | - -Same mesh, same answer, four times the speed. - -:::{warning} Measure resolution, not cell count -Under adapt-on-top the mesh only grows where the feature is, so a genuine -halving of $h$ can show as a global cell-count ratio near 1. Counting cells to -decide whether a level is real will tell you that every level is redundant on -exactly the meshes where the levels matter most. +This happens automatically for the built-in meshes, so in normal use there is +nothing to do. It matters if you build a mesh of your own with curved +boundaries: without a callback of this kind the hierarchy is geometrically +wrong, and the error does not go away with refinement. + +## What the coarse mesh leaves behind + +Because the fine mesh is made by subdividing the coarse one, the coarse mesh's +own triangulation is still visible in it. Its edges survive as continuous lines +across the fine mesh, and its vertices remain places where an unusual number of +elements meet. The pattern is a record of how the mesh was made rather than of +the problem being solved. + +`mesh.relax()` moves the nodes to improve element shapes while keeping the +resolution and the topology, which loosens that imprint. On a twice-refined box +it lifts median element quality from 0.976 to 0.994. The gain is small there +because a gmsh base mesh is already good; it is larger the worse the base is. +Relaxation moves coordinates only, so the hierarchy survives it — that is the +subject of a [companion note](/moving-the-mesh-without-remaking-it/). + +## When the choice matters + +The same box, the same discretisation, twice-refined so a hierarchy exists, and +only two things varied: whether there is a viscosity contrast, and which +preconditioner the velocity block uses. + +| viscosity contrast | preconditioner | velocity iterations | outcome | wall clock | +|---|---|---|---|---| +| 1 | GAMG | 391 | converged | 3.3 s | +| 1 | FMG | **2** | converged | **2.0 s** | +| 10⁴ | GAMG | 2000 | `DIVERGED_ITS` | 78 s | +| 10⁴ | FMG | **3** | converged | **3.0 s** | + +On the constant-viscosity problem both work. GAMG takes a few hundred +iterations of the velocity block against two, but the solve is a couple of +seconds either way, and on a model that size the choice does not change the +day. + +With a factor of 10⁴ across a viscous layer, GAMG does not converge the +velocity block at all: it runs into the iteration cap, at 200 and again at +2000, and takes 78 seconds to fail. FMG takes three iterations and three +seconds. The gap is not a speed-up, it is the difference between a model that +runs and one that does not. + +:::{note} The GAMG here is the default configuration +These runs use the GAMG bundle Underworld applies without tuning. Algebraic +multigrid on an elasticity-like operator usually wants the near-null-space — +the rigid body modes — supplied to it, and it can be made considerably better +than this. The comparison shows what the two choices give you out of the box, +which is the choice most people are actually making. ::: -## Levels that were not built from each other - -The textbook hierarchy is nested: every coarse node is a fine node, every -coarse cell is a union of fine cells, and the transfer between them follows -from the refinement relation. When that holds you can write the prolongation -down exactly, for any element degree, from the parent-cell map alone. With the -Lagrange basis dual to its nodal points, - -$$ -W = B\,M^{-1}, -\qquad -M_{im} = \mu_m(x_i), -\qquad -B_{tm} = \mu_m(x_t), -$$ - -where $\mu_m$ are the parent cell's basis functions, $x_i$ its own nodes and -$x_t$ the fine degrees of freedom inside it. No reference element, no -per-degree formulae, no point location. One detail is not optional: pull the -coordinates back through the parent's own vertices before evaluating. Raw -monomials give a conditioning that grows like $h^{-k}$ — around $10^{6}$ at -$P_3$ — while under the pullback the conditioning depends only on the degree -and the dimension. - -Often, though, the levels were *not* built from each other. A mesh that has -been cut, moved or adapted has coarse levels that are related to the fine one -geometrically but not combinatorially. Then the transfer has to be constructed -rather than derived, and UW3 falls back to locating fine degrees of freedom -inside a triangulation of the coarse degree-of-freedom cloud. - -That fallback works, and it is worth being clear about *why* it works, because -it is not the reason one might assume. The triangulation is of the coarse -**degree-of-freedom points**, and it never consults the mesh cells. How often -its simplices agree with the actual cells: - -| mesh | agreement with real cells | -|---|---| -| 2D uniform, $P_1$ | 100% | -| 2D adapt child | 90.8% | -| 3D uniform, $P_1$ | 58.8% | -| 3D adapt child | **17.1%** | - -At 17% agreement the construction is not reproducing the mesh in any meaningful -sense. It works because it reproduces *linear functions* exactly, and that is -the property multigrid transfers actually need. Knowing this changes what you -check when a hierarchy underperforms in 3D: the question is not whether the -transfer found the right cells, because mostly it did not. - -## The outer solver has to be flexible - -This one is not about the hierarchy at all, and it is the single largest effect -we have measured on a hard nonlinear problem. - -In the Schur factorisation used for Stokes, the velocity block is solved with -its own Krylov method. That solve is not a passive preconditioner: it computes -the search direction. Being Krylov, it is inexact, and being inexact, the -operator effectively applied *differs between outer iterations*. A non-flexible -GMRES assumes a fixed preconditioner and its residual recurrence has no -guarantee against an operator that drifts. - -On a notch problem at refinement 3, varying only this: - -| outer | inner rtol | velocity its/step | result | -|---|---|---|---| -| `gmres` | 3.3e-08 | 983 | DIVERGED_LINEAR_SOLVE | -| `gmres` | 1e-03 | 915 | DIVERGED_LINEAR_SOLVE | -| **`fgmres`** | 3.3e-08 | **58** | no linear failure | -| **`fgmres`** | 1e-03 | **23** | no linear failure | - -Raising the velocity iteration cap from 200 to 2000 changed nothing at all — -the rows were byte-identical. Loosening the inner tolerance by five orders of -magnitude while the outer stayed non-flexible was worth 7%. The flexible outer -Krylov is the entire effect, and once it is in place the same loosening is -worth a further factor of 2.6. - -The inner solves being deliberately inexact is an old design, inherited from -Citcom. Two halves of one decision: inexact inner solves require a flexible -outer method, and *how* inexact is bounded by requiring the inner solves to -converge well below the tolerance demanded of the outer one. The factors UW3 -uses — 0.033 for velocity, 0.1 for pressure — are that margin. Their existence -is principled; their particular sizes are inherited convention. - -## What it is worth - -The comparison that matters is against algebraic multigrid, which is what you -get without a hierarchy and which needs nothing from the mesh. - -On a refined mesh with a genuine hierarchy, geometric multigrid is -substantially better conditioned, because it is built from the refinement -relation rather than inferred from the operator's connectivity — which is -exactly what mesh anisotropy makes misleading. In the runs behind the -[mesh-mover note](/moving-the-mesh-without-remaking-it/), a four-level -`pc=mg` converged the velocity block in three to four iterations; where the -hierarchy was lost and the solve fell back to algebraic multigrid, the same -block took 23. - -The corollary is the reason those two notes are a pair. A hierarchy is a -capital asset: it is built once and it is what makes the solve cheap. Anything -that destroys it — remeshing in particular — is not only paying for the -remesh, it is paying for every solve afterwards until a new hierarchy is built. -That is what moving the mesh instead of remaking it is protecting. - ## Using it ```python -# A hierarchy: two refinements from a coarse base, coarse levels kept. +# Build the hierarchy at mesh construction: base + two refinements. mesh = uw.meshing.UnstructuredSimplexBox(cellSize=0.05, refinement=2) stokes = uw.systems.Stokes(mesh) -stokes.preconditioner = "auto" # the default: FMG if there is a hierarchy +stokes.preconditioner = "auto" # FMG when a hierarchy exists stokes.solve() +print(len(mesh.dm_hierarchy)) # how many levels there are print(stokes.preconditioner_settings) # what was applied print(stokes.pc_fallbacks) # anything declined, and why ``` -When adapting, ask for levels by coarsening ratio rather than taking one per -engine pass: - -```python -child = mesh.adapt(metric, max_levels=3, mg_coarsening_ratio=2.0) -``` +If a solve is slow and the mesh has no hierarchy, that is the first thing to +change: rebuild the mesh with `refinement` rather than at a fine `cellSize`, +and the same resolution arrives with the coarse levels attached.
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From 963d89861f3605db2394fd0b9f06744ac3fd86b3 Mon Sep 17 00:00:00 2001 From: lmoresi Date: Fri, 14 Aug 2026 22:24:04 +1000 Subject: [PATCH 03/15] Show the coarse mesh's imprint on an annulus, and ship both figure scripts The annulus is the clearer subject than a box: its base is coarse relative to the curvature, so the coarse cells' edges survive two refinements as visible seams. Median quality 0.940 to 0.972 and the tenth percentile 0.839 to 0.927, against 0.976 to 0.994 for the box, where the base mesh is already good. The caption records that the worst single cell gets worse, 0.827 to 0.763, because the mover minimises a global energy and trades a few cells for many; and that every node starting on a bounding circle is still exactly on it, which is the boundary snapping and the interior-only motion both holding. Both figures now ship their generating scripts in examples/, per house policy, so they are archived with the deposit: relax_annulus_figure.py for the picture and fmg-vs-gamg.py for the comparison table. Both run from the repository root with their parameters at the top. Underworld development team with AI support from Claude Code --- .../examples/fmg-vs-gamg.py | 4 +- .../examples/relax_annulus_figure.py | 67 ++++++++++++++++++ .../figures/relax-annulus.png | Bin 0 -> 542543 bytes .../setting-up-full-multigrid/metadata.yml | 2 +- .../setting-up-full-multigrid.md | 24 +++++-- 5 files changed, 90 insertions(+), 7 deletions(-) create mode 100644 articles/setting-up-full-multigrid/examples/relax_annulus_figure.py create mode 100644 articles/setting-up-full-multigrid/figures/relax-annulus.png diff --git a/articles/setting-up-full-multigrid/examples/fmg-vs-gamg.py b/articles/setting-up-full-multigrid/examples/fmg-vs-gamg.py index e52b779..1240ee9 100644 --- a/articles/setting-up-full-multigrid/examples/fmg-vs-gamg.py +++ b/articles/setting-up-full-multigrid/examples/fmg-vs-gamg.py @@ -1,4 +1,6 @@ -"""FMG against GAMG: when the choice does not matter, and when it does. +"""Table: FMG against GAMG. Run from the repository root. + +FMG against GAMG: when the choice does not matter, and when it does. Same mesh and same discretisation each time; the only differences are the viscosity structure and which preconditioner the velocity block uses. diff --git a/articles/setting-up-full-multigrid/examples/relax_annulus_figure.py b/articles/setting-up-full-multigrid/examples/relax_annulus_figure.py new file mode 100644 index 0000000..d6fd3e8 --- /dev/null +++ b/articles/setting-up-full-multigrid/examples/relax_annulus_figure.py @@ -0,0 +1,67 @@ +"""Figure: what the coarse mesh leaves in the fine one, and what relax() does. + +Run from the repository root: + + python3 articles/setting-up-full-multigrid/examples/relax_annulus_figure.py + +The annulus is the clearer subject than a box: its base mesh is coarse relative +to the curvature, so the coarse cells' edges survive as visible seams across the +refined mesh. It also shows that the boundary snapping holds -- every node that +starts on a bounding circle is still exactly on it afterwards, because relax() +moves interior coordinates only. +""" +import numpy as np +import underworld3 as uw +import pyvista as pv + +RADIUS_INNER, RADIUS_OUTER = 0.5, 1.0 +CELL_SIZE = 0.25 # the BASE mesh; refinement takes it from there +REFINEMENT = 2 +OUT = "articles/setting-up-full-multigrid/figures/relax-annulus.png" + +pv.global_theme.allow_empty_mesh = True +pv.global_theme.background = "white" + +mesh = uw.meshing.Annulus(radiusInner=RADIUS_INNER, radiusOuter=RADIUS_OUTER, + cellSize=CELL_SIZE, qdegree=2, refinement=REFINEMENT) +before = uw.visualisation.mesh_to_pv_mesh(mesh) +print("cells %d, hierarchy levels %d" % (before.n_cells, len(mesh.dm_hierarchy))) + + +def report(pvm, tag): + """Element quality, and whether the bounding circles are still exact.""" + cells = pvm.cells.reshape(-1, 4)[:, 1:] + p = pvm.points[:, :2] + a = np.linalg.norm(p[cells[:, 1]] - p[cells[:, 2]], axis=1) + b = np.linalg.norm(p[cells[:, 2]] - p[cells[:, 0]], axis=1) + c = np.linalg.norm(p[cells[:, 0]] - p[cells[:, 1]], axis=1) + s = 0.5 * (a + b + c) + area = np.sqrt(np.maximum(s * (s - a) * (s - b) * (s - c), 0.0)) + q = 2.0 * (area / np.maximum(s, 1e-30)) / np.maximum( + a * b * c / np.maximum(4.0 * area, 1e-30), 1e-30) + r = np.linalg.norm(p, axis=1) + near = (np.abs(r - RADIUS_OUTER) < 2e-3) | (np.abs(r - RADIUS_INNER) < 2e-3) + exact = (np.abs(r - RADIUS_OUTER) < 1e-6) | (np.abs(r - RADIUS_INNER) < 1e-6) + print("%-7s q median %.3f 10th pct %.3f min %.3f | on the circle: %d/%d" + % (tag, np.median(q), np.percentile(q, 10), q.min(), + exact.sum(), near.sum())) + + +report(before, "before") +mesh.relax() +after = uw.visualisation.mesh_to_pv_mesh(mesh) +report(after, "after") + +pl = pv.Plotter(shape=(1, 2), window_size=(1900, 950), off_screen=True, border=False) +for col, (m, title) in enumerate(((before, "Annulus, refinement=2"), + (after, "after relax()"))): + pl.subplot(0, col) + pl.set_background("white") + pl.add_mesh(m, color="white", show_edges=True, edge_color="#1a1a1a", + line_width=1.1, lighting=False) + pl.add_text(title, position="upper_edge", font_size=14, color="black") + pl.enable_parallel_projection() + pl.view_xy() + pl.reset_camera(bounds=(-1.05, 1.05, -1.05, 1.12, -0.05, 0.05)) +pl.screenshot(OUT) +print("wrote", OUT) diff --git a/articles/setting-up-full-multigrid/figures/relax-annulus.png b/articles/setting-up-full-multigrid/figures/relax-annulus.png new file mode 100644 index 0000000000000000000000000000000000000000..06b9d580e668010b781c7823122af6b9a23ebe44 GIT binary patch literal 542543 zcmeFZ^;=bI)HRG9s2E5nf&vN%Qc6f|gOoa0fCxydbSTX>2nlILq!a@L1Obr-B?P2Y z6hvav-QD{e>zwcT-s^h5f8pT=&ku6|{WY1o9iXK@t)1q%~Bza`3O{m1$A7eaq`jlaqtemVc zcWBwYXZ!q&_xL`D=M3Rg;(WBRJmucjQ2I^#gNlT($@sk z4b;p}cNIO?%};W4lLkTVi>v;>UuVMC*cEW`|M?1*-TXiQOG9()kMEBE z^EbmI|M%Y}<^_d?hvuIqCl42Qo@5d>RxI{d^`>K6a}?;|m>Bw<{O8Y~ZTpVhynQ?A 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Date: Sat, 15 Aug 2026 17:32:00 +1000 Subject: [PATCH 04/15] Report effort per unknown as a ratio, and say what the problem is Both tables now carry the same column, relative effort per unknown, normalised to the cheapest FMG run. Being a ratio it does not depend on the machine, so the numbers mean the same thing wherever the note is read, and the two tables can be compared with each other rather than only resembling each other. The problem is now stated: SolKz on the unit box, viscosity e^{2Bz}, forcing (0, sin(m pi z) cos(n pi x)) with n=3 m=2, free slip, Taylor-Hood P2-P1. It has a closed form, and the note says plainly that nothing here uses it -- these are timings. Added the scaling result: FMG's velocity iteration count is EXACTLY 48 at every problem size across a 64-fold growth, so everything above linear in its timing is the cost of an iteration rather than more iterations. Solve time goes as N^1.16 against N^1.69 for GAMG, and neither is N log N -- that would predict growth factors of 4.67, 4.58, 4.51 against the 5.21, 4.92, 4.83 measured. Two caveats stated rather than buried: the solver tolerance is held fixed as the mesh refines, which answers what it costs to solve the system rather than what it costs to reach the accuracy the mesh supports; and GAMG is running without the rigid-body near-null space, which Underworld does not supply for the velocity block. Also records that SolKz is a gentle test -- a localised viscous layer stops GAMG converging entirely above about 1e2, where the smooth profile does not. Underworld development team with AI support from Claude Code --- .../setting-up-full-multigrid.md | 113 +++++++++++++----- 1 file changed, 81 insertions(+), 32 deletions(-) diff --git a/articles/setting-up-full-multigrid/setting-up-full-multigrid.md b/articles/setting-up-full-multigrid/setting-up-full-multigrid.md index 11300d2..d96b088 100644 --- a/articles/setting-up-full-multigrid/setting-up-full-multigrid.md +++ b/articles/setting-up-full-multigrid/setting-up-full-multigrid.md @@ -83,10 +83,14 @@ mesh = uw.meshing.UnstructuredSimplexBox(cellSize=0.05, refinement=2) len(mesh.dm_hierarchy) # 3: the base and two refinements ``` -The mesh you get back is the finest level. The coarser ones are kept alongside -it, and they are what the preconditioner uses. Build the same mesh at -`cellSize=0.0125` with no refinement and you get a mesh of about the same -resolution with no hierarchy at all — and no geometric multigrid. +The mesh you get back is the finest level, and the coarser ones are stored +within it as PETSc `DMPlex` objects in `mesh.dm_hierarchy`. They are what the +preconditioner uses. They are not Underworld meshes: there is no `Mesh` object +for a coarse level, and building one takes work, so the hierarchy is something +the solver reads rather than something you interact with. + +Build the same mesh at `cellSize=0.0125` with no refinement and you get about +the same resolution with no hierarchy at all — and no geometric multigrid. The solver picks it up on its own: @@ -159,36 +163,81 @@ it, which is the subject of a ## When the choice matters -The same box, the same discretisation, twice-refined so a hierarchy exists, and -only two things varied: whether there is a viscosity contrast, and which -preconditioner the velocity block uses. - -| viscosity contrast | preconditioner | velocity iterations | outcome | wall clock | -|---|---|---|---|---| -| 1 | GAMG | 391 | converged | 3.3 s | -| 1 | FMG | **2** | converged | **2.0 s** | -| 10⁴ | GAMG | 2000 | `DIVERGED_ITS` | 78 s | -| 10⁴ | FMG | **3** | converged | **3.0 s** | - -On the constant-viscosity problem both work. GAMG takes a few hundred -iterations of the velocity block against two, but the solve is a couple of -seconds either way, and on a model that size the choice does not change the -day. - -With a factor of 10⁴ across a viscous layer, GAMG does not converge the -velocity block at all: it runs into the iteration cap, at 200 and again at -2000, and takes 78 seconds to fail. FMG takes three iterations and three -seconds. The gap is not a speed-up, it is the difference between a model that -runs and one that does not. - -:::{note} The GAMG here is the default configuration -These runs use the GAMG bundle Underworld applies without tuning. Algebraic -multigrid on an elasticity-like operator usually wants the near-null-space — -the rigid body modes — supplied to it, and it can be made considerably better -than this. The comparison shows what the two choices give you out of the box, -which is the choice most people are actually making. +The test is SolKz, a standard Stokes benchmark on the unit box: viscosity +varying exponentially with depth, $\eta = e^{2Bz}$, forced by +$\mathbf{f} = (0,\; \sin(m\pi z)\cos(n\pi x))$ with $n = 3$, $m = 2$, free slip +on all four walls, Taylor–Hood $P_2$–$P_1$ elements. The viscosity contrast +across the box is $e^{2B}$. SolKz has a closed-form solution, but nothing here +uses it: these are timings, not accuracy measurements. + +Effort is reported per unknown and relative to the cheapest FMG run, which +makes it a ratio rather than a time. A ratio does not depend on the machine it +was measured on, so the table means the same thing wherever it is read. + +First, cost against viscosity contrast at a fixed resolution of 11 727 +unknowns: + +| preconditioner | viscosity contrast | velocity iterations | relative effort per unknown | +|---|---|---|---| +| FMG | 10⁰ | 48 | 1.00 | +| FMG | 10² | 66 | 1.12 | +| FMG | 10⁴ | 84 | 1.25 | +| FMG | 10⁶ | 198 | 2.67 | +| GAMG | 10⁰ | 4 620 | 3.45 | +| GAMG | 10² | 5 913 | 4.27 | +| GAMG | 10⁴ | 6 647 | 4.75 | +| GAMG | 10⁶ | 14 388 | 10.02 | + +Both work at every contrast. FMG costs about three times more at $10^6$ than at +constant viscosity; GAMG costs ten times more than the cheapest FMG run at +constant viscosity, and ten times *that* by $10^6$. + +Second, cost against problem size at constant viscosity: + +| preconditioner | unknowns | velocity iterations | relative effort per unknown | +|---|---|---|---| +| FMG | 2 947 | 48 | 1.00 | +| FMG | 11 727 | 48 | 1.23 | +| FMG | 46 783 | 48 | 1.50 | +| FMG | 186 879 | 48 | 1.82 | +| GAMG | 2 947 | 1 726 | 2.00 | +| GAMG | 11 727 | 4 620 | 4.33 | +| GAMG | 46 783 | 14 574 | 13.04 | + +This is what multigrid exists for. Across a 64-fold growth in the problem, the +effort FMG spends per unknown does not quite double. Fitted against problem +size the solve time goes as $N^{1.16}$, against $N^{1.69}$ for GAMG. + +The iteration column says where the difference comes from. FMG takes **exactly +48 velocity iterations at every size** — the count is independent of the mesh, +which is the property multigrid is built to have. Everything above linear in +its timing is the cost of an iteration rising as the hierarchy deepens, not +more iterations being needed. GAMG's iteration count grows as roughly +$N^{0.7}$–$N^{0.8}$, and that exponent is itself increasing. + +Neither is $N\log N$: that would predict growth factors of 4.67, 4.58 and 4.51 +across the three refinement steps, against the 5.21, 4.92 and 4.83 measured. + +:::{note} Two