Generalized Optimal AMG Convergence Theory for Stokes Equations Using Smooth Aggregation and Vanka Relaxation Strategies

Ahsan Ali; Alexey Voronin; Ben S. Southworth; Jacob B. Schroder; James J. Brannick; Karsten Kahl; Oliver A. Krzysik

REVIEW 4 major objections 5 minor 20 references

Generalized Optimal AMG Convergence Theory for Stokes Equations Using Smooth Aggregation and Vanka Relaxation Strategies

T0 review · 4 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read A generalized optimal AMG convergence theory accurately predicts the convergence rate of a monolithic SA-AMG solver for the Stokes equations, as a lower bound.

desk verdict Honest application of the authors' own optimal two-grid theory to a monolithic Stokes solver; the predicted lower bound holds in small tests, but the diagonalizability condition for the nonsymmetric Vanka smoother is never verified. read the letter →

arxiv 2501.06621 v1 pith:U3LO7ZDW submitted 2025-01-11 math.NA cs.NA

Ahsan Ali , James J. Brannick , Karsten Kahl , Oliver A. Krzysik , Jacob B. Schroder , Ben S. Southworth , Alexey Voronin This is my paper · ORCID

classification math.NAcs.NA MSC 65N5565F1065N30

keywords Stokesequationsalgebraicmultigridsaddle-pointsystemsVankarelaxationsmoothedaggregationgeneralizedeigenvalueproblemconvergencetheoryTaylor-Hoodelements

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper tries to establish that the generalized optimal algebraic multigrid (AMG) convergence theory, originally built for nonsymmetric and indefinite problems, gives a reliable prediction of how fast a modern monolithic smoothed-aggregation AMG (SA-AMG) solver converges on the steady-state Stokes equations. The prediction takes the form of a rigorous lower bound on the spectral radius of the two-grid error-propagation operator: the bound is $|1-\lambda_{n_c+1}|$, where $\lambda_{n_c+1}$ is the next generalized eigenvalue of the matrix pencil $(K,M)$ formed by the Stokes operator $K$ and the Vanka relaxation operator $M$. The paper applies the theory to Taylor-Hood ($\mathbb{P}_2/\mathbb{P}_1$) discretizations and to both additive and multiplicative Vanka relaxation, finding that the predicted factors consistently sit below the geometric and asymptotic convergence factors observed in $V(1,0)$ cycles. A sympathetic reader would care because the theory converts a black-box solver's convergence into a spectrum-based quantity, which can guide coarsening and damping choices without brute-force parameter searches. If the framework is right, it opens a route to principled AMG design for saddle-point systems.

What carries the argument

The load-bearing object is the generalized eigenvalue problem $K v_i = \lambda_i M v_i$ for the system matrix $K$ and the relaxation operator $M$, together with the matrix-induced orthogonality between left and right eigenvectors. The optimal interpolation operator $P_\sharp$ is defined by spanning the $n_c$ right eigenvectors with largest $|1-\lambda_i|$, and the two-grid error-propagation operator $E_{TG}(P_\sharp)$ then has spectral radius exactly $|1-\lambda_{n_c+1}|$. This identity is the machinery that turns convergence prediction into an eigenvalue calculation. In the Stokes application, $K$ is the symmetric indefinite saddle-point matrix from the Taylor-Hood discretization, while $M$ is the additive Vanka relaxation operator (nonsymmetric because the partition of unity enters on one side) or the multiplicative Vanka operator (symmetric). The paper evaluates the bound by comparing $|1-\lambda_{n_c+1}|$ with measured geometric and asymptotic convergence factors.

What would settle it

Run a genuine two-grid $V(1,0)$ cycle on the 1095-by-1095 Stokes problem with interpolation $P_\sharp$ spanning the first 191 right generalized eigenvectors of $(K,M)$; if the measured convergence factor does not equal $|1-\lambda_{192}|$, or if any measured factor for the actual SA-AMG hierarchy falls below the bound, the theory's assumptions would be refuted.

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Extended reading notes

Core claim

The central claim is that the generalized optimal AMG theory, summarized by the identity $\rho(E_{TG}(P_\sharp)) = |1-\lambda_{n_c+1}|$ with $P_\sharp$ the optimal interpolation spanning the first $n_c$ right generalized eigenvectors of $Kv=\lambda M v$, carries over to the Stokes saddle-point setting. For the monolithic SA-AMG solver, the paper shows that this identity supplies a consistent lower bound for the observed convergence factors of a $V(1,0)$ cycle using additive Vanka relaxation, across a range of damping parameters and under both standard and aggressive coarsening. It also reports that multiplicative Vanka relaxation, which keeps the relaxation operator symmetric, reduces residuals faster than the nonsymmetric additive form. The paper's own summary is that the generalized theory consistently provides accurate lower bounds on the convergence rate and thereby offers a predictive tool for improving AMG solvers for saddle-point problems.