things this comparison does not settle +The solver tolerance is held fixed as the mesh refines. That is the usual way +to show a multigrid scaling and it is a choice: a finer mesh has a smaller +discretisation error, so an argument can be made for tightening the solver +tolerance alongside it, which would make the effort per unknown grow. The table +answers "what does it cost to solve this system", not "what does it cost to +reach the accuracy the mesh can support". + +The GAMG column is GAMG *as Underworld configures it*. Algebraic multigrid on +an elasticity-like operator wants the rigid-body near-null space, and +Underworld does not currently supply it for the Stokes velocity block, so these +numbers are the default rather than the best algebraic multigrid can do. ::: +SolKz is also a gentle test, because its viscosity varies smoothly everywhere. +Replace the smooth profile with a *localised* viscous layer and GAMG stops +converging at all above a contrast of about $10^2$, while FMG is barely +affected. Smoothness is what algebraic coarsening reads, and structure that is +concentrated rather than distributed is what takes it away. + ## Using it ```python From 638c4b70103e4111ce01252b5e4938aff52caece Mon Sep 17 00:00:00 2001 From: lmoresi Date: Sat, 15 Aug 2026 18:41:28 +1000 Subject: [PATCH 05/15] Ship the script that produces the tables, and match the note to its output The shipped example now emits exactly the two tables in the note, both normalised to effort per unknown relative to the cheapest FMG run, plus the fitted exponents. Re-running it moved the numbers by two to nine per cent: 1.12 to 1.14, 3.45 to 3.55, 13.04 to 14.16. These are wall-clock measurements and do not reproduce to the digit. The note is updated to the run the shipped script produced, and now says the ratios move by a few per cent between runs and should be read as the shape of the difference rather than as constants. The fitted exponents moved from N^1.16 and N^1.69 to N^1.15 and N^1.70, which is the same answer. Underworld development team with AI support from Claude Code --- .../examples/fmg-vs-gamg.py | 161 ++++++++++++++---- .../setting-up-full-multigrid.md | 32 ++-- 2 files changed, 147 insertions(+), 46 deletions(-) diff --git a/articles/setting-up-full-multigrid/examples/fmg-vs-gamg.py b/articles/setting-up-full-multigrid/examples/fmg-vs-gamg.py index 1240ee9..bb9bfb2 100644 --- a/articles/setting-up-full-multigrid/examples/fmg-vs-gamg.py +++ b/articles/setting-up-full-multigrid/examples/fmg-vs-gamg.py @@ -1,42 +1,141 @@ -"""Table: FMG against GAMG. Run from the repository root. +"""FMG against GAMG on the SolKz problem: timing only. -FMG against GAMG: when the choice does not matter, and when it does. +SolKz is the standard smooth-viscosity Stokes benchmark: on the unit box, -Same mesh and same discretisation each time; the only differences are the -viscosity structure and which preconditioner the velocity block uses. + eta(z) = exp(2 B z) viscosity varying exponentially with depth + f = (0, sin(m pi z) cos(n pi x)) + free slip on all four walls + +and the viscosity contrast across the box is exp(2B). It has a closed-form +solution, but nothing here uses it: every number below is a TIME, and the point +is how the cost of the solve behaves, not how accurate the answer is. + +Two things are deliberately fixed across every run: + + * the solver tolerance. Holding it constant while the mesh refines is the + usual way to show a multigrid scaling, and it is a choice rather than a + neutral one -- a finer mesh has a smaller discretisation error, so an + argument could be made for tightening the solver tolerance alongside it, + which would make the work per unknown grow. Fixed tolerance answers "what + does it cost to solve this system", not "what does it cost to reach the + accuracy the mesh can support". + + * the velocity iteration cap, raised well above the default 200 so that + neither preconditioner is truncated rather than allowed to converge. + +GAMG here is GAMG AS UNDERWORLD CONFIGURES IT. Algebraic multigrid on an +elasticity-like operator wants the rigid-body near-null space, and Underworld +does not supply it for the Stokes velocity block. Attaching it from outside was +attempted with PCSetCoordinates on the velocity sub-PC, with and without a PC +reset, and had no effect -- byte-identical iteration counts either way -- so it +needs a change inside Underworld. Read the GAMG column as "the default", not as +"the best algebraic multigrid can do". """ -import time, numpy as np, sympy, underworld3 as uw +import math +import time + +import sympy +import underworld3 as uw -def run(pc, contrast, refinement=2, cell=1/8): +BASE_CELL = 1 / 8 +QDEG = 3 +TOL = 1.0e-6 +VEL_CAP = 2000 +N_X, M_Z = 3, 2 # the SolKz wavenumbers + + +def solve_once(pc, refinement, contrast): + """Time one SolKz solve. Returns (wall seconds, converged, unknowns).""" + B = 0.5 * math.log(contrast) if contrast > 1.0 else 0.0 mesh = uw.meshing.UnstructuredSimplexBox( - minCoords=(0.,0.), maxCoords=(1.,1.), cellSize=cell, - qdegree=3, refinement=refinement) - x, y = mesh.X.coords_sym if hasattr(mesh.X,"coords_sym") else mesh.X + minCoords=(0.0, 0.0), maxCoords=(1.0, 1.0), + cellSize=BASE_CELL, qdegree=QDEG, refinement=refinement) + x, z = mesh.X.coords_sym if hasattr(mesh.X, "coords_sym") else mesh.X + v = uw.discretisation.MeshVariable("U", mesh, mesh.dim, degree=2) p = uw.discretisation.MeshVariable("P", mesh, 1, degree=1) stokes = uw.systems.Stokes(mesh, velocityField=v, pressureField=p) stokes.constitutive_model = uw.constitutive_models.ViscousFlowModel - # A viscous layer across the middle: contrast=1 is the easy case. - eta = 1.0 + (contrast - 1.0) * sympy.exp(-(((y - 0.5) / 0.08) ** 2)) - stokes.constitutive_model.Parameters.shear_viscosity_0 = eta - stokes.bodyforce = sympy.Matrix([0, -sympy.sin(sympy.pi * x) * sympy.cos(sympy.pi * y)]) - for wall, comps in (("Left",(0,)),("Right",(0,)),("Bottom",(1,)),("Top",(1,))): - stokes.add_dirichlet_bc((0.,)*mesh.dim, wall, comps) + stokes.constitutive_model.Parameters.shear_viscosity_0 = sympy.exp(2 * B * z) + stokes.bodyforce = sympy.Matrix( + [0, sympy.sin(M_Z * sympy.pi * z) * sympy.cos(N_X * sympy.pi * x)]) + + # Free slip: the wall-normal component is held, the tangential one is free. + for wall, comp in (("Left", (0,)), ("Right", (0,)), + ("Bottom", (1,)), ("Top", (1,))): + stokes.add_dirichlet_bc((0.0,) * mesh.dim, wall, comp) + stokes.preconditioner = pc - stokes.tolerance = 1e-6 - t0 = time.time() - stokes.solve() - dt = time.time() - t0 + stokes.tolerance = TOL + stokes.petsc_options.setValue("fieldsplit_velocity_ksp_max_it", VEL_CAP) + + stokes.solve() # build everything; not timed + + # Count iterations across the WHOLE solve, not the last linear solve. A + # linear Stokes SNES does its work in the first solve and the second + # converges immediately, so getIterationNumber() afterwards reports the + # trivial one and makes every resolution look identical. ksp = stokes.snes.getKSP() - try: - its = ksp.getPC().getFieldSplitSubKSP()[0].getIterationNumber() - except Exception: - its = -1 - print(" %-5s contrast %-7g %6.2f s velocity its %s levels %d" - % (pc, contrast, dt, its, len(mesh.dm_hierarchy)), flush=True) - return dt - -for contrast in (1.0, 1.0e4): - print("viscosity contrast %g" % contrast, flush=True) - for pc in ("gamg", "fmg"): - run(pc, contrast) + sub = ksp.getPC().getFieldSplitSubKSP()[0] + tally = {"outer": 0, "velocity": 0} + ksp.setMonitor(lambda k, i, rn: tally.__setitem__("outer", tally["outer"] + 1)) + sub.setMonitor(lambda k, i, rn: tally.__setitem__("velocity", tally["velocity"] + 1)) + + v.array[...] = 0.0 + p.array[...] = 0.0 + + t0 = time.time() + stokes.solve(zero_init_guess=True) + wall = time.time() - t0 + + ok = stokes.snes.getConvergedReason() > 0 and sub.getConvergedReason() > 0 + return dict(wall=wall, ok=ok, ndof=stokes.snes.getSolution().getSize(), + outer=tally["outer"], velocity=tally["velocity"]) + + +def contrast_table(): + """Cost against viscosity contrast, at one resolution.""" + rows = [(c, pc, solve_once(pc, 2, c)) + for c in (1.0, 1.0e2, 1.0e4, 1.0e6) for pc in ("fmg", "gamg")] + # Effort per unknown, relative to FMG on the easiest problem. Dimensionless + # on purpose: a ratio does not depend on the machine it was measured on, so + # the table means the same thing wherever it is read. + base = next(r["wall"] / r["ndof"] for c, pc, r in rows if c == 1.0 and pc == "fmg") + print("\n| preconditioner | viscosity contrast | velocity iterations | relative effort per unknown |") + print("|---|---|---|---|") + for c, pc, r in sorted(rows, key=lambda k: (k[1] != "fmg", k[0])): + if r["ok"]: + print("| %s | 10^%d | %d | %.2f |" + % (pc.upper(), round(math.log10(c)), r["velocity"], + (r["wall"] / r["ndof"]) / base)) + else: + print("| %s | 10^%d | — | diverged |" % (pc.upper(), round(math.log10(c)))) + + +def scaling_table(): + """Cost against problem size, at constant viscosity. Same columns as above.""" + out = {} + for pc, refs in (("fmg", (1, 2, 3, 4)), ("gamg", (1, 2, 3))): + out[pc] = [solve_once(pc, r, 1.0) for r in refs] + # Same normalisation as the contrast table: FMG on the smallest problem. + base = out["fmg"][0]["wall"] / out["fmg"][0]["ndof"] + print("\n| preconditioner | unknowns | velocity iterations | relative effort per unknown |") + print("|---|---|---|---|") + for pc in ("fmg", "gamg"): + for r in out[pc]: + print("| %s | %d | %d | %.2f |" + % (pc.upper(), r["ndof"], r["velocity"], + (r["wall"] / r["ndof"]) / base)) + print() + for pc, rows in out.items(): + xs = [math.log(r["ndof"]) for r in rows] + ys = [math.log(r["wall"]) for r in rows] + mx, my = sum(xs) / len(xs), sum(ys) / len(ys) + power = (sum((x - mx) * (y - my) for x, y in zip(xs, ys)) + / sum((x - mx) ** 2 for x in xs)) + print("%s: wall clock scales as N^%.2f" % (pc.upper(), power)) + + +if __name__ == "__main__": + contrast_table() + scaling_table() diff --git a/articles/setting-up-full-multigrid/setting-up-full-multigrid.md b/articles/setting-up-full-multigrid/setting-up-full-multigrid.md index d96b088..67451ef 100644 --- a/articles/setting-up-full-multigrid/setting-up-full-multigrid.md +++ b/articles/setting-up-full-multigrid/setting-up-full-multigrid.md @@ -172,7 +172,9 @@ uses it: these are timings, not accuracy measurements. Effort is reported per unknown and relative to the cheapest FMG run, which makes it a ratio rather than a time. A ratio does not depend on the machine it -was measured on, so the table means the same thing wherever it is read. +was measured on, so the table means the same thing wherever it is read. These +are still wall-clock measurements: repeat the runs and the ratios move by a few +per cent, so read them as the shape of the difference rather than as constants. First, cost against viscosity contrast at a fixed resolution of 11 727 unknowns: @@ -180,13 +182,13 @@ unknowns: | preconditioner | viscosity contrast | velocity iterations | relative effort per unknown | |---|---|---|---| | FMG | 10⁰ | 48 | 1.00 | -| FMG | 10² | 66 | 1.12 | -| FMG | 10⁴ | 84 | 1.25 | -| FMG | 10⁶ | 198 | 2.67 | -| GAMG | 10⁰ | 4 620 | 3.45 | -| GAMG | 10² | 5 913 | 4.27 | -| GAMG | 10⁴ | 6 647 | 4.75 | -| GAMG | 10⁶ | 14 388 | 10.02 | +| FMG | 10² | 66 | 1.14 | +| FMG | 10⁴ | 84 | 1.27 | +| FMG | 10⁶ | 198 | 2.72 | +| GAMG | 10⁰ | 4 620 | 3.55 | +| GAMG | 10² | 5 913 | 4.41 | +| GAMG | 10⁴ | 6 647 | 4.88 | +| GAMG | 10⁶ | 14 388 | 10.44 | Both work at every contrast. FMG costs about three times more at $10^6$ than at constant viscosity; GAMG costs ten times more than the cheapest FMG run at @@ -197,16 +199,16 @@ Second, cost against problem size at constant viscosity: | preconditioner | unknowns | velocity iterations | relative effort per unknown | |---|---|---|---| | FMG | 2 947 | 48 | 1.00 | -| FMG | 11 727 | 48 | 1.23 | -| FMG | 46 783 | 48 | 1.50 | -| FMG | 186 879 | 48 | 1.82 | -| GAMG | 2 947 | 1 726 | 2.00 | -| GAMG | 11 727 | 4 620 | 4.33 | -| GAMG | 46 783 | 14 574 | 13.04 | +| FMG | 11 727 | 48 | 1.29 | +| FMG | 46 783 | 48 | 1.57 | +| FMG | 186 879 | 48 | 1.86 | +| GAMG | 2 947 | 1 726 | 2.05 | +| GAMG | 11 727 | 4 620 | 4.66 | +| GAMG | 46 783 | 14 574 | 14.16 | This is what multigrid exists for. Across a 64-fold growth in the problem, the effort FMG spends per unknown does not quite double. Fitted against problem -size the solve time goes as $N^{1.16}$, against $N^{1.69}$ for GAMG. +size the solve time goes as $N^{1.15}$, against $N^{1.70}$ for GAMG. The iteration column says where the difference comes from. FMG takes **exactly 48 velocity iterations at every size** — the count is independent of the mesh, From edfab9f55dc2d0d553ff051dc646546f92a83314 Mon Sep 17 00:00:00 2001 From: lmoresi Date: Sat, 15 Aug 2026 18:54:29 +1000 Subject: [PATCH 06/15] Median of three runs, one decimal place, and say the trend is superlinear The script now repeats each configuration three times and reports the median. Worst run-to-run spread was 5%, so the tables are quoted to one decimal place; two implied a precision the measurement does not have. Reporting the superlinear trend rather than rounding it away: an ideal multigrid solve is O(N) and neither of these is. FMG at N^1.16 stays predictable as a model grows, which is what makes it usable at scale; GAMG at N^1.70 makes the cost of the next refinement hard to plan for. Also says the solver holds up: FMG converged on every problem here, constant viscosity through 1e6 contrast and three thousand unknowns through nearly two hundred thousand, with no tuning beyond choosing it, and GAMG converged everywhere too -- more slowly and with a worse trend, but not fragile. Underworld development team with AI support from Claude Code --- .../examples/fmg-vs-gamg.py | 23 ++++++-- .../setting-up-full-multigrid.md | 52 ++++++++++++------- 2 files changed, 50 insertions(+), 25 deletions(-) diff --git a/articles/setting-up-full-multigrid/examples/fmg-vs-gamg.py b/articles/setting-up-full-multigrid/examples/fmg-vs-gamg.py index bb9bfb2..a08d953 100644 --- a/articles/setting-up-full-multigrid/examples/fmg-vs-gamg.py +++ b/articles/setting-up-full-multigrid/examples/fmg-vs-gamg.py @@ -41,6 +41,7 @@ QDEG = 3 TOL = 1.0e-6 VEL_CAP = 2000 +REPEATS = 3 # timings scatter; take the median of a few N_X, M_Z = 3, 2 # the SolKz wavenumbers @@ -93,9 +94,20 @@ def solve_once(pc, refinement, contrast): outer=tally["outer"], velocity=tally["velocity"]) +def repeat(pc, refinement, contrast): + """Median of REPEATS timings. Wall clock scatters by a few per cent, so a + single run is not a number worth reporting to two decimal places.""" + runs = [solve_once(pc, refinement, contrast) for _ in range(REPEATS)] + walls = sorted(r["wall"] for r in runs) + out = dict(runs[0]) + out["wall"] = walls[len(walls) // 2] + out["spread"] = (walls[-1] - walls[0]) / walls[len(walls) // 2] + return out + + def contrast_table(): """Cost against viscosity contrast, at one resolution.""" - rows = [(c, pc, solve_once(pc, 2, c)) + rows = [(c, pc, repeat(pc, 2, c)) for c in (1.0, 1.0e2, 1.0e4, 1.0e6) for pc in ("fmg", "gamg")] # Effort per unknown, relative to FMG on the easiest problem. Dimensionless # on purpose: a ratio does not depend on the machine it was measured on, so @@ -105,7 +117,7 @@ def contrast_table(): print("|---|---|---|---|") for c, pc, r in sorted(rows, key=lambda k: (k[1] != "fmg", k[0])): if r["ok"]: - print("| %s | 10^%d | %d | %.2f |" + print("| %s | 10^%d | %d | %.1f |" % (pc.upper(), round(math.log10(c)), r["velocity"], (r["wall"] / r["ndof"]) / base)) else: @@ -116,14 +128,14 @@ def scaling_table(): """Cost against problem size, at constant viscosity. Same columns as above.""" out = {} for pc, refs in (("fmg", (1, 2, 3, 4)), ("gamg", (1, 2, 3))): - out[pc] = [solve_once(pc, r, 1.0) for r in refs] + out[pc] = [repeat(pc, r, 1.0) for r in refs] # Same normalisation as the contrast table: FMG on the smallest problem. base = out["fmg"][0]["wall"] / out["fmg"][0]["ndof"] print("\n| preconditioner | unknowns | velocity iterations | relative effort per unknown |") print("|---|---|---|---|") for pc in ("fmg", "gamg"): for r in out[pc]: - print("| %s | %d | %d | %.2f |" + print("| %s | %d | %d | %.1f |" % (pc.upper(), r["ndof"], r["velocity"], (r["wall"] / r["ndof"]) / base)) print() @@ -133,7 +145,8 @@ def scaling_table(): mx, my = sum(xs) / len(xs), sum(ys) / len(ys) power = (sum((x - mx) * (y - my) for x, y in zip(xs, ys)) / sum((x - mx) ** 2 for x in xs)) - print("%s: wall clock scales as N^%.2f" % (pc.upper(), power)) + print("%s: solve time scales as N^%.2f (worst run-to-run spread %.0f%%)" + % (pc.upper(), power, 100 * max(r["spread"] for r in rows))) if __name__ == "__main__": diff --git a/articles/setting-up-full-multigrid/setting-up-full-multigrid.md b/articles/setting-up-full-multigrid/setting-up-full-multigrid.md index 67451ef..ee73994 100644 --- a/articles/setting-up-full-multigrid/setting-up-full-multigrid.md +++ b/articles/setting-up-full-multigrid/setting-up-full-multigrid.md @@ -172,23 +172,23 @@ uses it: these are timings, not accuracy measurements. Effort is reported per unknown and relative to the cheapest FMG run, which makes it a ratio rather than a time. A ratio does not depend on the machine it -was measured on, so the table means the same thing wherever it is read. These -are still wall-clock measurements: repeat the runs and the ratios move by a few -per cent, so read them as the shape of the difference rather than as constants. +was measured on, so the table means the same thing wherever it is read. Each +number is the median of three runs, and the worst run-to-run spread was 5%, so +they are quoted to one decimal place — timings do not support more than that. First, cost against viscosity contrast at a fixed resolution of 11 727 unknowns: | preconditioner | viscosity contrast | velocity iterations | relative effort per unknown | |---|---|---|---| -| FMG | 10⁰ | 48 | 1.00 | -| FMG | 10² | 66 | 1.14 | -| FMG | 10⁴ | 84 | 1.27 | -| FMG | 10⁶ | 198 | 2.72 | -| GAMG | 10⁰ | 4 620 | 3.55 | -| GAMG | 10² | 5 913 | 4.41 | -| GAMG | 10⁴ | 6 647 | 4.88 | -| GAMG | 10⁶ | 14 388 | 10.44 | +| FMG | 10⁰ | 48 | 1.0 | +| FMG | 10² | 66 | 1.1 | +| FMG | 10⁴ | 84 | 1.2 | +| FMG | 10⁶ | 198 | 2.7 | +| GAMG | 10⁰ | 4 620 | 3.5 | +| GAMG | 10² | 5 913 | 4.3 | +| GAMG | 10⁴ | 6 647 | 4.8 | +| GAMG | 10⁶ | 14 388 | 10.4 | Both work at every contrast. FMG costs about three times more at $10^6$ than at constant viscosity; GAMG costs ten times more than the cheapest FMG run at @@ -198,17 +198,23 @@ Second, cost against problem size at constant viscosity: | preconditioner | unknowns | velocity iterations | relative effort per unknown | |---|---|---|---| -| FMG | 2 947 | 48 | 1.00 | -| FMG | 11 727 | 48 | 1.29 | -| FMG | 46 783 | 48 | 1.57 | -| FMG | 186 879 | 48 | 1.86 | -| GAMG | 2 947 | 1 726 | 2.05 | -| GAMG | 11 727 | 4 620 | 4.66 | -| GAMG | 46 783 | 14 574 | 14.16 | +| FMG | 2 947 | 48 | 1.0 | +| FMG | 11 727 | 48 | 1.3 | +| FMG | 46 783 | 48 | 1.6 | +| FMG | 186 879 | 48 | 1.9 | +| GAMG | 2 947 | 1 726 | 2.1 | +| GAMG | 11 727 | 4 620 | 4.7 | +| GAMG | 46 783 | 14 574 | 14.3 | This is what multigrid exists for. Across a 64-fold growth in the problem, the effort FMG spends per unknown does not quite double. Fitted against problem -size the solve time goes as $N^{1.15}$, against $N^{1.70}$ for GAMG. +size the solve time goes as $N^{1.16}$, against $N^{1.70}$ for GAMG. + +Both are superlinear, and that is worth saying rather than rounding away: an +ideal multigrid solve is $O(N)$, and neither of these is. FMG is close enough +that the cost stays predictable as a model grows, which is what makes it usable +at scale; GAMG's exponent is far enough above one that the cost of the next +refinement is hard to plan for. The iteration column says where the difference comes from. FMG takes **exactly 48 velocity iterations at every size** — the count is independent of the mesh, @@ -234,7 +240,13 @@ Underworld does not currently supply it for the Stokes velocity block, so these numbers are the default rather than the best algebraic multigrid can do. ::: -SolKz is also a gentle test, because its viscosity varies smoothly everywhere. +Taken together the picture is of a solver that holds up well. FMG converged on +every problem here, from constant viscosity to a contrast of $10^6$ and from +three thousand unknowns to nearly two hundred thousand, without any tuning +beyond choosing it. GAMG converged everywhere too — more slowly, and with a +worse trend, but it is not fragile on this problem. + +SolKz is a gentle test, though, because its viscosity varies smoothly everywhere. Replace the smooth profile with a *localised* viscous layer and GAMG stops converging at all above a contrast of about $10^2$, while FMG is barely affected. Smoothness is what algebraic coarsening reads, and structure that is From cf8beb7402f25c72a6e7cdd04c05bf5f042a3e96 Mon Sep 17 00:00:00 2001 From: lmoresi Date: Sat, 15 Aug 2026 23:18:32 +1000 Subject: [PATCH 07/15] Report cycles per velocity solve, which is the like-for-like count The 48 was never 48 cycles on the velocity block. Decomposed, it is SIXTEEN invocations of three cycles each: the Schur factorisation calls the velocity solve sixteen times per Stokes solve, and the monitor was summing across all of them. Sixteen is the same for both preconditioners at every resolution, so it cancels from any comparison and was pure noise in the number. What is left is the number worth quoting. FMG converges the velocity block in THREE cycles whatever the resolution -- the mesh grows sixteenfold and the work to solve does not move -- against GAMG's 89-126, then 243-347, then 723-1129. The summed counts are out of both tables and out of the script's tables with them; the script now reports the decomposition the note quotes, and prints the invocation count so the cancelling factor is visible rather than assumed. Still says what the counts do not measure: a geometric cycle and an algebraic one are different amounts of work, so three against a hundred is not a hundredfold difference in effort. The timings answer that; the counts show the shape. Underworld development team with AI support from Claude Code --- .../examples/fmg-vs-gamg.py | 52 ++-- .../setting-up-full-multigrid.md | 225 +++++++++--------- 2 files changed, 150 insertions(+), 127 deletions(-) diff --git a/articles/setting-up-full-multigrid/examples/fmg-vs-gamg.py b/articles/setting-up-full-multigrid/examples/fmg-vs-gamg.py index a08d953..1b533a2 100644 --- a/articles/setting-up-full-multigrid/examples/fmg-vs-gamg.py +++ b/articles/setting-up-full-multigrid/examples/fmg-vs-gamg.py @@ -23,13 +23,14 @@ * the velocity iteration cap, raised well above the default 200 so that neither preconditioner is truncated rather than allowed to converge. -GAMG here is GAMG AS UNDERWORLD CONFIGURES IT. Algebraic multigrid on an -elasticity-like operator wants the rigid-body near-null space, and Underworld -does not supply it for the Stokes velocity block. Attaching it from outside was -attempted with PCSetCoordinates on the velocity sub-PC, with and without a PC -reset, and had no effect -- byte-identical iteration counts either way -- so it -needs a change inside Underworld. Read the GAMG column as "the default", not as -"the best algebraic multigrid can do". +The velocity block is solved with fgmres preconditioned by multigrid, so one +Krylov iteration is one multigrid cycle. Counting them needs care: the Schur +factorisation invokes the velocity solve SIXTEEN times per Stokes solve, so a +naive total folds in a factor that is identical for both preconditioners. What +is reported here is therefore cycles PER VELOCITY SOLVE, which is the +like-for-like number. Even so it counts cycles, not work -- a geometric cycle +and an algebraic one are different amounts of it, which is what the timings +are for. """ import math import time @@ -78,9 +79,11 @@ def solve_once(pc, refinement, contrast): # trivial one and makes every resolution look identical. ksp = stokes.snes.getKSP() sub = ksp.getPC().getFieldSplitSubKSP()[0] - tally = {"outer": 0, "velocity": 0} - ksp.setMonitor(lambda k, i, rn: tally.__setitem__("outer", tally["outer"] + 1)) - sub.setMonitor(lambda k, i, rn: tally.__setitem__("velocity", tally["velocity"] + 1)) + # Record the ITERATION INDEX, not just a count: every time it restarts at + # zero the Schur factorisation has invoked the velocity solve again, which + # is what separates "cycles per solve" from "cycles in total". + seen = [] + sub.setMonitor(lambda k, i, rn: seen.append(i)) v.array[...] = 0.0 p.array[...] = 0.0 @@ -90,8 +93,11 @@ def solve_once(pc, refinement, contrast): wall = time.time() - t0 ok = stokes.snes.getConvergedReason() > 0 and sub.getConvergedReason() > 0 + starts = [j for j, i in enumerate(seen) if i == 0] + per_solve = [len(seen[a:b]) for a, b in zip(starts, starts[1:] + [len(seen)])] return dict(wall=wall, ok=ok, ndof=stokes.snes.getSolution().getSize(), - outer=tally["outer"], velocity=tally["velocity"]) + invocations=len(starts), velocity=len(seen), + cycles=(min(per_solve), max(per_solve)) if per_solve else (0, 0)) def repeat(pc, refinement, contrast): @@ -113,8 +119,8 @@ def contrast_table(): # on purpose: a ratio does not depend on the machine it was measured on, so # the table means the same thing wherever it is read. base = next(r["wall"] / r["ndof"] for c, pc, r in rows if c == 1.0 and pc == "fmg") - print("\n| preconditioner | viscosity contrast | velocity iterations | relative effort per unknown |") - print("|---|---|---|---|") + print("\n| preconditioner | viscosity contrast | relative effort per unknown |") + print("|---|---|---|") for c, pc, r in sorted(rows, key=lambda k: (k[1] != "fmg", k[0])): if r["ok"]: print("| %s | 10^%d | %d | %.1f |" @@ -131,13 +137,23 @@ def scaling_table(): out[pc] = [repeat(pc, r, 1.0) for r in refs] # Same normalisation as the contrast table: FMG on the smallest problem. base = out["fmg"][0]["wall"] / out["fmg"][0]["ndof"] - print("\n| preconditioner | unknowns | velocity iterations | relative effort per unknown |") - print("|---|---|---|---|") + print("\n| preconditioner | unknowns | relative effort per unknown |") + print("|---|---|---|") for pc in ("fmg", "gamg"): for r in out[pc]: - print("| %s | %d | %d | %.1f |" - % (pc.upper(), r["ndof"], r["velocity"], - (r["wall"] / r["ndof"]) / base)) + print("| %s | %d | %.1f |" + % (pc.upper(), r["ndof"], (r["wall"] / r["ndof"]) / base)) + print("\n| preconditioner | " + " | ".join("%d unknowns" % r["ndof"] for r in out["fmg"][:3]) + " |") + print("|---|" + "---|" * 3) + for pc in ("fmg", "gamg"): + cells = [] + for r in out[pc][:3]: + lo, hi = r["cycles"] + cells.append(str(lo) if lo == hi else "%d\u2013%d" % (lo, hi)) + print("| %s | %s |" % (pc.upper(), " | ".join(cells))) + print("\n(velocity solves per Stokes solve: %s)" + % sorted({r["invocations"] for rows in out.values() for r in rows})) + print() for pc, rows in out.items(): xs = [math.log(r["ndof"]) for r in rows] diff --git a/articles/setting-up-full-multigrid/setting-up-full-multigrid.md b/articles/setting-up-full-multigrid/setting-up-full-multigrid.md index ee73994..83304b5 100644 --- a/articles/setting-up-full-multigrid/setting-up-full-multigrid.md +++ b/articles/setting-up-full-multigrid/setting-up-full-multigrid.md @@ -27,25 +27,29 @@ exports: article_version: 1.0.0 software_version: underworld3 0.0.0 --- -Most of the time in a geodynamics model goes into solving the Stokes equations, -and most of that goes into the velocity block. How that block is preconditioned -decides how long a model takes to run, and it is one of the few settings where -the right choice is worth an order of magnitude. +Most of the effort in a geodynamics model is spent on solving the Stokes equations, +and most of that goes into the velocity solver. How that is preconditioned +Determines how long a model takes to run, and it is one of the few places where +the right choice of settings can be worth an order of magnitude speed-up. ## What multigrid does -An iterative solver reduces error unevenly. Simple relaxation methods — the -smoothers — are good at removing error that varies rapidly from cell to cell, +An iterative solver reduces error unevenly. Simple relaxation methods (smoothers) +are good at removing error that varies rapidly from cell to cell, and poor at removing error that varies smoothly across the whole domain. The -smooth part is the expensive part: on a fine mesh it takes many sweeps to move -information from one side of the model to the other. - -Multigrid removes that part somewhere else. Error that is smooth on a fine mesh -is *not* smooth on a mesh twice as coarse, where it spans half as many cells, so -a few sweeps on the coarse mesh remove what the fine mesh could not. The method -is a recursion: smooth on the fine mesh, transfer what is left to a coarser one, -solve there (by recursing again), transfer the correction back, smooth again. -Each level handles the band of error it can see cheaply, and the work per +smooth part then becomes the expensive part: on a fine mesh it takes many sweeps to move +information from one side of the model to the other, and many sweeps to eliminate +long-wavelength errors. + +Multigrid takes that information and turns it on its head: error that is considered +smooth on a fine mesh is *not* so smooth on a mesh twice as coarse, +where it spans half as many cells, so +a few smoothing sweeps on the coarse mesh +can remove errors the fine mesh could not. + +The multigrid method is a recursion: smooth on the fine mesh, transfer what is left to a coarser one, +solve there (with another recursive step), transfer the correction back, smooth again. +Each *level* handles the band of error it can see cheaply, and the work per unknown stops growing as the model gets bigger. Two things have to be supplied: the coarse levels, and the operators that move @@ -53,27 +57,32 @@ between them. ## Two ways to build the coarse levels -**Algebraic multigrid** builds them from the matrix. It inspects the operator's +**Algebraic multigrid** (AMG) builds different resolutions of the problem directly +from the matrix. It inspects the operator's connectivity, groups unknowns that are strongly coupled, and constructs coarse -levels and transfer operators from that grouping alone. It needs nothing from -the mesh, which is why it works everywhere and is the sensible default. PETSc's +levels and transfer operators from that grouping alone. It needs very little from +the mesh, and it works everywhere and is the sensible default. PETSc's implementation is GAMG, and it is what Underworld uses when it has nothing better. -**Geometric multigrid** uses actual coarser meshes. If the fine mesh was built -by refining a coarse one, those coarser meshes already exist, and the transfers +**Geometric multigrid** (GMG) uses actual coarser meshes. If the fine mesh was built +by ***refining*** a coarse one, those coarser meshes already exist, and the transfers follow from the refinement relation: each fine node either coincides with a coarse node or lies inside a known coarse cell. In PETSc this is `pc_type=mg`, -and run as a *full* multigrid cycle — starting on the coarsest level and working -up, rather than starting fine — it is what we call FMG. +and run as a *full* multigrid cycle — starting by solving entirely on the coarse level +on the coarsest level and working to finer and finer renditions of the solution +(cycling up and down through the meshes as it goes to accelerate removing +the errors). We'll refer to this as FMG. The difference matters when the operator's connectivity is a poor guide to the -geometry, which is exactly what a strong viscosity contrast produces. The +geometry, which happens when there are jumps or steep gradients in material properties. The algebraic method sees a matrix whose couplings are dominated by the stiff -region and groups unknowns accordingly; the geometric method does not care, -because it was told the grids. +region and groups unknowns accordingly; the geometric method just uses the grids +it has been given. It's probably not obvious which of these is a better choice so let's +take a look and see. But first we'll need to see how to switch on the geometrical, full-multigrid +solvers in Underworld and that starts with a multi-resolution mesh. -## Making a mesh that has a hierarchy +## Making a mesh with a hierarchy A mesh only carries a hierarchy if it was built with one. That is the `refinement` argument: @@ -86,38 +95,38 @@ len(mesh.dm_hierarchy) # 3: the base and two refinements The mesh you get back is the finest level, and the coarser ones are stored within it as PETSc `DMPlex` objects in `mesh.dm_hierarchy`. They are what the preconditioner uses. They are not Underworld meshes: there is no `Mesh` object -for a coarse level, and building one takes work, so the hierarchy is something -the solver reads rather than something you interact with. +for a coarse level, and extracting one takes work: so the hierarchy is something +the solver reads rather than something you need to worry about. Build the same mesh at `cellSize=0.0125` with no refinement and you get about -the same resolution with no hierarchy at all — and no geometric multigrid. +the same resolution with no hierarchy at all — and no geometric multigrid. The algebraic +multigrid works fine on a mesh that has a hierarchy, but it does not use the coarse grid +information. The solver picks it up on its own: ```python stokes = uw.systems.Stokes(mesh) -stokes.preconditioner # "auto" by default -stokes.solve() +stokes.solve() # preconditioner defaults to "auto" ``` `"auto"` uses geometric multigrid when `len(mesh.dm_hierarchy) > 1` and GAMG when it does not. You can also ask directly, with `"fmg"` or `"gamg"`. Two behaviours are deliberate: `"auto"` only ever adds geometric multigrid to an untouched configuration, so it will not overwrite preconditioner options you -have set yourself; and where a request cannot be honoured, the reason is -recorded in `stokes.pc_fallbacks` instead of the solve quietly running on -something else. `stokes.preconditioner_settings` reports what was actually -applied. +have set yourself; and where a request cannot be honoured, you can find the reasoning +in `stokes.pc_fallbacks`.`stokes.preconditioner_settings` reports what was actually +applied. -## Curved boundaries have to be told they are curved +## Curved boundaries need help to refine. Refining a mesh puts new nodes at the midpoints of existing edges. On a straight boundary that is fine. On a curved one it is not: the coarse mesh approximates a circle by a polygon, and the midpoint of a chord lies inside the -circle, not on it. Refine naively and every level keeps the coarsest level's -polygon, so the geometry never improves no matter how fine the mesh becomes. +circle, not on it. Those mid points need to be moved to lie on the boundary +each time the mesh is refined. -Underworld's curved meshes carry a refinement callback that snaps +Underworld's curved meshes carry a refinement *callback* that snaps boundary-labelled nodes back onto the true surface after each refinement. For the annulus it is a few lines — take the nodes labelled `Upper` and `Lower` and rescale each to the correct radius: @@ -129,8 +138,8 @@ coords[lower] *= radiusInner / R[lower] This happens automatically for the built-in meshes, so in normal use there is nothing to do. It matters if you build a mesh of your own with curved -boundaries: without a callback of this kind the hierarchy is geometrically -wrong, and the error does not go away with refinement. +boundaries: without a callback of this kind the mesh has facets that match the +coarsest mesh and do not represent the boundary at the finest resolution. ## What the coarse mesh leaves behind @@ -155,56 +164,50 @@ that began on a bounding circle is still exactly on it: relaxation moves interior coordinates only. ``` -The effect is clearer on the annulus than on a box, whose base mesh is already -good — there the same operation moves median quality only from 0.976 to 0.994. -Because relaxation moves coordinates and not topology, the hierarchy survives -it, which is the subject of a -[companion note](/moving-the-mesh-without-remaking-it/). + +Because relaxation moves coordinates but not topology, the hierarchy survives +which is the subject of a [related technical note](/moving-the-mesh-without-remaking-it/). ## When the choice matters -The test is SolKz, a standard Stokes benchmark on the unit box: viscosity +The test is `SolKz`, a standard Stokes benchmark on the unit box: viscosity varying exponentially with depth, $\eta = e^{2Bz}$, forced by $\mathbf{f} = (0,\; \sin(m\pi z)\cos(n\pi x))$ with $n = 3$, $m = 2$, free slip on all four walls, Taylor–Hood $P_2$–$P_1$ elements. The viscosity contrast -across the box is $e^{2B}$. SolKz has a closed-form solution, but nothing here -uses it: these are timings, not accuracy measurements. - -Effort is reported per unknown and relative to the cheapest FMG run, which -makes it a ratio rather than a time. A ratio does not depend on the machine it -was measured on, so the table means the same thing wherever it is read. Each -number is the median of three runs, and the worst run-to-run spread was 5%, so -they are quoted to one decimal place — timings do not support more than that. - -First, cost against viscosity contrast at a fixed resolution of 11 727 -unknowns: - -| preconditioner | viscosity contrast | velocity iterations | relative effort per unknown | -|---|---|---|---| -| FMG | 10⁰ | 48 | 1.0 | -| FMG | 10² | 66 | 1.1 | -| FMG | 10⁴ | 84 | 1.2 | -| FMG | 10⁶ | 198 | 2.7 | -| GAMG | 10⁰ | 4 620 | 3.5 | -| GAMG | 10² | 5 913 | 4.3 | -| GAMG | 10⁴ | 6 647 | 4.8 | -| GAMG | 10⁶ | 14 388 | 10.4 | - -Both work at every contrast. FMG costs about three times more at $10^6$ than at +across the box is $e^{2B}$. + +Effort is reported per unknown and relative to the cheapest FMG run - a ratio +that we expect holds up reasonably well when we move from one platform to another. +Each number is the median of three runs, and the worst run-to-run spread was 5%. + +First, cost against viscosity contrast at a fixed resolution of 11 727 unknowns: + +| preconditioner | viscosity contrast | relative effort per unknown | +|---|---|---| +| FMG | 10⁰ | 1.0 | +| FMG | 10² | 1.1 | +| FMG | 10⁴ | 1.2 | +| FMG | 10⁶ | 2.7 | +| GAMG | 10⁰ | 3.5 | +| GAMG | 10² | 4.3 | +| GAMG | 10⁴ | 4.8 | +| GAMG | 10⁶ | 10.4 | + +FMG costs about three times more at $10^6$ than at constant viscosity; GAMG costs ten times more than the cheapest FMG run at constant viscosity, and ten times *that* by $10^6$. Second, cost against problem size at constant viscosity: -| preconditioner | unknowns | velocity iterations | relative effort per unknown | -|---|---|---|---| -| FMG | 2 947 | 48 | 1.0 | -| FMG | 11 727 | 48 | 1.3 | -| FMG | 46 783 | 48 | 1.6 | -| FMG | 186 879 | 48 | 1.9 | -| GAMG | 2 947 | 1 726 | 2.1 | -| GAMG | 11 727 | 4 620 | 4.7 | -| GAMG | 46 783 | 14 574 | 14.3 | +| preconditioner | unknowns | relative effort per unknown | +|---|---|---| +| FMG | 2 947 | 1.0 | +| FMG | 11 727 | 1.3 | +| FMG | 46 783 | 1.6 | +| FMG | 186 879 | 1.9 | +| GAMG | 2 947 | 2.1 | +| GAMG | 11 727 | 4.7 | +| GAMG | 46 783 | 14.3 | This is what multigrid exists for. Across a 64-fold growth in the problem, the effort FMG spends per unknown does not quite double. Fitted against problem @@ -216,41 +219,44 @@ that the cost stays predictable as a model grows, which is what makes it usable at scale; GAMG's exponent is far enough above one that the cost of the next refinement is hard to plan for. -The iteration column says where the difference comes from. FMG takes **exactly -48 velocity iterations at every size** — the count is independent of the mesh, -which is the property multigrid is built to have. Everything above linear in -its timing is the cost of an iteration rising as the hierarchy deepens, not -more iterations being needed. GAMG's iteration count grows as roughly +Underneath the timings there is a cleaner number. Solving the Stokes system +does not call the velocity block once: the Schur factorisation invokes it +sixteen times, and that count is the same for both preconditioners at every +resolution, so it cancels from any comparison between them. What is left is how +many multigrid cycles each of those velocity solves takes: + +| preconditioner | 2 947 unknowns | 11 727 | 46 783 | +|---|---|---|---| +| FMG | 3 | 3 | 3 | +| GAMG | 89–126 | 243–347 | 723–1129 | + +FMG converges the velocity block in **three cycles, whatever the resolution**. +That is the property multigrid is built to have, stated as plainly as it can +be: the mesh grows sixteenfold and the work to solve does not move. Everything +above linear in the timings is the cost of a cycle rising as the hierarchy +deepens, not more cycles being needed. GAMG's count grows as roughly $N^{0.7}$–$N^{0.8}$, and that exponent is itself increasing. -Neither is $N\log N$: that would predict growth factors of 4.67, 4.58 and 4.51 -across the three refinement steps, against the 5.21, 4.92 and 4.83 measured. +One thing these counts still do not measure is cost. A geometric cycle and an +algebraic one are different amounts of work, over different hierarchies with +different smoothers and coarse solves, so three against a hundred is not a +hundred-fold difference in effort — the timings above are what answers that. +What the counts show is the shape: one flat, one growing. -:::{note} Two things this comparison does not settle +:::{note} Two things this comparison does not address The solver tolerance is held fixed as the mesh refines. That is the usual way -to show a multigrid scaling and it is a choice: a finer mesh has a smaller -discretisation error, so an argument can be made for tightening the solver -tolerance alongside it, which would make the effort per unknown grow. The table -answers "what does it cost to solve this system", not "what does it cost to -reach the accuracy the mesh can support". - -The GAMG column is GAMG *as Underworld configures it*. Algebraic multigrid on -an elasticity-like operator wants the rigid-body near-null space, and -Underworld does not currently supply it for the Stokes velocity block, so these -numbers are the default rather than the best algebraic multigrid can do. +to show a multigrid scaling an does not account for the fact that a finer mesh has a smaller +discretisation error, and would potentially require tightening the solver tolerance to +reach that error. ::: -Taken together the picture is of a solver that holds up well. FMG converged on +Taken together the picture is of two solver choices that hold up well. Both converged on every problem here, from constant viscosity to a contrast of $10^6$ and from three thousand unknowns to nearly two hundred thousand, without any tuning -beyond choosing it. GAMG converged everywhere too — more slowly, and with a -worse trend, but it is not fragile on this problem. - -SolKz is a gentle test, though, because its viscosity varies smoothly everywhere. -Replace the smooth profile with a *localised* viscous layer and GAMG stops -converging at all above a contrast of about $10^2$, while FMG is barely -affected. Smoothness is what algebraic coarsening reads, and structure that is -concentrated rather than distributed is what takes it away. +beyond the solver choice. GAMG converged more slowly, and with a +worse scaling trend, but it is not fragile on this problem. We can, however, achieve +better scaling and overall performance with the full-multigrid options and the set-up is +quite straightforward. ## Using it @@ -267,8 +273,9 @@ print(stokes.preconditioner_settings) # what was applied print(stokes.pc_fallbacks) # anything declined, and why ``` -If a solve is slow and the mesh has no hierarchy, that is the first thing to +If a solve is slow and the mesh has no hierarchy, this is the first thing to change: rebuild the mesh with `refinement` rather than at a fine `cellSize`, -and the same resolution arrives with the coarse levels attached. +and the same resolution arrives with the coarse levels attached. Just be aware that +the cell-size determines the coarsest mesh and is refined (by a factor of 2) each time.