Load-bearing premise

The comparison rests on assuming that the full multilevel solver's convergence is controlled by the two-grid error-propagation operator with the same Vanka relaxation, so the optimal-interpolation bound is the quantity that should match the observed factors.

Editorial extensions

If this is right

If the theory is correct, the convergence rate of a standalone SA-AMG $V(1,0)$ cycle on Stokes problems is bounded below by a single eigenvalue gap, so damping parameters can be selected by inspecting the spectrum of $M^{-1}K$ rather than by brute-force search.
The same identity applies to symmetric indefinite saddle-point systems generally, extending optimal AMG theory beyond the symmetric positive-definite case where energy-norm arguments were previously used.
Aggressive coarsening (larger $n_c$) is predicted to yield smaller asymptotic convergence factors, which matches the paper's observation and suggests coarsening can be tuned to improve speed.
Multiplicative Vanka relaxation, by preserving symmetry of $M$, fits the Hermitian version of the theory and converges in fewer iterations than additive Vanka in the paper's tests, pointing to symmetric relaxation as a design principle.
The gap between the predicted lower bound and observed convergence factors indicates the actual monolithic interpolation is not optimal; moving it closer to $P_\sharp$ could bring observed rates down toward the bound.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A natural next test, not performed in the paper, is to replace the monolithic interpolation with the truly optimal $P_\sharp$ in a two-grid cycle; if the measured factor then equals $|1-\lambda_{n_c+1}|$, the lower-bound gap would be confirmed as interpolation loss rather than a multilevel effect.
The paper does not verify that $M^{-1}K$ is diagonalizable for the nonsymmetric additive Vanka operator; if near-defective spectra occur, the identity may need a Jordan-block formulation, and the lower-bound interpretation could change in those regimes.
The same machinery should be testable on Oseen or Navier-Stokes systems and on three-dimensional Stokes problems, where eigenvalue scatter and coarsening ratios differ; such tests would show whether the bound remains tight outside the two-dimensional Taylor-Hood setting.

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Formalized claims in Lean

Claim #1: The central claim is that the generalized optimal AMG theory, summarized by the identity $\rho(E_{TG}(P_\sharp)) = |1-\lambda_{n_c+1}|$ with $P_\sharp$ the optimal interpolation spanning the first $n_c$ right generalized eigenvectors of $Kv=\lambda M v$, carries over to the Stokes saddle-point setting. For the monolithic SA-AMG solver, the paper shows that this identity supplies a consistent lower

/-- @claim 1 The central claim is that the generalized optimal AMG theory, summarized by the identity $\rho(E_{TG}(P_\sharp)) = |1-\lambda_{n_c+1}|$ with $P_\sharp$ the optimal interpolation spanning the first $n_c$ right generalized eigenvectors of $Kv=\lambda M v$, carries over to the Stokes saddle-point setting. For the monolithic SA-AMG solver, the paper shows that this identity supplies a consistent lower -/
def central_claim : Prop :=

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

The thing to know about this paper is that it is exactly what the title says: an application of the authors' generalized optimal AMG theory (from their preprint arXiv:2401.11146) to the monolithic SA-AMG Stokes solver of Voronin et al. (arXiv:2306.06795). It does not prove a new theorem. What it does is take the identity ρ(E_TG(P♯)) = |1−λ_{n_c+1}| and check whether the number |1−λ_{n_c+1}|, computed from the generalized eigenvalue problem (4.3), is a valid lower bound on the observed convergence factors of V(1,0) cycles. The numerical results in Section 5 show exactly that: the predicted value is consistently below both the geometric and asymptotic factors for the two small problems tested (1095 and 1891 unknowns). That is a legitimate, if modest, contribution, and the paper is clearly written about what was actually done.

The paper does well to state the theorem with its hypotheses, including the requirement that M^{-1}K be diagonalizable. But that is also where the main soft spot sits. For the additive Vanka operator, M is nonsymmetric—the paper says so—and the authors never verify that M^{-1}K is diagonalizable for their tested problems. The stress-test note raises exactly this: without diagonalizability, the eigenvector-based optimal interpolation P♯ need not exist and the spectral radius identity is not guaranteed. This is not a minor technicality; it is a condition of the theorem being used. I believe the authors should be asked to provide a check (or at least a numerical argument) for the specific M and K they use.