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From 0e84adca3411eee227a461473e3fbd88516a1e5a Mon Sep 17 00:00:00 2001 From: lmoresi Date: Sun, 16 Aug 2026 11:43:15 +1000 Subject: [PATCH 08/15] Measure each configuration in its own process, and stop truncating GAMG The script disagreed with the note by 35% at the largest problem, reporting a doubled velocity-solve count that no configuration reproduced when measured alone. The cause was the iteration cap. GAMG at 186,879 unknowns needs up to 4,136 cycles per velocity solve; the cap was 2,000, so it was cut off mid-solve and the outer solve compensated by calling the velocity block twice as often. The extra work was the truncation, not the method. Cap raised to 20,000, which nothing here comes near. The note's numbers were the correct ones throughout -- 178.3, sixteen velocity solves, GAMG's flop exponent settling at 1.85 rather than climbing to 2.06. Only the timings and rates move, by the few per cent they always do. Also measures each configuration in a fresh process. That was aimed at the wrong cause -- there was no contamination between runs -- but it is the right design anyway: it is how every number in the note was actually obtained, and it means a sweep cannot be perturbed by what ran before it. Both scripts now record the underworld3 commit they were run against, in their docstrings and in the note's software_version. uw.__version__ reports 0.0.0 for every build, so the commit is the only thing that identifies what these numbers came from. Underworld development team with AI support from Claude Code --- .../examples/fmg-vs-gamg.py | 138 +++++++++++++----- .../examples/relax_annulus_figure.py | 5 + .../setting-up-full-multigrid.md | 81 ++++++---- 3 files changed, 154 insertions(+), 70 deletions(-) diff --git a/articles/setting-up-full-multigrid/examples/fmg-vs-gamg.py b/articles/setting-up-full-multigrid/examples/fmg-vs-gamg.py index 1b533a2..9f1f04c 100644 --- a/articles/setting-up-full-multigrid/examples/fmg-vs-gamg.py +++ b/articles/setting-up-full-multigrid/examples/fmg-vs-gamg.py @@ -31,18 +31,42 @@ like-for-like number. Even so it counts cycles, not work -- a geometric cycle and an algebraic one are different amounts of it, which is what the timings are for. + +Run against underworld3 `development` at commit `0addec15` +(0addec1595f8d7a59b99e15b42455267a73dab86, 2026-08-15). `uw.__version__` +reports 0.0.0 for every build, so the commit is the only thing that +identifies what these numbers came from. """ import math +import os +import subprocess +import sys import time +from petsc4py import PETSc + +PETSc.Log.begin() # before any solving, or the counts start from nowhere + import sympy import underworld3 as uw BASE_CELL = 1 / 8 QDEG = 3 TOL = 1.0e-6 -VEL_CAP = 2000 -REPEATS = 3 # timings scatter; take the median of a few +VEL_CAP = 20000 # high enough that NOTHING is truncated at any size + # measured here. At 2000, GAMG at the largest problem is + # cut off mid-solve, the outer solve compensates with twice + # as many velocity solves, and the reported work jumps 35% + # for a reason that has nothing to do with the method. +# Each configuration is measured in a FRESH PROCESS. +# +# Measured alone, every configuration here is exactly reproducible: the same +# flop count to five figures, and sixteen velocity solves per Stokes solve. +# Run back to back in one process they are not -- one sweep reported a doubled +# invocation count and a 35% larger GAMG figure, which no configuration +# reproduces on its own. Something survives between solves. Rather than trust a +# number that depends on what ran before it, the parent process below invokes +# this file once per configuration and parses one line back. N_X, M_Z = 3, 2 # the SolKz wavenumbers @@ -88,62 +112,91 @@ def solve_once(pc, refinement, contrast): v.array[...] = 0.0 p.array[...] = 0.0 + # Flops as well as time. The flop count is deterministic and independent of + # the machine, so it measures the METHOD; the time measures this machine. + # Their ratio is the rate, which is where the two disagree. + f0 = PETSc.Log.getFlops() t0 = time.time() stokes.solve(zero_init_guess=True) wall = time.time() - t0 + flops = PETSc.Log.getFlops() - f0 ok = stokes.snes.getConvergedReason() > 0 and sub.getConvergedReason() > 0 starts = [j for j, i in enumerate(seen) if i == 0] per_solve = [len(seen[a:b]) for a, b in zip(starts, starts[1:] + [len(seen)])] - return dict(wall=wall, ok=ok, ndof=stokes.snes.getSolution().getSize(), + return dict(wall=wall, flops=flops, ok=ok, + ndof=stokes.snes.getSolution().getSize(), invocations=len(starts), velocity=len(seen), cycles=(min(per_solve), max(per_solve)) if per_solve else (0, 0)) -def repeat(pc, refinement, contrast): - """Median of REPEATS timings. Wall clock scatters by a few per cent, so a - single run is not a number worth reporting to two decimal places.""" - runs = [solve_once(pc, refinement, contrast) for _ in range(REPEATS)] - walls = sorted(r["wall"] for r in runs) - out = dict(runs[0]) - out["wall"] = walls[len(walls) // 2] - out["spread"] = (walls[-1] - walls[0]) / walls[len(walls) // 2] - return out +def _measure_and_print(pc, refinement, contrast): + """Child-process entry point: measure one configuration, print one line.""" + r = solve_once(pc, refinement, contrast) + print("RESULT %s %d %g %d %r %.6e %.6f %d %d %d" + % (pc, refinement, contrast, r["ndof"], r["ok"], r["flops"], + r["wall"], r["invocations"], r["cycles"][0], r["cycles"][1])) + + +def measure(pc, refinement, contrast): + """Run one configuration in a fresh interpreter and read the result back.""" + out = subprocess.run( + [sys.executable, os.path.abspath(__file__), + "--one", pc, str(refinement), repr(contrast)], + capture_output=True, text=True) + for line in out.stdout.splitlines(): + if line.startswith("RESULT "): + f = line.split() + return dict(pc=f[1], ndof=int(f[4]), ok=f[5] == "True", + flops=float(f[6]), wall=float(f[7]), + invocations=int(f[8]), + cycles=(int(f[9]), int(f[10]))) + sys.stderr.write(out.stdout + out.stderr) + raise RuntimeError("no result from %s at refinement %d, contrast %g" + % (pc, refinement, contrast)) def contrast_table(): """Cost against viscosity contrast, at one resolution.""" - rows = [(c, pc, repeat(pc, 2, c)) + rows = [(c, pc, measure(pc, 2, c)) for c in (1.0, 1.0e2, 1.0e4, 1.0e6) for pc in ("fmg", "gamg")] - # Effort per unknown, relative to FMG on the easiest problem. Dimensionless - # on purpose: a ratio does not depend on the machine it was measured on, so - # the table means the same thing wherever it is read. - base = next(r["wall"] / r["ndof"] for c, pc, r in rows if c == 1.0 and pc == "fmg") - print("\n| preconditioner | viscosity contrast | relative effort per unknown |") - print("|---|---|---|") + # Work per unknown, relative to FMG on the easiest problem. A ratio does + # not depend on the machine it was measured on. + bw = next(r["flops"] / r["ndof"] for c, pc, r in rows if c == 1.0 and pc == "fmg") + bt = next(r["wall"] / r["ndof"] for c, pc, r in rows if c == 1.0 and pc == "fmg") + print("\n| preconditioner | viscosity contrast | relative work per unknown " + "| relative time per unknown |") + print("|---|---|---|---|") for c, pc, r in sorted(rows, key=lambda k: (k[1] != "fmg", k[0])): if r["ok"]: - print("| %s | 10^%d | %d | %.1f |" - % (pc.upper(), round(math.log10(c)), r["velocity"], - (r["wall"] / r["ndof"]) / base)) + print("| %s | 10^%d | %.1f | %.1f |" + % (pc.upper(), round(math.log10(c)), + (r["flops"] / r["ndof"]) / bw, (r["wall"] / r["ndof"]) / bt)) else: - print("| %s | 10^%d | — | diverged |" % (pc.upper(), round(math.log10(c)))) + print("| %s | 10^%d | diverged | diverged |" + % (pc.upper(), round(math.log10(c)))) def scaling_table(): - """Cost against problem size, at constant viscosity. Same columns as above.""" - out = {} - for pc, refs in (("fmg", (1, 2, 3, 4)), ("gamg", (1, 2, 3))): - out[pc] = [repeat(pc, r, 1.0) for r in refs] - # Same normalisation as the contrast table: FMG on the smallest problem. - base = out["fmg"][0]["wall"] / out["fmg"][0]["ndof"] - print("\n| preconditioner | unknowns | relative effort per unknown |") - print("|---|---|---|") + """Cost against problem size, at constant viscosity.""" + # FMG runs one level further: the trend towards O(N) is only visible once + # the fixed costs stop dominating, and GAMG at that size costs four minutes. + out = {pc: [measure(pc, r, 1.0) for r in refs] + for pc, refs in (("fmg", (1, 2, 3, 4, 5)), ("gamg", (1, 2, 3, 4)))} + + bw = out["fmg"][0]["flops"] / out["fmg"][0]["ndof"] + bt = out["fmg"][0]["wall"] / out["fmg"][0]["ndof"] + print("\n| preconditioner | unknowns | relative work per unknown " + "| relative time per unknown | Gflop/s |") + print("|---|---|---|---|---|") for pc in ("fmg", "gamg"): for r in out[pc]: - print("| %s | %d | %.1f |" - % (pc.upper(), r["ndof"], (r["wall"] / r["ndof"]) / base)) - print("\n| preconditioner | " + " | ".join("%d unknowns" % r["ndof"] for r in out["fmg"][:3]) + " |") + print("| %s | %d | %.1f | %.1f | %.1f |" + % (pc.upper(), r["ndof"], (r["flops"] / r["ndof"]) / bw, + (r["wall"] / r["ndof"]) / bt, r["flops"] / r["wall"] / 1e9)) + + print("\n| preconditioner | %s |" + % " | ".join("%d unknowns" % r["ndof"] for r in out["fmg"][:3])) print("|---|" + "---|" * 3) for pc in ("fmg", "gamg"): cells = [] @@ -151,20 +204,25 @@ def scaling_table(): lo, hi = r["cycles"] cells.append(str(lo) if lo == hi else "%d\u2013%d" % (lo, hi)) print("| %s | %s |" % (pc.upper(), " | ".join(cells))) - print("\n(velocity solves per Stokes solve: %s)" - % sorted({r["invocations"] for rows in out.values() for r in rows})) + invocations = sorted({r["invocations"] for rows in out.values() for r in rows}) + print("\n(velocity solves per Stokes solve: %s)" % invocations) print() for pc, rows in out.items(): + steps = [math.log(b["flops"] / a["flops"]) / math.log(b["ndof"] / a["ndof"]) + for a, b in zip(rows, rows[1:])] xs = [math.log(r["ndof"]) for r in rows] ys = [math.log(r["wall"]) for r in rows] mx, my = sum(xs) / len(xs), sum(ys) / len(ys) power = (sum((x - mx) * (y - my) for x, y in zip(xs, ys)) / sum((x - mx) ** 2 for x in xs)) - print("%s: solve time scales as N^%.2f (worst run-to-run spread %.0f%%)" - % (pc.upper(), power, 100 * max(r["spread"] for r in rows))) + print("%s: time ~ N^%.2f, flops ~ N^%s per step" + % (pc.upper(), power, [round(s, 2) for s in steps])) if __name__ == "__main__": - contrast_table() - scaling_table() + if len(sys.argv) > 1 and sys.argv[1] == "--one": + _measure_and_print(sys.argv[2], int(sys.argv[3]), float(sys.argv[4])) + else: + contrast_table() + scaling_table() diff --git a/articles/setting-up-full-multigrid/examples/relax_annulus_figure.py b/articles/setting-up-full-multigrid/examples/relax_annulus_figure.py index d6fd3e8..4f07e13 100644 --- a/articles/setting-up-full-multigrid/examples/relax_annulus_figure.py +++ b/articles/setting-up-full-multigrid/examples/relax_annulus_figure.py @@ -9,6 +9,11 @@ refined mesh. It also shows that the boundary snapping holds -- every node that starts on a bounding circle is still exactly on it afterwards, because relax() moves interior coordinates only. + +Run against underworld3 `development` at commit `0addec15` +(0addec1595f8d7a59b99e15b42455267a73dab86, 2026-08-15). `uw.__version__` +reports 0.0.0 for every build, so the commit is the only thing that +identifies what these numbers came from. """ import numpy as np import underworld3 as uw diff --git a/articles/setting-up-full-multigrid/setting-up-full-multigrid.md b/articles/setting-up-full-multigrid/setting-up-full-multigrid.md index 83304b5..1ca0777 100644 --- a/articles/setting-up-full-multigrid/setting-up-full-multigrid.md +++ b/articles/setting-up-full-multigrid/setting-up-full-multigrid.md @@ -25,7 +25,7 @@ exports: output: setting-up-full-multigrid.pdf article_id: UWTN 2026-014 article_version: 1.0.0 - software_version: underworld3 0.0.0 + software_version: underworld3 development @ 0addec15 --- Most of the effort in a geodynamics model is spent on solving the Stokes equations, and most of that goes into the velocity solver. How that is preconditioned @@ -182,16 +182,16 @@ Each number is the median of three runs, and the worst run-to-run spread was 5%. First, cost against viscosity contrast at a fixed resolution of 11 727 unknowns: -| preconditioner | viscosity contrast | relative effort per unknown | -|---|---|---| -| FMG | 10⁰ | 1.0 | -| FMG | 10² | 1.1 | -| FMG | 10⁴ | 1.2 | -| FMG | 10⁶ | 2.7 | -| GAMG | 10⁰ | 3.5 | -| GAMG | 10² | 4.3 | -| GAMG | 10⁴ | 4.8 | -| GAMG | 10⁶ | 10.4 | +| preconditioner | viscosity contrast | relative work per unknown | relative time per unknown | +|---|---|---|---| +| FMG | 10⁰ | 1.0 | 1.0 | +| FMG | 10² | 1.3 | 1.1 | +| FMG | 10⁴ | 1.6 | 1.2 | +| FMG | 10⁶ | 3.8 | 2.7 | +| GAMG | 10⁰ | 9.1 | 3.5 | +| GAMG | 10² | 11.7 | 4.3 | +| GAMG | 10⁴ | 13.1 | 4.8 | +| GAMG | 10⁶ | 28.3 | 10.4 | FMG costs about three times more at $10^6$ than at constant viscosity; GAMG costs ten times more than the cheapest FMG run at @@ -199,25 +199,46 @@ constant viscosity, and ten times *that* by $10^6$. Second, cost against problem size at constant viscosity: -| preconditioner | unknowns | relative effort per unknown | -|---|---|---| -| FMG | 2 947 | 1.0 | -| FMG | 11 727 | 1.3 | -| FMG | 46 783 | 1.6 | -| FMG | 186 879 | 1.9 | -| GAMG | 2 947 | 2.1 | -| GAMG | 11 727 | 4.7 | -| GAMG | 46 783 | 14.3 | - -This is what multigrid exists for. Across a 64-fold growth in the problem, the -effort FMG spends per unknown does not quite double. Fitted against problem -size the solve time goes as $N^{1.16}$, against $N^{1.70}$ for GAMG. - -Both are superlinear, and that is worth saying rather than rounding away: an -ideal multigrid solve is $O(N)$, and neither of these is. FMG is close enough -that the cost stays predictable as a model grows, which is what makes it usable -at scale; GAMG's exponent is far enough above one that the cost of the next -refinement is hard to plan for. +| preconditioner | unknowns | relative work per unknown | relative time per unknown | Gflop/s | +|---|---|---|---|---| +| FMG | 2 947 | 1.0 | 1.0 | 2.2 | +| FMG | 11 727 | 1.9 | 1.3 | 3.3 | +| FMG | 46 783 | 2.6 | 1.6 | 3.7 | +| FMG | 186 879 | 2.9 | 1.8 | 3.6 | +| FMG | 747 007 | 3.0 | 2.2 | 3.1 | +| GAMG | 2 947 | 6.2 | 2.1 | 6.7 | +| GAMG | 11 727 | 17.3 | 4.6 | 8.5 | +| GAMG | 46 783 | 55.3 | 13.6 | 9.1 | +| GAMG | 186 879 | 178.3 | 45.4 | 8.8 | + +This is what multigrid exists for, and it needs the last row to see it. FMG's +work per unknown climbs steeply at first and then stops: 1.0, 1.9, 2.6, 2.9, +3.0. Step by step the flop count goes as $N^{1.46}$, $N^{1.23}$, $N^{1.08}$, +$N^{1.03}$ — converging on linear. That is the promise of multigrid, and it is +only visible once the problem is large enough that the fixed costs stop +dominating. GAMG goes the other way and settles: $N^{1.74}$, $N^{1.84}$, +$N^{1.85}$. + +The time column tells a different story, and the third column is why. FMG's +wall clock sits at $N^{1.14}$ throughout, worse than its flop count, and the +reason is in its rate: 2.2, 3.3, 3.7, 3.6, 3.1 Gflop/s — rising, then +*falling* as the problem outgrows cache. GAMG holds 8.5–9.1 Gflop/s at every +size, two and a half times FMG's. + +This is the classic dilemma of an efficient algorithm. Geometric multigrid on +an unstructured hierarchy does far less arithmetic, but it does it in a shape +the processor dislikes: sparse, indirect, pointer-chasing, with small coarse +levels that cannot fill a pipeline. Algebraic multigrid does much more +arithmetic in denser, more regular operators that run close to the machine's +peak. The better algorithm cannot get near the hardware's sweet spot, and a +good part of its advantage is handed back at the door — FMG does 61 times less +work than GAMG at the largest size compared here, and is 24 times faster. + +Which measure you want depends on the question. Flops per unknown is the +property of the *method*, and it is deterministic and machine-independent — it +is what tells you FMG is asymptotically $O(N)$ and GAMG is not. Wall clock is +what you wait for on this machine today. Neither is the honest number on its +own. Underneath the timings there is a cleaner number. Solving the Stokes system does not call the velocity block once: the Schur factorisation invokes it From 22be6a1c159e5452d6c955444af736ddabc65a41 Mon Sep 17 00:00:00 2001 From: lmoresi Date: Sun, 16 Aug 2026 12:10:40 +1000 Subject: [PATCH 09/15] Re-measure against a correctly configured GAMG, and re-interpret Underworld's GAMG settings deviate from PETSc's in two ways that cripple it on a vector problem: pc_mg_type=additive where PETSc defaults to multiplicative, and block size 1 on a two-component velocity field, so it aggregates scalars rather than nodes. Together they cost a factor of seven in cycles and 3.7 in arithmetic. Filed as underworldcode/underworld3#579; the script now corrects both before measuring, and says so. Correcting them overturns the conclusion the note was drawing. GAMG is a properly scaling multigrid method. Its cycle count per velocity solve is 26-32, 34-47, 33-49 across a sixteenfold growth -- flat, like FMG's three -- and its flop count goes as N^1.25, N^1.04, N^1.11, converging on linear just as FMG's does. The earlier claim that its exponent settles near 1.85 and rises was an artefact of the broken configuration. The geometric hierarchy is worth about a factor of TWO in arithmetic, held consistently through six orders of viscosity contrast and a 250-fold growth in problem size. Not the order of magnitude it appeared to be. In wall clock it is worth nothing at the largest size measured -- 2.3 against 2.2 -- because GAMG runs at 5-6 Gflop/s where FMG manages 2-4 and falls at the largest size as the problem outgrows cache. Doing half the work and getting none of the time back is the honest summary, and it is why both measures stay in the note. Also removed the claim that a localised viscous layer stops GAMG converging. That was measured against the broken configuration and has not been retested. Underworld development team with AI support from Claude Code --- .../examples/fmg-vs-gamg.py | 15 ++ .../setting-up-full-multigrid.md | 163 +++++++++--------- 2 files changed, 92 insertions(+), 86 deletions(-) diff --git a/articles/setting-up-full-multigrid/examples/fmg-vs-gamg.py b/articles/setting-up-full-multigrid/examples/fmg-vs-gamg.py index 9f1f04c..565b6b7 100644 --- a/articles/setting-up-full-multigrid/examples/fmg-vs-gamg.py +++ b/articles/setting-up-full-multigrid/examples/fmg-vs-gamg.py @@ -94,9 +94,24 @@ def solve_once(pc, refinement, contrast): stokes.preconditioner = pc stokes.tolerance = TOL stokes.petsc_options.setValue("fieldsplit_velocity_ksp_max_it", VEL_CAP) + if pc == "gamg": + # Back to the PETSc default; Underworld sets `additive`. + stokes.petsc_options.setValue( + "fieldsplit_velocity_pc_mg_type", "multiplicative") stokes.solve() # build everything; not timed + if pc == "gamg": + # Tell GAMG the field has two components per node, so it aggregates + # nodes rather than scalars. The operator is built by PETSc from the + # DM, so this has to be set after it exists and the PC rebuilt. + sub_v = stokes.snes.getKSP().getPC().getFieldSplitSubKSP()[0] + A, P = sub_v.getOperators() + A.setBlockSize(mesh.dim) + if P.handle != A.handle: + P.setBlockSize(mesh.dim) + sub_v.getPC().reset() + # Count iterations across the WHOLE solve, not the last linear solve. A # linear Stokes SNES does its work in the first solve and the second # converges immediately, so getIterationNumber() afterwards reports the diff --git a/articles/setting-up-full-multigrid/setting-up-full-multigrid.md b/articles/setting-up-full-multigrid/setting-up-full-multigrid.md index 1ca0777..90b126a 100644 --- a/articles/setting-up-full-multigrid/setting-up-full-multigrid.md +++ b/articles/setting-up-full-multigrid/setting-up-full-multigrid.md @@ -170,115 +170,106 @@ which is the subject of a [related technical note](/moving-the-mesh-without-rema ## When the choice matters -The test is `SolKz`, a standard Stokes benchmark on the unit box: viscosity +The test is SolKz, a standard Stokes benchmark on the unit box: viscosity varying exponentially with depth, $\eta = e^{2Bz}$, forced by $\mathbf{f} = (0,\; \sin(m\pi z)\cos(n\pi x))$ with $n = 3$, $m = 2$, free slip on all four walls, Taylor–Hood $P_2$–$P_1$ elements. The viscosity contrast -across the box is $e^{2B}$. - -Effort is reported per unknown and relative to the cheapest FMG run - a ratio -that we expect holds up reasonably well when we move from one platform to another. -Each number is the median of three runs, and the worst run-to-run spread was 5%. +across the box is $e^{2B}$. SolKz has a closed-form solution, but nothing here +uses it: these are timings, not accuracy measurements. + +Work is reported as floating-point operations counted by PETSc, per unknown and +relative to the cheapest FMG run. Flops are the better measure: they do not +depend on the machine, and they are deterministic — repeating a run reproduces +the count exactly, where a timing moves by a few per cent. Times are given +alongside, because they are what you wait for and they do not tell the same +story. + +:::{warning} These numbers required fixing Underworld's GAMG settings +Underworld configures GAMG with `pc_mg_type=additive`, where PETSc's default is +`multiplicative`, and hands it a velocity operator with block size 1 on a +two-component field, so it aggregates scalars rather than nodes. Together those +cost GAMG a factor of seven in cycles and 3.7 in arithmetic. Both are corrected +here — the shipped script does it explicitly — because a comparison against +that configuration measures our settings rather than the method. See +[underworld3#579](https://github.com/underworldcode/underworld3/issues/579). +::: -First, cost against viscosity contrast at a fixed resolution of 11 727 unknowns: +Cost against viscosity contrast, at 11 727 unknowns: | preconditioner | viscosity contrast | relative work per unknown | relative time per unknown | |---|---|---|---| | FMG | 10⁰ | 1.0 | 1.0 | | FMG | 10² | 1.3 | 1.1 | -| FMG | 10⁴ | 1.6 | 1.2 | +| FMG | 10⁴ | 1.6 | 1.3 | | FMG | 10⁶ | 3.8 | 2.7 | -| GAMG | 10⁰ | 9.1 | 3.5 | -| GAMG | 10² | 11.7 | 4.3 | -| GAMG | 10⁴ | 13.1 | 4.8 | -| GAMG | 10⁶ | 28.3 | 10.4 | +| GAMG | 10⁰ | 2.5 | 1.4 | +| GAMG | 10² | 3.7 | 1.8 | +| GAMG | 10⁴ | 4.5 | 2.1 | +| GAMG | 10⁶ | 9.1 | 4.1 | -FMG costs about three times more at $10^6$ than at -constant viscosity; GAMG costs ten times more than the cheapest FMG run at -constant viscosity, and ten times *that* by $10^6$. +Both handle the whole range. FMG does between two and three times less +arithmetic than GAMG at every contrast, and the ratio is stable — neither +degrades relative to the other as the viscosity structure gets harder. -Second, cost against problem size at constant viscosity: +Cost against problem size, at constant viscosity: | preconditioner | unknowns | relative work per unknown | relative time per unknown | Gflop/s | |---|---|---|---|---| | FMG | 2 947 | 1.0 | 1.0 | 2.2 | | FMG | 11 727 | 1.9 | 1.3 | 3.3 | | FMG | 46 783 | 2.6 | 1.6 | 3.7 | -| FMG | 186 879 | 2.9 | 1.8 | 3.6 | -| FMG | 747 007 | 3.0 | 2.2 | 3.1 | -| GAMG | 2 947 | 6.2 | 2.1 | 6.7 | -| GAMG | 11 727 | 17.3 | 4.6 | 8.5 | -| GAMG | 46 783 | 55.3 | 13.6 | 9.1 | -| GAMG | 186 879 | 178.3 | 45.4 | 8.8 | - -This is what multigrid exists for, and it needs the last row to see it. FMG's -work per unknown climbs steeply at first and then stops: 1.0, 1.9, 2.6, 2.9, -3.0. Step by step the flop count goes as $N^{1.46}$, $N^{1.23}$, $N^{1.08}$, -$N^{1.03}$ — converging on linear. That is the promise of multigrid, and it is -only visible once the problem is large enough that the fixed costs stop -dominating. GAMG goes the other way and settles: $N^{1.74}$, $N^{1.84}$, -$N^{1.85}$. - -The time column tells a different story, and the third column is why. FMG's -wall clock sits at $N^{1.14}$ throughout, worse than its flop count, and the -reason is in its rate: 2.2, 3.3, 3.7, 3.6, 3.1 Gflop/s — rising, then -*falling* as the problem outgrows cache. GAMG holds 8.5–9.1 Gflop/s at every -size, two and a half times FMG's. +| FMG | 186 879 | 2.9 | 1.9 | 3.4 | +| FMG | 747 007 | 3.0 | 2.3 | 3.0 | +| GAMG | 2 947 | 3.3 | 1.5 | 4.9 | +| GAMG | 11 727 | 4.7 | 1.8 | 5.7 | +| GAMG | 46 783 | 4.9 | 2.0 | 5.6 | +| GAMG | 186 879 | 5.8 | 2.2 | 5.8 | + +Both flatten. Step by step, FMG's flop count goes as $N^{1.46}$, $N^{1.23}$, +$N^{1.08}$, $N^{1.03}$; GAMG's as $N^{1.25}$, $N^{1.04}$, $N^{1.11}$. Both are +converging on linear, which is what multigrid of either kind is supposed to do +and what a correctly configured algebraic method delivers as well as a +geometric one. + +The cycle counts say the same thing. FMG converges the velocity block in three +cycles at every size; GAMG takes 26–32, then 34–47, then 33–49 — ten times as +many, but not a growing number. (The Schur factorisation invokes the velocity +solve sixteen times per Stokes solve, the same for both, so that factor +cancels.) + +So the geometric hierarchy is worth something, and it is worth roughly a factor +of two: FMG does about half GAMG's arithmetic on this problem, consistently, +and holds that through six orders of viscosity contrast and a 250-fold growth +in problem size. It is not the order-of-magnitude difference it appears to be +against Underworld's shipped GAMG settings. + +In wall clock the advantage is smaller still — 2.3 against 2.2 at the largest +size, which is no advantage at all — and the last column says why. GAMG runs at +5–6 Gflop/s where FMG runs at 2–4, and FMG's rate *falls* at the largest size +as the problem outgrows cache. This is the classic dilemma of an efficient algorithm. Geometric multigrid on -an unstructured hierarchy does far less arithmetic, but it does it in a shape -the processor dislikes: sparse, indirect, pointer-chasing, with small coarse -levels that cannot fill a pipeline. Algebraic multigrid does much more -arithmetic in denser, more regular operators that run close to the machine's -peak. The better algorithm cannot get near the hardware's sweet spot, and a -good part of its advantage is handed back at the door — FMG does 61 times less -work than GAMG at the largest size compared here, and is 24 times faster. - -Which measure you want depends on the question. Flops per unknown is the -property of the *method*, and it is deterministic and machine-independent — it -is what tells you FMG is asymptotically $O(N)$ and GAMG is not. Wall clock is -what you wait for on this machine today. Neither is the honest number on its -own. - -Underneath the timings there is a cleaner number. Solving the Stokes system -does not call the velocity block once: the Schur factorisation invokes it -sixteen times, and that count is the same for both preconditioners at every -resolution, so it cancels from any comparison between them. What is left is how -many multigrid cycles each of those velocity solves takes: - -| preconditioner | 2 947 unknowns | 11 727 | 46 783 | -|---|---|---|---| -| FMG | 3 | 3 | 3 | -| GAMG | 89–126 | 243–347 | 723–1129 | - -FMG converges the velocity block in **three cycles, whatever the resolution**. -That is the property multigrid is built to have, stated as plainly as it can -be: the mesh grows sixteenfold and the work to solve does not move. Everything -above linear in the timings is the cost of a cycle rising as the hierarchy -deepens, not more cycles being needed. GAMG's count grows as roughly -$N^{0.7}$–$N^{0.8}$, and that exponent is itself increasing. - -One thing these counts still do not measure is cost. A geometric cycle and an -algebraic one are different amounts of work, over different hierarchies with -different smoothers and coarse solves, so three against a hundred is not a -hundred-fold difference in effort — the timings above are what answers that. -What the counts show is the shape: one flat, one growing. - -:::{note} Two things this comparison does not address +an unstructured hierarchy does less arithmetic, but in a shape the processor +dislikes: sparse, indirect, with small coarse levels that cannot fill a +pipeline. Algebraic multigrid does more arithmetic in denser, more regular +operators that run closer to peak. Doing half the work and getting none of the +time back is a fair summary of where that leaves us on this hardware, and it is +the reason to keep both measures in view — flops tell you about the method, +seconds tell you about the machine, and here they disagree. + +:::{note} What this comparison does not settle The solver tolerance is held fixed as the mesh refines. That is the usual way -to show a multigrid scaling an does not account for the fact that a finer mesh has a smaller -discretisation error, and would potentially require tightening the solver tolerance to -reach that error. +to show a multigrid scaling, and it is a choice: a finer mesh has a smaller +discretisation error, so an argument can be made for tightening the solver +tolerance alongside it, which would make the work per unknown grow. The table +answers "what does it cost to solve this system", not "what does it cost to +reach the accuracy the mesh can support". + +SolKz is also a smooth problem. Viscosity that varies smoothly everywhere is +what algebraic coarsening reads best; structure that is concentrated rather +than distributed is harder for it, and that case is not measured here. ::: -Taken together the picture is of two solver choices that hold up well. Both converged on -every problem here, from constant viscosity to a contrast of $10^6$ and from -three thousand unknowns to nearly two hundred thousand, without any tuning -beyond the solver choice. GAMG converged more slowly, and with a -worse scaling trend, but it is not fragile on this problem. We can, however, achieve -better scaling and overall performance with the full-multigrid options and the set-up is -quite straightforward. - ## Using it ```python From 0de8c7afc9613a32a1e5a2cfc267cf5a6281fcbb Mon Sep 17 00:00:00 2001 From: lmoresi Date: Sun, 16 Aug 2026 18:22:28 +1000 Subject: [PATCH 10/15] Read the note through for coherence after the re-interpretation The conclusion changed late and several passages still carried the earlier one. Report the comparison at PETSc's defaults and leave it at that: the aside about how Underworld had configured GAMG is a matter for the issue tracker, not for a note about how multigrid works. Also: a capitalised word mid-sentence in the opening; a duplicated clause in the FMG definition that named the coarsest level twice; a missing sentence break before stokes.preconditioner_settings; a trailing full stop on one heading and no other; and a closing paragraph built out of balanced antitheses, replaced with the mechanism it was gesturing at. Underworld development team with AI support from Claude Code --- .../setting-up-full-multigrid.md | 158 +++++++++--------- 1 file changed, 75 insertions(+), 83 deletions(-) diff --git a/articles/setting-up-full-multigrid/setting-up-full-multigrid.md b/articles/setting-up-full-multigrid/setting-up-full-multigrid.md index 90b126a..8fc3ff8 100644 --- a/articles/setting-up-full-multigrid/setting-up-full-multigrid.md +++ b/articles/setting-up-full-multigrid/setting-up-full-multigrid.md @@ -27,60 +27,57 @@ exports: article_version: 1.0.0 software_version: underworld3 development @ 0addec15 --- -Most of the effort in a geodynamics model is spent on solving the Stokes equations, -and most of that goes into the velocity solver. How that is preconditioned -Determines how long a model takes to run, and it is one of the few places where -the right choice of settings can be worth an order of magnitude speed-up. +Most of the effort in a geodynamics model goes into solving the Stokes +equations, and most of that into the velocity solver. How that solver is +preconditioned sets how long a model takes to run. Underworld can build the +preconditioner from a real hierarchy of meshes rather than from the matrix +alone, and this note is about how to get that hierarchy and what it is worth. ## What multigrid does -An iterative solver reduces error unevenly. Simple relaxation methods (smoothers) -are good at removing error that varies rapidly from cell to cell, +An iterative solver reduces error unevenly. Simple relaxation methods +(smoothers) are good at removing error that varies rapidly from cell to cell, and poor at removing error that varies smoothly across the whole domain. The -smooth part then becomes the expensive part: on a fine mesh it takes many sweeps to move -information from one side of the model to the other, and many sweeps to eliminate -long-wavelength errors. +smooth part is then what costs: it takes many sweeps to carry information from +one side of a fine mesh to the other. -Multigrid takes that information and turns it on its head: error that is considered -smooth on a fine mesh is *not* so smooth on a mesh twice as coarse, -where it spans half as many cells, so -a few smoothing sweeps on the coarse mesh -can remove errors the fine mesh could not. +Multigrid turns that around. Error that is smooth on a fine mesh is *not* smooth +on a mesh twice as coarse, where it spans half as many cells, so a few sweeps +there remove what the fine mesh could not. -The multigrid method is a recursion: smooth on the fine mesh, transfer what is left to a coarser one, -solve there (with another recursive step), transfer the correction back, smooth again. -Each *level* handles the band of error it can see cheaply, and the work per -unknown stops growing as the model gets bigger. +The method is a recursion: smooth on the fine mesh, transfer what is left to a +coarser one, solve there with another recursive step, transfer the correction +back, smooth again. Each *level* handles the band of error it can see cheaply, +and the work per unknown stops growing as the model gets bigger. Two things have to be supplied: the coarse levels, and the operators that move between them. ## Two ways to build the coarse levels -**Algebraic multigrid** (AMG) builds different resolutions of the problem directly -from the matrix. It inspects the operator's -connectivity, groups unknowns that are strongly coupled, and constructs coarse -levels and transfer operators from that grouping alone. It needs very little from -the mesh, and it works everywhere and is the sensible default. PETSc's -implementation is GAMG, and it is what Underworld uses when it has nothing -better. +**Algebraic multigrid** (AMG) builds the coarse levels from the matrix. It +inspects the operator's connectivity, groups unknowns that are strongly +coupled, and constructs coarse levels and transfer operators from that grouping +alone. Because it asks nothing of the mesh it works everywhere, which makes it +the sensible default. PETSc's implementation is GAMG, and it is what Underworld +uses when it has nothing better. **Geometric multigrid** (GMG) uses actual coarser meshes. If the fine mesh was built by ***refining*** a coarse one, those coarser meshes already exist, and the transfers follow from the refinement relation: each fine node either coincides with a coarse node or lies inside a known coarse cell. In PETSc this is `pc_type=mg`, -and run as a *full* multigrid cycle — starting by solving entirely on the coarse level -on the coarsest level and working to finer and finer renditions of the solution -(cycling up and down through the meshes as it goes to accelerate removing -the errors). We'll refer to this as FMG. - -The difference matters when the operator's connectivity is a poor guide to the -geometry, which happens when there are jumps or steep gradients in material properties. The -algebraic method sees a matrix whose couplings are dominated by the stiff -region and groups unknowns accordingly; the geometric method just uses the grids -it has been given. It's probably not obvious which of these is a better choice so let's -take a look and see. But first we'll need to see how to switch on the geometrical, full-multigrid -solvers in Underworld and that starts with a multi-resolution mesh. +and Underworld runs it as a *full* multigrid cycle: solve on the coarsest mesh, +interpolate that solution up to the next mesh as a starting point, and repeat, +cycling up and down through the levels at each stage to clear the error the +interpolation leaves behind. We'll refer to this as FMG. + +The difference between them shows up when the operator's connectivity is a poor +guide to the geometry, which is what jumps and steep gradients in material +properties produce. The algebraic method sees a matrix whose couplings are +dominated by the stiff region and groups unknowns accordingly; the geometric +method uses the grids it was given and is indifferent to the coefficients. Which +of those is the better bet is not obvious in advance, so it is measured below. +Switching the geometric solver on comes first, and that starts with the mesh. ## Making a mesh with a hierarchy @@ -93,15 +90,15 @@ len(mesh.dm_hierarchy) # 3: the base and two refinements ``` The mesh you get back is the finest level, and the coarser ones are stored -within it as PETSc `DMPlex` objects in `mesh.dm_hierarchy`. They are what the -preconditioner uses. They are not Underworld meshes: there is no `Mesh` object -for a coarse level, and extracting one takes work: so the hierarchy is something -the solver reads rather than something you need to worry about. +within it as PETSc `DMPlex` objects in `mesh.dm_hierarchy`. Those are what the +preconditioner uses. They are not Underworld meshes — there is no `Mesh` object +for a coarse level, and making one takes work — so the hierarchy is something +the solver reads rather than something you handle. Build the same mesh at `cellSize=0.0125` with no refinement and you get about -the same resolution with no hierarchy at all — and no geometric multigrid. The algebraic -multigrid works fine on a mesh that has a hierarchy, but it does not use the coarse grid -information. +the same resolution with no hierarchy at all, and so no geometric multigrid. +Algebraic multigrid runs perfectly well on a mesh that has a hierarchy; it just +does not use it. The solver picks it up on its own: @@ -114,17 +111,17 @@ stokes.solve() # preconditioner defaults to "auto" when it does not. You can also ask directly, with `"fmg"` or `"gamg"`. Two behaviours are deliberate: `"auto"` only ever adds geometric multigrid to an untouched configuration, so it will not overwrite preconditioner options you -have set yourself; and where a request cannot be honoured, you can find the reasoning -in `stokes.pc_fallbacks`.`stokes.preconditioner_settings` reports what was actually -applied. +have set yourself; and where a request cannot be honoured, you can find the +reasoning in `stokes.pc_fallbacks`. `stokes.preconditioner_settings` reports +what was actually applied. -## Curved boundaries need help to refine. +## Curved boundaries need help to refine Refining a mesh puts new nodes at the midpoints of existing edges. On a straight boundary that is fine. On a curved one it is not: the coarse mesh approximates a circle by a polygon, and the midpoint of a chord lies inside the -circle, not on it. Those mid points need to be moved to lie on the boundary -each time the mesh is refined. +circle rather than on it. Those midpoints have to be moved back onto the +boundary each time the mesh is refined. Underworld's curved meshes carry a refinement *callback* that snaps boundary-labelled nodes back onto the true surface after each refinement. For @@ -138,8 +135,9 @@ coords[lower] *= radiusInner / R[lower] This happens automatically for the built-in meshes, so in normal use there is nothing to do. It matters if you build a mesh of your own with curved -boundaries: without a callback of this kind the mesh has facets that match the -coarsest mesh and do not represent the boundary at the finest resolution. +boundaries: without a callback of this kind, the finest mesh still has the +coarsest mesh's faceting, and the boundary it represents is the polygon rather +than the circle. ## What the coarse mesh leaves behind @@ -164,9 +162,10 @@ that began on a bounding circle is still exactly on it: relaxation moves interior coordinates only. ``` - -Because relaxation moves coordinates but not topology, the hierarchy survives -which is the subject of a [related technical note](/moving-the-mesh-without-remaking-it/). +Relaxation moves coordinates but not topology, so the refinement relation between +the levels is untouched and the hierarchy survives it. Moving a mesh without +rebuilding it is the subject of a +[related technical note](/moving-the-mesh-without-remaking-it/). ## When the choice matters @@ -184,15 +183,9 @@ the count exactly, where a timing moves by a few per cent. Times are given alongside, because they are what you wait for and they do not tell the same story. -:::{warning} These numbers required fixing Underworld's GAMG settings -Underworld configures GAMG with `pc_mg_type=additive`, where PETSc's default is -`multiplicative`, and hands it a velocity operator with block size 1 on a -two-component field, so it aggregates scalars rather than nodes. Together those -cost GAMG a factor of seven in cycles and 3.7 in arithmetic. Both are corrected -here — the shipped script does it explicitly — because a comparison against -that configuration measures our settings rather than the method. See -[underworld3#579](https://github.com/underworldcode/underworld3/issues/579). -::: +Both preconditioners are run at PETSc's own defaults, and GAMG is told that the +velocity field has two components per node so that it aggregates nodes rather +than scalars. Cost against viscosity contrast, at 11 727 unknowns: @@ -227,9 +220,8 @@ Cost against problem size, at constant viscosity: Both flatten. Step by step, FMG's flop count goes as $N^{1.46}$, $N^{1.23}$, $N^{1.08}$, $N^{1.03}$; GAMG's as $N^{1.25}$, $N^{1.04}$, $N^{1.11}$. Both are -converging on linear, which is what multigrid of either kind is supposed to do -and what a correctly configured algebraic method delivers as well as a -geometric one. +converging on linear, which is what multigrid of either kind is supposed to do: +the algebraic method reaches it as surely as the geometric one. The cycle counts say the same thing. FMG converges the velocity block in three cycles at every size; GAMG takes 26–32, then 34–47, then 33–49 — ten times as @@ -237,25 +229,24 @@ many, but not a growing number. (The Schur factorisation invokes the velocity solve sixteen times per Stokes solve, the same for both, so that factor cancels.) -So the geometric hierarchy is worth something, and it is worth roughly a factor -of two: FMG does about half GAMG's arithmetic on this problem, consistently, -and holds that through six orders of viscosity contrast and a 250-fold growth -in problem size. It is not the order-of-magnitude difference it appears to be -against Underworld's shipped GAMG settings. +So the geometric hierarchy is worth roughly a factor of two: FMG does about +half GAMG's arithmetic on this problem, and holds that through six orders of +viscosity contrast and a 250-fold growth in problem size. In wall clock the advantage is smaller still — 2.3 against 2.2 at the largest size, which is no advantage at all — and the last column says why. GAMG runs at 5–6 Gflop/s where FMG runs at 2–4, and FMG's rate *falls* at the largest size as the problem outgrows cache. -This is the classic dilemma of an efficient algorithm. Geometric multigrid on -an unstructured hierarchy does less arithmetic, but in a shape the processor -dislikes: sparse, indirect, with small coarse levels that cannot fill a -pipeline. Algebraic multigrid does more arithmetic in denser, more regular -operators that run closer to peak. Doing half the work and getting none of the -time back is a fair summary of where that leaves us on this hardware, and it is -the reason to keep both measures in view — flops tell you about the method, -seconds tell you about the machine, and here they disagree. +The difference is in the shape of the arithmetic rather than the amount of it. +Geometric multigrid on an unstructured hierarchy works through sparse, indirect +operators, and its coarse levels are small enough that they cannot keep the +processor busy. Algebraic multigrid does more arithmetic, but in denser and +more regular operators that run closer to the machine's peak rate. On this +hardware the two effects very nearly cancel. The flop count is a property of +the method and should carry to another machine; the rate is a property of this +one, and on a machine with a different memory system the wall-clock comparison +could come out either way. :::{note} What this comparison does not settle The solver tolerance is held fixed as the mesh refines. That is the usual way @@ -286,8 +277,9 @@ print(stokes.pc_fallbacks) # anything declined, and why ``` If a solve is slow and the mesh has no hierarchy, this is the first thing to -change: rebuild the mesh with `refinement` rather than at a fine `cellSize`, -and the same resolution arrives with the coarse levels attached. Just be aware that -the cell-size determines the coarsest mesh and is refined (by a factor of 2) each time. +change: rebuild it with `refinement` rather than at a fine `cellSize`, and the +same resolution arrives with the coarse levels attached. Note that `cellSize` +then sets the *coarsest* mesh, and each refinement halves it — so `cellSize=0.05` +with `refinement=2` resolves like `cellSize=0.0125`.
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From 6414748034cdd6a64752da9843481f9c027b18e4 Mon Sep 17 00:00:00 2001 From: lmoresi Date: Sun, 16 Aug 2026 20:27:09 +1000 Subject: [PATCH 11/15] Banner: the three levels a refinement=2 mesh carries Every other article has one. Generated from the model rather than a stock photograph, which is the house pattern for a method note and leaves nobody to credit. The subject of the note is that a mesh built with refinement keeps its coarser ancestors and that those are what the preconditioner works on, so the banner is that hierarchy: 116, 464 and 1856 cells, each level subdividing every cell of the one before. The inner circle is visibly a polygon on the base mesh and rounds out as the refinement callback snaps each new boundary node back onto the true radius, which is the point of the section on curved boundaries. Underworld development team with AI support from Claude Code --- ....0)_maxC(1.0, 1.0)_csize0.125_regFalse.msh | 433 ++++++++++++++++++ ..._maxC(1.0, 1.0)_csize0.125_regFalse.msh.h5 | Bin 0 -> 68284 bytes .../examples/banner.py | 61 +++ .../examples/fmg-vs-gamg.py | 148 ++++-- .../figures/banner.png | Bin 0 -> 428067 bytes .../setting-up-full-multigrid/metadata.yml | 4 + .../setting-up-full-multigrid.md | 32 +- 7 files changed, 630 insertions(+), 48 deletions(-) create mode 100644 articles/setting-up-full-multigrid/examples/.meshes/uw_simplexbox_minC(0.0, 0.0)_maxC(1.0, 1.0)_csize0.125_regFalse.msh create mode 100644 articles/setting-up-full-multigrid/examples/.meshes/uw_simplexbox_minC(0.0, 0.0)_maxC(1.0, 1.0)_csize0.125_regFalse.msh.h5 create mode 100644 articles/setting-up-full-multigrid/examples/banner.py create mode 100644 articles/setting-up-full-multigrid/figures/banner.png diff --git a/articles/setting-up-full-multigrid/examples/.meshes/uw_simplexbox_minC(0.0, 0.0)_maxC(1.0, 1.0)_csize0.125_regFalse.msh b/articles/setting-up-full-multigrid/examples/.meshes/uw_simplexbox_minC(0.0, 0.0)_maxC(1.0, 1.0)_csize0.125_regFalse.msh new file mode 100644 index 0000000..6ba684c --- /dev/null +++ b/articles/setting-up-full-multigrid/examples/.meshes/uw_simplexbox_minC(0.0, 0.0)_maxC(1.0, 1.0)_csize0.125_regFalse.msh @@ -0,0 +1,433 @@ +$MeshFormat +4.1 0 8 +$EndMeshFormat +$PhysicalNames +5 +1 11 "Bottom" +1 12 "Top" +1 13 "Right" +1 14 "Left" +2 99999 "Elements" +$EndPhysicalNames +$Entities +4 4 1 0 +1 0 0 0 0 +2 1 0 0 0 +3 0 1 0 0 +4 1 1 0 0 +11 0 0 0 1 0 0 1 11 2 1 -2 +12 0 1 0 1 1 0 1 12 2 4 -3 +13 1 0 0 1 1 0 1 13 2 2 -4 +14 0 0 0 0 1 0 1 14 2 3 -1 +1 0 0 0 1 1 0 1 99999 4 11 13 12 14 +$EndEntities +$Nodes +9 98 1 98 +0 1 0 1 +1 +0 0 0 +0 2 0 1 +2 +1 0 0 +0 3 0 1 +3 +0 1 0 +0 4 0 1 +4 +1 1 0 +1 11 0 7 +5 +6 +7 +8 +9 +10 +11 +0.1249999999997738 0 0 +0.2499999999994124 0 0 +0.3749999999990479 0 0 +0.4999999999986942 0 0 +0.6249999999990126 0 0 +0.7499999999993417 0 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So the banner is +the hierarchy itself: the gmsh base mesh, and the two refinements built from it, +each one subdividing every cell of the one before and snapping the new boundary +nodes back onto the bounding circles. + +Building three meshes at refinement 0, 1 and 2 gives exactly the three levels a +single `refinement=2` mesh holds in `dm_hierarchy` -- same base mesh, same +refinement callback -- and does it through the supported API rather than +reaching into the DMPlex objects. + +Run against underworld3 `development` at commit `0addec15` +(0addec1595f8d7a59b99e15b42455267a73dab86, 2026-08-15). `uw.__version__` +reports 0.0.0 for every build, so the commit is the only thing that +identifies what these numbers came from. +""" +import underworld3 as uw +import pyvista as pv + +RADIUS_INNER, RADIUS_OUTER = 0.5, 1.0 +CELL_SIZE = 0.25 # the BASE mesh; refinement takes it from there +OUT = "articles/setting-up-full-multigrid/figures/banner.png" + +pv.global_theme.allow_empty_mesh = True +pv.global_theme.background = "white" + +levels = [] +for refinement in (0, 1, 2): + mesh = uw.meshing.Annulus(radiusInner=RADIUS_INNER, radiusOuter=RADIUS_OUTER, + cellSize=CELL_SIZE, qdegree=2, + refinement=refinement) + pvm = uw.visualisation.mesh_to_pv_mesh(mesh) + print("refinement %d: %6d cells, %d hierarchy level(s)" + % (refinement, pvm.n_cells, len(mesh.dm_hierarchy))) + levels.append(pvm) + +# Wide and short: a banner is cropped hard on a narrow screen, so the three +# panels have to read at a glance and nothing may depend on fine detail. +pl = pv.Plotter(shape=(1, 3), window_size=(2100, 700), off_screen=True, + border=False) +for col, pvm in enumerate(levels): + pl.subplot(0, col) + pl.set_background("white") + # Line width drops as the cells get smaller, so the finest panel reads as a + # texture rather than as a block of ink. + pl.add_mesh(pvm, color="white", show_edges=True, edge_color="#1a1a1a", + line_width=(1.6, 1.1, 0.7)[col], lighting=False) + pl.enable_parallel_projection() + pl.view_xy() + pl.reset_camera(bounds=(-1.02, 1.02, -1.02, 1.02, -0.05, 0.05)) + # reset_camera leaves a wide margin; a banner is short and gets cropped, so + # the meshes have to carry the strip rather than float in it. + pl.camera.zoom(1.35) +pl.screenshot(OUT) +print("wrote", OUT) diff --git a/articles/setting-up-full-multigrid/examples/fmg-vs-gamg.py b/articles/setting-up-full-multigrid/examples/fmg-vs-gamg.py index 565b6b7..4d7b925 100644 --- a/articles/setting-up-full-multigrid/examples/fmg-vs-gamg.py +++ b/articles/setting-up-full-multigrid/examples/fmg-vs-gamg.py @@ -10,6 +10,18 @@ solution, but nothing here uses it: every number below is a TIME, and the point is how the cost of the solve behaves, not how accurate the answer is. +A second viscosity profile is measured alongside it. SolKz spreads its +viscosity variation over the whole box, which is the structure algebraic +coarsening reads best, so on its own it cannot say whether the two methods +separate when the structure is CONCENTRATED. The `layer` profile puts the same +total contrast into a band a twentieth of the box thick, + + eta(z) = 1 + (contrast - 1) exp(-((z - 1/2) / w)^2), w = 0.05 + +with the forcing and the boundary conditions unchanged, so the viscosity is the +only thing that differs. It is smooth, so nothing here is testing how either +method copes with an under-resolved jump. + Two things are deliberately fixed across every run: * the solver tolerance. Holding it constant while the mesh refines is the @@ -68,11 +80,21 @@ # number that depends on what ran before it, the parent process below invokes # this file once per configuration and parses one line back. N_X, M_Z = 3, 2 # the SolKz wavenumbers +LAYER_W = 0.05 # half-width of the concentrated viscosity band + + +def viscosity(profile, contrast, z): + """The viscosity field, as a sympy expression in the depth coordinate.""" + if profile == "solkz": + B = 0.5 * math.log(contrast) if contrast > 1.0 else 0.0 + return sympy.exp(2 * B * z) + if profile == "layer": + return 1 + (contrast - 1) * sympy.exp(-(((z - 0.5) / LAYER_W) ** 2)) + raise ValueError("unknown viscosity profile %r" % profile) -def solve_once(pc, refinement, contrast): - """Time one SolKz solve. Returns (wall seconds, converged, unknowns).""" - B = 0.5 * math.log(contrast) if contrast > 1.0 else 0.0 +def solve_once(pc, refinement, contrast, profile="solkz"): + """Time one solve. Returns wall seconds, flops, cycles and unknowns.""" mesh = uw.meshing.UnstructuredSimplexBox( minCoords=(0.0, 0.0), maxCoords=(1.0, 1.0), cellSize=BASE_CELL, qdegree=QDEG, refinement=refinement) @@ -82,7 +104,8 @@ def solve_once(pc, refinement, contrast): p = uw.discretisation.MeshVariable("P", mesh, 1, degree=1) stokes = uw.systems.Stokes(mesh, velocityField=v, pressureField=p) stokes.constitutive_model = uw.constitutive_models.ViscousFlowModel - stokes.constitutive_model.Parameters.shear_viscosity_0 = sympy.exp(2 * B * z) + stokes.constitutive_model.Parameters.shear_viscosity_0 = viscosity( + profile, contrast, z) stokes.bodyforce = sympy.Matrix( [0, sympy.sin(M_Z * sympy.pi * z) * sympy.cos(N_X * sympy.pi * x)]) @@ -94,12 +117,43 @@ def solve_once(pc, refinement, contrast): stokes.preconditioner = pc stokes.tolerance = TOL stokes.petsc_options.setValue("fieldsplit_velocity_ksp_max_it", VEL_CAP) + + # The OUTER Krylov, set explicitly for both preconditioners so this script + # measures the same thing whatever Underworld's default happens to be. + # + # It has to be flexible. Both sub-blocks are Krylov solves run to a + # tolerance, so inside the Schur factorisation the operator the outer method + # applies differs from one iteration to the next, and plain gmres assumes it + # does not. On a smooth problem the outer converges in about two iterations + # and the choice is invisible; on a concentrated viscosity structure the + # inner solves are genuinely inexact and it is not. Leaving this to the + # default would measure that choice rather than the preconditioner. + stokes.petsc_options.setValue("ksp_type", "fgmres") if pc == "gamg": - # Back to the PETSc default; Underworld sets `additive`. + # PETSc's own default cycle for GAMG, set explicitly so that this + # script measures the same thing whatever Underworld ships. stokes.petsc_options.setValue( "fieldsplit_velocity_pc_mg_type", "multiplicative") - stokes.solve() # build everything; not timed + # Build everything; not timed, and deliberately not converged. This solve + # exists to make PETSc assemble the operators and set up the fieldsplit, so + # that the block size can be corrected below -- the ANSWER is thrown away. + # + # It has to be capped. For GAMG this runs at the operator's default block + # size of 1, which is the configuration the correction below exists to undo, + # and on a concentrated viscosity structure at high contrast that costs + # HOURS while the measured solve costs seconds. Capped at one iteration it + # costs nothing and builds exactly the same objects. + stokes.petsc_options.setValue("fieldsplit_velocity_ksp_max_it", 1) + stokes.petsc_options.setValue("ksp_max_it", 1) + stokes.petsc_options.setValue("snes_max_it", 1) + try: + stokes.solve() + except (RuntimeError, PETSc.Error): + pass # not converging is the point + stokes.petsc_options.setValue("fieldsplit_velocity_ksp_max_it", VEL_CAP) + stokes.petsc_options.delValue("ksp_max_it") + stokes.petsc_options.delValue("snes_max_it") if pc == "gamg": # Tell GAMG the field has two components per node, so it aggregates @@ -145,59 +199,72 @@ def solve_once(pc, refinement, contrast): cycles=(min(per_solve), max(per_solve)) if per_solve else (0, 0)) -def _measure_and_print(pc, refinement, contrast): +def _measure_and_print(pc, refinement, contrast, profile): """Child-process entry point: measure one configuration, print one line.""" - r = solve_once(pc, refinement, contrast) - print("RESULT %s %d %g %d %r %.6e %.6f %d %d %d" - % (pc, refinement, contrast, r["ndof"], r["ok"], r["flops"], + r = solve_once(pc, refinement, contrast, profile) + print("RESULT %s %d %g %s %d %r %.6e %.6f %d %d %d" + % (pc, refinement, contrast, profile, r["ndof"], r["ok"], r["flops"], r["wall"], r["invocations"], r["cycles"][0], r["cycles"][1])) -def measure(pc, refinement, contrast): +def measure(pc, refinement, contrast, profile="solkz"): """Run one configuration in a fresh interpreter and read the result back.""" out = subprocess.run( [sys.executable, os.path.abspath(__file__), - "--one", pc, str(refinement), repr(contrast)], + "--one", pc, str(refinement), repr(contrast), profile], capture_output=True, text=True) for line in out.stdout.splitlines(): if line.startswith("RESULT "): f = line.split() - return dict(pc=f[1], ndof=int(f[4]), ok=f[5] == "True", - flops=float(f[6]), wall=float(f[7]), - invocations=int(f[8]), - cycles=(int(f[9]), int(f[10]))) + # Echo each configuration as it lands. The tables are assembled at + # the end, and a sweep that only prints then loses everything if it + # is interrupted -- which matters when one configuration can run for + # an hour. + print(" " + line, flush=True) + return dict(pc=f[1], ndof=int(f[5]), ok=f[6] == "True", + flops=float(f[7]), wall=float(f[8]), + invocations=int(f[9]), + cycles=(int(f[10]), int(f[11]))) sys.stderr.write(out.stdout + out.stderr) - raise RuntimeError("no result from %s at refinement %d, contrast %g" - % (pc, refinement, contrast)) + raise RuntimeError("no result from %s at refinement %d, contrast %g (%s)" + % (pc, refinement, contrast, profile)) -def contrast_table(): +def contrast_table(profile="solkz"): """Cost against viscosity contrast, at one resolution.""" - rows = [(c, pc, measure(pc, 2, c)) + rows = [(c, pc, measure(pc, 2, c, profile)) for c in (1.0, 1.0e2, 1.0e4, 1.0e6) for pc in ("fmg", "gamg")] # Work per unknown, relative to FMG on the easiest problem. A ratio does # not depend on the machine it was measured on. bw = next(r["flops"] / r["ndof"] for c, pc, r in rows if c == 1.0 and pc == "fmg") bt = next(r["wall"] / r["ndof"] for c, pc, r in rows if c == 1.0 and pc == "fmg") + print("\n%s: cost against viscosity contrast" % profile) print("\n| preconditioner | viscosity contrast | relative work per unknown " - "| relative time per unknown |") - print("|---|---|---|---|") + "| relative time per unknown | cycles per velocity solve |") + print("|---|---|---|---|---|") for c, pc, r in sorted(rows, key=lambda k: (k[1] != "fmg", k[0])): + lo, hi = r["cycles"] + cycles = str(lo) if lo == hi else "%d–%d" % (lo, hi) if r["ok"]: - print("| %s | 10^%d | %.1f | %.1f |" + print("| %s | 10^%d | %.1f | %.1f | %s |" % (pc.upper(), round(math.log10(c)), - (r["flops"] / r["ndof"]) / bw, (r["wall"] / r["ndof"]) / bt)) + (r["flops"] / r["ndof"]) / bw, (r["wall"] / r["ndof"]) / bt, + cycles)) else: - print("| %s | 10^%d | diverged | diverged |" - % (pc.upper(), round(math.log10(c)))) + # Not converged. Say so rather than reporting the cost of a solve + # that did not finish -- and report the cycle count anyway, because + # a count sitting exactly on VEL_CAP means truncation, not stall. + print("| %s | 10^%d | did not converge | did not converge | %s |" + % (pc.upper(), round(math.log10(c)), cycles)) def scaling_table(): """Cost against problem size, at constant viscosity.""" - # FMG runs one level further: the trend towards O(N) is only visible once - # the fixed costs stop dominating, and GAMG at that size costs four minutes. + # BOTH run to refinement 5. GAMG at that size is expensive, but stopping it + # a level short leaves the largest FMG row with nothing to compare against, + # and the wall-clock comparison is the one that turns on the largest size. out = {pc: [measure(pc, r, 1.0) for r in refs] - for pc, refs in (("fmg", (1, 2, 3, 4, 5)), ("gamg", (1, 2, 3, 4)))} + for pc, refs in (("fmg", (1, 2, 3, 4, 5)), ("gamg", (1, 2, 3, 4, 5)))} bw = out["fmg"][0]["flops"] / out["fmg"][0]["ndof"] bt = out["fmg"][0]["wall"] / out["fmg"][0]["ndof"] @@ -210,12 +277,15 @@ def scaling_table(): % (pc.upper(), r["ndof"], (r["flops"] / r["ndof"]) / bw, (r["wall"] / r["ndof"]) / bt, r["flops"] / r["wall"] / 1e9)) + # Cycles at EVERY size, not the first few: the claim being made is that the + # count stops growing, and a table that stops short of the largest problem + # cannot support it. print("\n| preconditioner | %s |" - % " | ".join("%d unknowns" % r["ndof"] for r in out["fmg"][:3])) - print("|---|" + "---|" * 3) + % " | ".join("%d unknowns" % r["ndof"] for r in out["fmg"])) + print("|---|" + "---|" * len(out["fmg"])) for pc in ("fmg", "gamg"): cells = [] - for r in out[pc][:3]: + for r in out[pc]: lo, hi = r["cycles"] cells.append(str(lo) if lo == hi else "%d\u2013%d" % (lo, hi)) print("| %s | %s |" % (pc.upper(), " | ".join(cells))) @@ -237,7 +307,17 @@ def scaling_table(): if __name__ == "__main__": if len(sys.argv) > 1 and sys.argv[1] == "--one": - _measure_and_print(sys.argv[2], int(sys.argv[3]), float(sys.argv[4])) + _measure_and_print(sys.argv[2], int(sys.argv[3]), float(sys.argv[4]), + sys.argv[5]) + elif len(sys.argv) > 1: + # One table at a time, so a re-measurement does not mean re-running + # everything: `fmg-vs-gamg.py solkz`, `layer`, or `scaling`. + for name in sys.argv[1:]: + if name == "scaling": + scaling_table() + else: + contrast_table(name) else: - contrast_table() + contrast_table("solkz") + contrast_table("layer") scaling_table() diff --git a/articles/setting-up-full-multigrid/figures/banner.png 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b/articles/setting-up-full-multigrid/metadata.yml @@ -31,4 +31,8 @@ methods: ghost_tags: - Underworld Code figures: 1 +# Generated from the model rather than a stock photograph, so there is nobody to +# credit. `figures` counts the figures in the body; the banner is not one. +banner: figures/banner.png +banner_credit: null source: native diff --git a/articles/setting-up-full-multigrid/setting-up-full-multigrid.md b/articles/setting-up-full-multigrid/setting-up-full-multigrid.md index 8fc3ff8..152681c 100644 --- a/articles/setting-up-full-multigrid/setting-up-full-multigrid.md +++ b/articles/setting-up-full-multigrid/setting-up-full-multigrid.md @@ -12,6 +12,7 @@ authors: affiliations: - Australian National University license: CC-BY-4.0 +banner: figures/banner.png keywords: - Underworld Code - Tricks of the Trade @@ -27,6 +28,8 @@ exports: article_version: 1.0.0 software_version: underworld3 development @ 0addec15 --- +

+ Most of the effort in a geodynamics model goes into solving the Stokes equations, and most of that into the velocity solver. How that solver is preconditioned sets how long a model takes to run. Underworld can build the @@ -189,20 +192,21 @@ than scalars. Cost against viscosity contrast, at 11 727 unknowns: -| preconditioner | viscosity contrast | relative work per unknown | relative time per unknown | -|---|---|---|---| -| FMG | 10⁰ | 1.0 | 1.0 | -| FMG | 10² | 1.3 | 1.1 | -| FMG | 10⁴ | 1.6 | 1.3 | -| FMG | 10⁶ | 3.8 | 2.7 | -| GAMG | 10⁰ | 2.5 | 1.4 | -| GAMG | 10² | 3.7 | 1.8 | -| GAMG | 10⁴ | 4.5 | 2.1 | -| GAMG | 10⁶ | 9.1 | 4.1 | - -Both handle the whole range. FMG does between two and three times less -arithmetic than GAMG at every contrast, and the ratio is stable — neither -degrades relative to the other as the viscosity structure gets harder. +| preconditioner | viscosity contrast | relative work per unknown | relative time per unknown | cycles per velocity solve | +|---|---|---|---|---| +| FMG | 10⁰ | 1.0 | 1.0 | 3 | +| FMG | 10² | 1.3 | 1.1 | 3 | +| FMG | 10⁴ | 1.6 | 1.2 | 3 | +| FMG | 10⁶ | 1.8 | 1.3 | 3 | +| GAMG | 10⁰ | 2.2 | 1.2 | 34–47 | +| GAMG | 10² | 3.3 | 1.5 | 39–49 | +| GAMG | 10⁴ | 4.0 | 1.7 | 36–48 | +| GAMG | 10⁶ | 4.0 | 1.6 | 30–44 | + +Both handle the whole range, and neither degrades relative to the other as the +viscosity gets harder. GAMG does a little over twice FMG's arithmetic — the +ratio runs 2.2, 2.5, 2.5, 2.2 — and FMG holds three cycles per velocity solve +throughout where GAMG needs thirty to fifty. Cost against problem size, at constant viscosity: From 0ec29ea362e54dec7b85258632e3e57cbd41d69f Mon Sep 17 00:00:00 2001 From: lmoresi Date: Mon, 17 Aug 2026 08:15:03 +1000 Subject: [PATCH 12/15] Measure the concentrated-viscosity case, and let the figure carry it The comparison so far used SolKz, whose viscosity varies smoothly across the whole box -- the distribution algebraic coarsening reads best. Putting the same total contrast into a band a twentieth of the box thick, and changing nothing else, separates the two methods: FMG is almost indifferent, GAMG's cost climbs with the contrast, and at 1e6 it does not converge within 20 000 multigrid cycles per velocity solve. Not an under-resolution artefact. The band spans about six cells here, and resolving it better does not rescue GAMG -- at four and sixteen times the cell count its disadvantage relative to the smooth problem at the same size goes 5.5, 7.9, 8.0. Comparing band against smooth at equal size is what makes that a control: the problem size cancels. Also in this pass: - The scaling table now runs both preconditioners to 747 007 unknowns. It previously compared FMG at that size against GAMG one level down, which is not a comparison; the wall-clock reading is 1.8 against 1.9, at the same size. - Cycle counts are gone. A geometric cycle and an algebraic one are different amounts of work, so the two columns never compared and only invited the reader to divide them. - The code sample for selecting a preconditioner shows how to select one. - SolKz now points at the companion note on testing against exact solutions. - The passage on flops against wall clock says what it depends on and leaves the reader to time their own machine. Underworld development team with AI support from Claude Code --- .../examples/layer_figure.py | 104 +++++++++ .../figures/layer-vs-smooth.png | Bin 0 -> 126097 bytes .../setting-up-full-multigrid/metadata.yml | 2 +- .../setting-up-full-multigrid.md | 213 ++++++++++-------- 4 files changed, 219 insertions(+), 100 deletions(-) create mode 100644 articles/setting-up-full-multigrid/examples/layer_figure.py create mode 100644 articles/setting-up-full-multigrid/figures/layer-vs-smooth.png diff --git a/articles/setting-up-full-multigrid/examples/layer_figure.py b/articles/setting-up-full-multigrid/examples/layer_figure.py new file mode 100644 index 0000000..5180ca0 --- /dev/null +++ b/articles/setting-up-full-multigrid/examples/layer_figure.py @@ -0,0 +1,104 @@ +"""Figure: what happens to the comparison when the viscosity structure is +concentrated rather than spread through the box. + +Run from the repository root: + + python3 articles/setting-up-full-multigrid/examples/layer_figure.py + +The numbers are those printed by `fmg-vs-gamg.py solkz layer` and are repeated +here so the figure has a single, checkable source. Both panels are relative to +the same baseline -- FMG on the constant-viscosity problem -- so the two +profiles can be read against each other. At contrast 1 the two profiles ARE the +same problem (eta = 1 everywhere) and the runs agree to the last digit, which is +why the curves start together. + +Colour carries the preconditioner and line style carries the viscosity profile, +so neither is distinguished by colour alone. The palette is checked for +colour-vision separation (worst pair dE 24.7, against a target of 8). + +Run against underworld3 `development` at commit `0addec15` +(0addec1595f8d7a59b99e15b42455267a73dab86, 2026-08-15). `uw.__version__` +reports 0.0.0 for every build, so the commit is the only thing that +identifies what these numbers came from. +""" +import matplotlib +matplotlib.use("Agg") +import matplotlib.pyplot as plt + +OUT = "articles/setting-up-full-multigrid/figures/layer-vs-smooth.png" + +CONTRAST = [1e0, 1e2, 1e4, 1e6] + +# Relative to FMG at contrast 1. `None` = did not converge: GAMG on the band at +# 1e6 ran past 20 000 multigrid cycles per velocity solve without reaching the +# tolerance, so there is no cost to plot -- reporting the work it managed to do +# before being stopped would be reporting a cap, not a method. +WORK = { + ("fmg", "smooth"): [1.0, 1.3, 1.6, 1.8], + ("gamg", "smooth"): [2.2, 3.3, 4.0, 4.0], + ("fmg", "band"): [1.0, 1.9, 2.1, 2.5], + ("gamg", "band"): [2.2, 8.1, 28.1, None], +} +TIME = { + ("fmg", "smooth"): [1.0, 1.1, 1.2, 1.3], + ("gamg", "smooth"): [1.2, 1.5, 1.7, 1.6], + ("fmg", "band"): [1.0, 1.3, 1.4, 1.5], + ("gamg", "band"): [1.2, 2.7, 7.7, None], +} + +COLOUR = {"fmg": "#2a78d6", "gamg": "#eb6834"} # validated categorical 1, 2 +STYLE = {"smooth": "-", "band": "--"} +INK = "#0b0b0b" +INK_MUTED = "#52514e" +GRID = "#e4e3df" + + +def panel(ax, data, title): + for (pc, profile), y in data.items(): + xs = [x for x, v in zip(CONTRAST, y) if v is not None] + ys = [v for v in y if v is not None] + ax.plot(xs, ys, STYLE[profile], color=COLOUR[pc], linewidth=2.0, + marker="o", markersize=5, markerfacecolor="white", + markeredgewidth=1.6, zorder=3, + label="%s, %s" % (pc.upper(), profile)) + # An open marker at the last point that converged, with the failure + # named rather than left as a gap the reader has to interpret. + if y[-1] is None: + # Say what the gap means. An unexplained stop reads as missing data. + ax.annotate("does not converge\nbeyond this point", + xy=(xs[-1], ys[-1]), xytext=(10, -2), + textcoords="offset points", ha="left", va="top", + fontsize=8.5, color=INK_MUTED, linespacing=1.35) + ax.set_xscale("log") + ax.set_yscale("log") + ax.set_xlabel("viscosity contrast", fontsize=9.5, color=INK_MUTED) + ax.set_title(title, fontsize=10.5, color=INK, pad=10) + ax.grid(True, which="major", color=GRID, linewidth=0.8, zorder=0) + ax.set_axisbelow(True) + for side in ("top", "right"): + ax.spines[side].set_visible(False) + for side in ("left", "bottom"): + ax.spines[side].set_color(GRID) + ax.tick_params(colors=INK_MUTED, labelsize=9) + # Same tick VALUES in both panels so the two can be read against each + # other, but each panel framed on its own data rather than padded to a + # shared top -- the time differences are genuinely smaller and should look + # it without half the panel being empty. + ax.set_yticks([1, 1.5, 2, 3, 5, 10, 20, 30]) + ax.set_yticklabels(["1", "1.5", "2", "3", "5", "10", "20", "30"]) + finite = [v for y in data.values() for v in y if v is not None] + ax.set_ylim(min(finite) * 0.88, max(finite) * 1.45) + ax.minorticks_off() + + +fig, axes = plt.subplots(1, 2, figsize=(9.6, 4.1), sharex=True) +fig.patch.set_facecolor("white") +panel(axes[0], WORK, "work per unknown, relative to FMG at contrast 1") +panel(axes[1], TIME, "time per unknown, relative to FMG at contrast 1") +# handlelength long enough that the dashed entries are visibly dashed -- the +# line style is half the encoding, so a legend that hides it breaks identity. +axes[0].legend(frameon=False, fontsize=9, labelcolor=INK_MUTED, + loc="upper left", handlelength=3.4, borderaxespad=0.2) +fig.tight_layout() +fig.savefig(OUT, dpi=200, facecolor="white") +print("wrote", OUT) diff --git a/articles/setting-up-full-multigrid/figures/layer-vs-smooth.png b/articles/setting-up-full-multigrid/figures/layer-vs-smooth.png new file mode 100644 index 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--- a/articles/setting-up-full-multigrid/metadata.yml +++ b/articles/setting-up-full-multigrid/metadata.yml @@ -30,7 +30,7 @@ methods: - parallel-hpc ghost_tags: - Underworld Code -figures: 1 +figures: 2 # Generated from the model rather than a stock photograph, so there is nobody to # credit. `figures` counts the figures in the body; the banner is not one. banner: figures/banner.png diff --git a/articles/setting-up-full-multigrid/setting-up-full-multigrid.md b/articles/setting-up-full-multigrid/setting-up-full-multigrid.md index 152681c..9e978bd 100644 --- a/articles/setting-up-full-multigrid/setting-up-full-multigrid.md +++ b/articles/setting-up-full-multigrid/setting-up-full-multigrid.md @@ -31,10 +31,11 @@ exports:

Most of the effort in a geodynamics model goes into solving the Stokes -equations, and most of that into the velocity solver. How that solver is -preconditioned sets how long a model takes to run. Underworld can build the -preconditioner from a real hierarchy of meshes rather than from the matrix -alone, and this note is about how to get that hierarchy and what it is worth. +equations, and most of that is spent on the velocity solver. How that solver is +preconditioned sets how long a model takes to run. Multigrid methods accelerate elliptic +solvers using a hierarchy of mesh resolutions. Underworld can build the +preconditioner from a real hierarchy of meshes or directly from the matrix +alone, and this note is about how to build that hierarchy and what it is worth. ## What multigrid does @@ -46,24 +47,25 @@ one side of a fine mesh to the other. Multigrid turns that around. Error that is smooth on a fine mesh is *not* smooth on a mesh twice as coarse, where it spans half as many cells, so a few sweeps -there remove what the fine mesh could not. +there can quickly reduce error that the fine grid struggles with. -The method is a recursion: smooth on the fine mesh, transfer what is left to a +The method is a recursion on that idea: smooth on the fine mesh, transfer what is left to a coarser one, solve there with another recursive step, transfer the correction -back, smooth again. Each *level* handles the band of error it can see cheaply, -and the work per unknown stops growing as the model gets bigger. +back, smooth again. Each *level* handles the band of error it can see, +and the work per unknown flattens out as the model gets bigger. -Two things have to be supplied: the coarse levels, and the operators that move +The two things we need: the coarse levels, and the operators that move between them. ## Two ways to build the coarse levels -**Algebraic multigrid** (AMG) builds the coarse levels from the matrix. It -inspects the operator's connectivity, groups unknowns that are strongly -coupled, and constructs coarse levels and transfer operators from that grouping -alone. Because it asks nothing of the mesh it works everywhere, which makes it -the sensible default. PETSc's implementation is GAMG, and it is what Underworld -uses when it has nothing better. +**Algebraic multigrid** (AMG) builds the coarse levels from the matrix +representation of the problem directly. It inspects the operator's connectivity, +groups unknowns that are strongly +coupled, and constructs coarse representations and transfer operators from that grouping +alone. Because it asks nothing of the mesh, it works everywhere. +PETSc's implementation is GAMG, and it is what Underworld +uses by default. **Geometric multigrid** (GMG) uses actual coarser meshes. If the fine mesh was built by ***refining*** a coarse one, those coarser meshes already exist, and the transfers @@ -75,12 +77,13 @@ cycling up and down through the levels at each stage to clear the error the interpolation leaves behind. We'll refer to this as FMG. The difference between them shows up when the operator's connectivity is a poor -guide to the geometry, which is what jumps and steep gradients in material -properties produce. The algebraic method sees a matrix whose couplings are +guide to the geometry, which can happen when there are jumps or steep gradients in material +properties. The algebraic method sees a matrix whose couplings are dominated by the stiff region and groups unknowns accordingly; the geometric method uses the grids it was given and is indifferent to the coefficients. Which -of those is the better bet is not obvious in advance, so it is measured below. -Switching the geometric solver on comes first, and that starts with the mesh. +of those is the better bet is not obvious in advance, so we'll measure it and see what works. +Switching the geometric solver on comes first, and we start with +the hierarchy of meshes. ## Making a mesh with a hierarchy @@ -94,9 +97,9 @@ len(mesh.dm_hierarchy) # 3: the base and two refinements The mesh you get back is the finest level, and the coarser ones are stored within it as PETSc `DMPlex` objects in `mesh.dm_hierarchy`. Those are what the -preconditioner uses. They are not Underworld meshes — there is no `Mesh` object -for a coarse level, and making one takes work — so the hierarchy is something -the solver reads rather than something you handle. +preconditioner uses. They are not Underworld meshes, there is no `Mesh` object +for a coarse level, and making one takes work. The hierarchy is something +the solver knows about rather than something you handle. Build the same mesh at `cellSize=0.0125` with no refinement and you get about the same resolution with no hierarchy at all, and so no geometric multigrid. @@ -108,6 +111,9 @@ The solver picks it up on its own: ```python stokes = uw.systems.Stokes(mesh) stokes.solve() # preconditioner defaults to "auto" + +stokes.preconditioner = "fmg" # or "gamg", or back to "auto" +stokes.solve() ``` `"auto"` uses geometric multigrid when `len(mesh.dm_hierarchy) > 1` and GAMG @@ -118,17 +124,17 @@ have set yourself; and where a request cannot be honoured, you can find the reasoning in `stokes.pc_fallbacks`. `stokes.preconditioner_settings` reports what was actually applied. -## Curved boundaries need help to refine +## Curved boundaries need help Refining a mesh puts new nodes at the midpoints of existing edges. On a -straight boundary that is fine. On a curved one it is not: the coarse mesh +straight boundary that works well. On a curved one it is not enough: the coarse mesh approximates a circle by a polygon, and the midpoint of a chord lies inside the circle rather than on it. Those midpoints have to be moved back onto the boundary each time the mesh is refined. Underworld's curved meshes carry a refinement *callback* that snaps boundary-labelled nodes back onto the true surface after each refinement. For -the annulus it is a few lines — take the nodes labelled `Upper` and `Lower` and +the annulus it is a few lines: take the nodes labelled `Upper` and `Lower` and rescale each to the correct radius: ```python @@ -137,7 +143,7 @@ coords[lower] *= radiusInner / R[lower] ``` This happens automatically for the built-in meshes, so in normal use there is -nothing to do. It matters if you build a mesh of your own with curved +nothing you need to do. It does matter if you build a mesh of your own with curved boundaries: without a callback of this kind, the finest mesh still has the coarsest mesh's faceting, and the boundary it represents is the polygon rather than the circle. @@ -147,11 +153,10 @@ than the circle. Because the fine mesh is made by subdividing the coarse one, the coarse mesh's own triangulation is still visible in it. Its edges survive as continuous lines across the fine mesh, and its vertices remain places where an unusual number of -elements meet. The pattern is a record of how the mesh was made rather than of -the problem being solved. +elements meet and the element density is inherited from the coarse triangulation. `mesh.relax()` moves the nodes to improve element shapes while keeping the -resolution and the topology, which loosens that imprint. +resolution and the topology, and it helps to loosen that imprint. ```{figure} figures/relax-annulus.png :alt: Two wireframe views of a twice-refined annulus mesh. On the left, long straight seams from the coarse base mesh run across the refined mesh and meet at vertices where an unusual number of elements converge. On the right the same mesh after relaxation, with the seams much less apparent and the element sizes more even. @@ -167,8 +172,7 @@ interior coordinates only. Relaxation moves coordinates but not topology, so the refinement relation between the levels is untouched and the hierarchy survives it. Moving a mesh without -rebuilding it is the subject of a -[related technical note](/moving-the-mesh-without-remaking-it/). +rebuilding it is the subject of a [related technical note](/moving-the-mesh-without-remaking-it/). ## When the choice matters @@ -176,81 +180,92 @@ The test is SolKz, a standard Stokes benchmark on the unit box: viscosity varying exponentially with depth, $\eta = e^{2Bz}$, forced by $\mathbf{f} = (0,\; \sin(m\pi z)\cos(n\pi x))$ with $n = 3$, $m = 2$, free slip on all four walls, Taylor–Hood $P_2$–$P_1$ elements. The viscosity contrast -across the box is $e^{2B}$. SolKz has a closed-form solution, but nothing here -uses it: these are timings, not accuracy measurements. +across the box is $e^{2B}$. SolKz has a closed-form solution, and Underworld +carries it in `uw.analytic` — putting it to its usual use, checking that a +solver converges to the right answer, is the subject of a +[companion note](/testing-a-solver-against-exact-solutions/). Nothing here uses +it: these are cost measurements, not accuracy measurements. -Work is reported as floating-point operations counted by PETSc, per unknown and -relative to the cheapest FMG run. Flops are the better measure: they do not -depend on the machine, and they are deterministic — repeating a run reproduces -the count exactly, where a timing moves by a few per cent. Times are given -alongside, because they are what you wait for and they do not tell the same -story. +*Computational work* is reported as floating-point operations (counted by PETSc) per unknown and +relative to the constant-viscosity FMG run with a fixed target iteration tolerance across the board. Flops are a useful measure: they do not depend on the machine, and they are generally deterministic. Times are given alongside, because they are what you wait for and they reveal that different algorithms may be more or less efficient on particular hardware. -Both preconditioners are run at PETSc's own defaults, and GAMG is told that the -velocity field has two components per node so that it aggregates nodes rather -than scalars. +First we examine the cost against viscosity contrast, at 11 727 unknowns: -Cost against viscosity contrast, at 11 727 unknowns: +| preconditioner | viscosity contrast | relative work per unknown | relative time per unknown | +|---|---|---|---| +| FMG | 10⁰ | 1.0 | 1.0 | +| FMG | 10² | 1.3 | 1.1 | +| FMG | 10⁴ | 1.6 | 1.2 | +| FMG | 10⁶ | 1.8 | 1.3 | +| GAMG | 10⁰ | 2.2 | 1.2 | +| GAMG | 10² | 3.3 | 1.5 | +| GAMG | 10⁴ | 4.0 | 1.7 | +| GAMG | 10⁶ | 4.0 | 1.6 | -| preconditioner | viscosity contrast | relative work per unknown | relative time per unknown | cycles per velocity solve | -|---|---|---|---|---| -| FMG | 10⁰ | 1.0 | 1.0 | 3 | -| FMG | 10² | 1.3 | 1.1 | 3 | -| FMG | 10⁴ | 1.6 | 1.2 | 3 | -| FMG | 10⁶ | 1.8 | 1.3 | 3 | -| GAMG | 10⁰ | 2.2 | 1.2 | 34–47 | -| GAMG | 10² | 3.3 | 1.5 | 39–49 | -| GAMG | 10⁴ | 4.0 | 1.7 | 36–48 | -| GAMG | 10⁶ | 4.0 | 1.6 | 30–44 | - -Both handle the whole range, and neither degrades relative to the other as the -viscosity gets harder. GAMG does a little over twice FMG's arithmetic — the -ratio runs 2.2, 2.5, 2.5, 2.2 — and FMG holds three cycles per velocity solve -throughout where GAMG needs thirty to fifty. - -Cost against problem size, at constant viscosity: +Both solver configurations handle the whole range in the viscosity gradient sweep and there is almost +no change in the amount of work required or time taken to solve. GAMG does slightly more work but does so a little more efficiently. + +Next, let us look at the cost (per unknown) as we change the problem size, at constant viscosity: | preconditioner | unknowns | relative work per unknown | relative time per unknown | Gflop/s | |---|---|---|---|---| -| FMG | 2 947 | 1.0 | 1.0 | 2.2 | -| FMG | 11 727 | 1.9 | 1.3 | 3.3 | -| FMG | 46 783 | 2.6 | 1.6 | 3.7 | -| FMG | 186 879 | 2.9 | 1.9 | 3.4 | -| FMG | 747 007 | 3.0 | 2.3 | 3.0 | -| GAMG | 2 947 | 3.3 | 1.5 | 4.9 | -| GAMG | 11 727 | 4.7 | 1.8 | 5.7 | -| GAMG | 46 783 | 4.9 | 2.0 | 5.6 | -| GAMG | 186 879 | 5.8 | 2.2 | 5.8 | - -Both flatten. Step by step, FMG's flop count goes as $N^{1.46}$, $N^{1.23}$, -$N^{1.08}$, $N^{1.03}$; GAMG's as $N^{1.25}$, $N^{1.04}$, $N^{1.11}$. Both are +| FMG | 2 947 | 1.0 | 1.0 | 1.6 | +| FMG | 11 727 | 1.7 | 1.3 | 2.1 | +| FMG | 46 783 | 2.2 | 1.4 | 2.4 | +| FMG | 186 879 | 2.5 | 1.6 | 2.4 | +| FMG | 747 007 | 2.6 | 1.8 | 2.2 | +| GAMG | 2 947 | 2.7 | 1.3 | 3.4 | +| GAMG | 11 727 | 3.8 | 1.5 | 3.8 | +| GAMG | 46 783 | 4.0 | 1.6 | 3.9 | +| GAMG | 186 879 | 4.6 | 1.7 | 4.2 | +| GAMG | 747 007 | 5.5 | 1.9 | 4.6 | + +Both flatten (which is good). Step by step, FMG's flop count goes as $N^{1.38}$, $N^{1.20}$, +$N^{1.07}$, $N^{1.03}$; GAMG's as $N^{1.24}$, $N^{1.04}$, $N^{1.11}$, $N^{1.12}$. Both are converging on linear, which is what multigrid of either kind is supposed to do: the algebraic method reaches it as surely as the geometric one. -The cycle counts say the same thing. FMG converges the velocity block in three -cycles at every size; GAMG takes 26–32, then 34–47, then 33–49 — ten times as -many, but not a growing number. (The Schur factorisation invokes the velocity -solve sixteen times per Stokes solve, the same for both, so that factor -cancels.) - -So the geometric hierarchy is worth roughly a factor of two: FMG does about -half GAMG's arithmetic on this problem, and holds that through six orders of -viscosity contrast and a 250-fold growth in problem size. - -In wall clock the advantage is smaller still — 2.3 against 2.2 at the largest -size, which is no advantage at all — and the last column says why. GAMG runs at -5–6 Gflop/s where FMG runs at 2–4, and FMG's rate *falls* at the largest size -as the problem outgrows cache. - -The difference is in the shape of the arithmetic rather than the amount of it. -Geometric multigrid on an unstructured hierarchy works through sparse, indirect -operators, and its coarse levels are small enough that they cannot keep the -processor busy. Algebraic multigrid does more arithmetic, but in denser and -more regular operators that run closer to the machine's peak rate. On this -hardware the two effects very nearly cancel. The flop count is a property of -the method and should carry to another machine; the rate is a property of this -one, and on a machine with a different memory system the wall-clock comparison -could come out either way. +So the geometric hierarchy is worth roughly a factor of two in throughput: FMG does about +half GAMG's arithmetic on this problem and holds on to that advantage through six orders of viscosity contrast and a 250-fold growth in problem size. + +In wall clock the advantage is smaller — 1.8 against 1.9 at the largest +size, which is no practical advantage at all. This is because GAMG runs at +3.4–4.6 Gflop/s where FMG runs at 1.6–2.4, and FMG's rate *falls* at the largest +size as the problem outgrows cache. + +The shape of the arithmetic matters as much as the amount. FMG's operators are +sparse and indirect and its coarse levels are too small to keep a processor +busy; GAMG's are denser and more regular and run closer to peak. How those +trade off is a property of the machine, so time both on yours and use whichever +wins. + +## When the viscosity structure is concentrated + +SolKz spreads its viscosity variation across the whole box, and smoothly varying +coefficients are what algebraic coarsening reads best. So the second test puts +the same total contrast into a band a twentieth of the box thick, + +$$\eta(z) = 1 + (\eta_0 - 1)\,e^{-\left((z - 1/2)/w\right)^2}, \qquad w = 0.05$$ + +and changes nothing else: same forcing, same boundary conditions, same mesh. At +contrast 1 the two problems are the same problem, and the runs agree to the last +digit. + +```{figure} figures/layer-vs-smooth.png +:alt: Two log-log panels of solver cost against viscosity contrast, work on the left and time on the right. Solid lines are the smooth SolKz viscosity, dashed lines the concentrated band; blue is FMG and orange GAMG. All four curves start together at contrast one. The three curves for FMG-smooth, FMG-band and GAMG-smooth stay nearly flat across six orders of contrast. The GAMG-band curve climbs steeply in both panels and stops at contrast ten thousand, labelled as not converging beyond that point. + +Cost against viscosity contrast for the two viscosity distributions, at 11 727 +unknowns. FMG is almost indifferent to the change. GAMG is not: on the band its +cost climbs with the contrast, and at $10^6$ it does not converge at all — 20 000 +multigrid cycles per velocity solve without reaching the tolerance. This is not +an artefact of an under-resolved band. At this resolution the band spans about +six cells, and resolving it better does not rescue GAMG: at four and sixteen +times the cell count, GAMG's disadvantage against FMG on the band grows from +5.5 to 7.9 to 8.0 times its disadvantage on the smooth problem at the same size. +``` + +The geometric hierarchy is indifferent to the coefficients because it never +consults them. That is the case for keeping a hierarchy when you can. :::{note} What this comparison does not settle The solver tolerance is held fixed as the mesh refines. That is the usual way @@ -260,9 +275,9 @@ tolerance alongside it, which would make the work per unknown grow. The table answers "what does it cost to solve this system", not "what does it cost to reach the accuracy the mesh can support". -SolKz is also a smooth problem. Viscosity that varies smoothly everywhere is -what algebraic coarsening reads best; structure that is concentrated rather -than distributed is harder for it, and that case is not measured here. +Both preconditioners are also run in serial, on one machine. GAMG's coarsening +is the part of it that changes most under parallel decomposition, and none of +that is measured here. ::: ## Using it From 84f4688f8dbcb87f56a2dcb38de2578566ada378 Mon Sep 17 00:00:00 2001 From: lmoresi Date: Mon, 17 Aug 2026 09:35:07 +1000 Subject: [PATCH 13/15] Alt text describes the figure, not the caption Alt text stands in for the image when it cannot be seen, so it has to carry what the image shows. The previous text described the SHAPE of the curves and left the reader with no way to know what any of them said. The band measurements appear nowhere else in the note -- the tables cover the smooth case only -- so the alt text is the sole route to those numbers for a reader who is not looking at the picture, and it now reads them out. Underworld development team with AI support from Claude Code --- .../setting-up-full-multigrid.md | 11 +++++------ 1 file changed, 5 insertions(+), 6 deletions(-) diff --git a/articles/setting-up-full-multigrid/setting-up-full-multigrid.md b/articles/setting-up-full-multigrid/setting-up-full-multigrid.md index 9e978bd..f67f2e6 100644 --- a/articles/setting-up-full-multigrid/setting-up-full-multigrid.md +++ b/articles/setting-up-full-multigrid/setting-up-full-multigrid.md @@ -252,7 +252,7 @@ contrast 1 the two problems are the same problem, and the runs agree to the last digit. ```{figure} figures/layer-vs-smooth.png -:alt: Two log-log panels of solver cost against viscosity contrast, work on the left and time on the right. Solid lines are the smooth SolKz viscosity, dashed lines the concentrated band; blue is FMG and orange GAMG. All four curves start together at contrast one. The three curves for FMG-smooth, FMG-band and GAMG-smooth stay nearly flat across six orders of contrast. The GAMG-band curve climbs steeply in both panels and stops at contrast ten thousand, labelled as not converging beyond that point. +:alt: Two side-by-side log-log panels. Both have viscosity contrast on the horizontal axis, marked at 1, 10^2, 10^4 and 10^6. The left panel plots relative work per unknown, the right relative time per unknown. Each panel carries four curves with open circular markers at those four contrasts: blue for FMG and orange for GAMG, solid for the smooth SolKz viscosity and dashed for the concentrated band. Reading the left panel, FMG smooth runs 1.0, 1.3, 1.6, 1.8; FMG band runs 1.0, 1.9, 2.1, 2.5; GAMG smooth runs 2.2, 3.3, 4.0, 4.0. The dashed orange GAMG band curve leaves the others behind — 2.2, 8.1, 28.1 — and then stops after 10^4, with the text "does not converge beyond this point" beside its last marker. In the right panel the first three curves are packed into a narrow span between 1.0 and 1.7, while GAMG band again separates, running 1.2, 2.7, 7.7 before stopping at the same place. All four curves meet at the left-hand edge, because at contrast 1 the two viscosity distributions are the same problem. Cost against viscosity contrast for the two viscosity distributions, at 11 727 unknowns. FMG is almost indifferent to the change. GAMG is not: on the band its @@ -267,17 +267,16 @@ times the cell count, GAMG's disadvantage against FMG on the band grows from The geometric hierarchy is indifferent to the coefficients because it never consults them. That is the case for keeping a hierarchy when you can. -:::{note} What this comparison does not settle +:::{note} What this comparison does not address The solver tolerance is held fixed as the mesh refines. That is the usual way -to show a multigrid scaling, and it is a choice: a finer mesh has a smaller +to show a multigrid scaling, but a finer mesh has a smaller discretisation error, so an argument can be made for tightening the solver tolerance alongside it, which would make the work per unknown grow. The table -answers "what does it cost to solve this system", not "what does it cost to +should be read as "what does it cost to solve this system", not "what does it cost to reach the accuracy the mesh can support". Both preconditioners are also run in serial, on one machine. GAMG's coarsening -is the part of it that changes most under parallel decomposition, and none of -that is measured here. +is the part of it that changes most under parallel decomposition and is not measured here. ::: ## Using it From d30e4f7ae4c4136ad42c60070e1e03bb475f10d7 Mon Sep 17 00:00:00 2001 From: lmoresi Date: Mon, 17 Aug 2026 09:43:07 +1000 Subject: [PATCH 14/15] Mark UWTN 2026-014 published, dated 2026-08-17 Follows UWTN 2026-011, which carried status and date in the merge commit and took its DOI at the deposit step afterwards. Underworld development team with AI support from Claude Code --- articles/setting-up-full-multigrid/metadata.yml | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/articles/setting-up-full-multigrid/metadata.yml b/articles/setting-up-full-multigrid/metadata.yml index ad029c5..bc250dd 100644 --- a/articles/setting-up-full-multigrid/metadata.yml +++ b/articles/setting-up-full-multigrid/metadata.yml @@ -6,12 +6,12 @@ id: UWTN 2026-014 slug: setting-up-full-multigrid title: Setting Up Full Multigrid article_type: technical-note -status: review +status: published authors: - name: Louis Moresi orcid: 0000-0003-3685-174X affiliation: Australian National University -publication_date: null +publication_date: 2026-08-17 version: 1.0.0 # The deposit writes archive_doi and repository_record_id when the note is # published; leave them out until then. `doi` and `doi_registrant` were here From 6a5b02a207e33ded4bc9d877105a831a9fbc5fb0 Mon Sep 17 00:00:00 2001 From: lmoresi Date: Mon, 17 Aug 2026 10:38:59 +1000 Subject: [PATCH 15/15] One section for the comparison, so the heading is answered rather than contradicted 'When the choice matters' sat over the smooth case, whose finding is that it mostly does not, and the case where it does had a heading of its own further down. Dropping the second heading puts both under the first, which the section then answers by the end. Underworld development team with AI support from Claude Code --- .../setting-up-full-multigrid.md | 8 +++----- 1 file changed, 3 insertions(+), 5 deletions(-) diff --git a/articles/setting-up-full-multigrid/setting-up-full-multigrid.md b/articles/setting-up-full-multigrid/setting-up-full-multigrid.md index f67f2e6..fb87464 100644 --- a/articles/setting-up-full-multigrid/setting-up-full-multigrid.md +++ b/articles/setting-up-full-multigrid/setting-up-full-multigrid.md @@ -239,11 +239,9 @@ busy; GAMG's are denser and more regular and run closer to peak. How those trade off is a property of the machine, so time both on yours and use whichever wins. -## When the viscosity structure is concentrated - -SolKz spreads its viscosity variation across the whole box, and smoothly varying -coefficients are what algebraic coarsening reads best. So the second test puts -the same total contrast into a band a twentieth of the box thick, +So far the viscosity has varied smoothly across the whole box, which is the +distribution algebraic coarsening reads best. The second test puts the same +total contrast into a band a twentieth of the box thick, $$\eta(z) = 1 + (\eta_0 - 1)\,e^{-\left((z - 1/2)/w\right)^2}, \qquad w = 0.05$$