The other soft spots are more routine. The experimental details are thin: no mesh description, boundary conditions, tolerances, or iteration counts beyond a fixed max of 100. No code or data are supplied. The abstract says 'precise predictions,' but the paper itself shows only lower bounds, which is an overstatement. And the theory is two-grid while the experiments are full multilevel V(1,0) cycles; the authors do not discuss how much multilevel effects might change the picture. None of these are fatal, but they should be addressed.

Who is this for? AMG researchers, especially those working on saddle-point systems and Vanka smoothers. It would be a useful data point for someone trying to use the optimal theory in practice. I would send it to peer review, asking for the diagonalizability check and better experimental reporting. It is not paradigm-shifting, but it is honest and the central claim holds up.

Referee Report

4 major / 5 minor

Summary. The paper applies the generalized optimal AMG convergence theory of Ali et al. [2] to the monolithic smooth-aggregation AMG (SA-AMG) Stokes solver of Voronin et al. [19], using Taylor-Hood P2/P1 finite elements and additive or multiplicative Vanka relaxation. The stated central claim is that the two-grid error-propagation spectral radius is predicted by the quantity |1 - λ_{nc+1}|, where λ_{nc+1} is the next neglected generalized eigenvalue of the pair (K, M), and that this predicted value consistently lower-bounds the observed geometric and asymptotic convergence factors of the actual SA-AMG solver for a small 1095- and 1891-degree-of-freedom Stokes problem (Section 5, Figure 5.2). The manuscript also reports damping-parameter sensitivity for V(1,0), V(2,0), V(2,2), and V(4,0) cycles in Table 5.1.

Significance. If the claimed predictive relationship is correct, the result would provide a useful diagnostic tool for designing AMG methods for saddle-point systems, since it connects a cheap generalized eigenvalue computation to the expected two-grid performance. The strength of the paper is that the predictor is not fitted to the observed convergence rates; it is computed from the actual K and M via the generalized eigenvalue problem (4.3), and the underlying identity is an independent theorem from [2], not a numerical coincidence. The paper also honestly distinguishes the symmetric multiplicative Vanka case from the nonsymmetric additive Vanka case. However, the incremental contribution is mainly empirical application of an existing theory, and the empirical demonstration is currently too under-specified and rests on an unverified technical hypothesis (diagonalizability of M^{-1}K) to support the advertised conclusion.

major comments (4)

[Section 4, Eqs. (4.1)-(4.2)] The central theorem is stated only under the explicit hypothesis that M^{-1}K is diagonalizable. For the additive Vanka relaxation used in Section 5, M is nonsymmetric because of the one-sided partition of unity, and the paper provides no verification, either numerical or structural, that M^{-1}K is diagonalizable for the 1095 x 1095 problem. The paper should report a concrete check, for example the condition number of the computed eigenvector matrix or the size of the largest Jordan block, for each damping parameter omega and mesh size reported in Figure 5.2. Without this check, Eq. (4.2) does not connect the scalar |1 - lambda_{nc+1}| to the spectral radius of any two-grid operator in the additive Vanka setting, and the agreement in Figure 5.2 could be coincidental.
[Section 4 and Section 5, Figure 5.2] Eq. (4.2) is an equality for the optimal interpolation operator P♯ whose range is spanned by the first nc right generalized eigenvectors. The SA-AMG prolongator P used in Figure 5.2 is not this optimal operator, and the manuscript nowhere states or proves the minimax result that converts the equality for P♯ into a lower bound for arbitrary P. The reader is left to infer that |1 - lambda_{nc+1}| is a rigorous lower bound for non-optimal interpolation, but that inference is not established in the text. The paper should either cite and state the exact lower-bound corollary from [2] with its hypotheses, or explicitly qualify the comparison as an empirical heuristic.
[Section 5, Figure 5.2 and Table 5.1] The numerical validation is under-specified to the point of being non-reproducible. The manuscript does not state the domain, mesh, boundary conditions, right-hand side, convergence tolerance, definition of the reported geometric and asymptotic convergence factors, or the number of levels in the hierarchy. The figures, in particular Figures 5.1-5.3, are not present in the submitted text beyond their captions, so no quantitative data underlying the claimed comparison is available to the reader. The paper should include the actual plots, provide the definitions and the numerical values used in the comparison, and either include a data/code repository or tabulate the relevant convergence factors.
[Section 5, Table 5.1] Several rows of Table 5.1 do not correspond to converged runs under the stated 100-iteration maximum: for V(1,0) at omega=0.30 the residual is only 2.88e-09, and at omega=0.84 it is 3.57e-07. Extracting a meaningful 'convergence factor' from a non-converged or stagnating run is not well-defined unless the authors specify whether only the converged portion of the iteration history is used, or how stagnation is handled. The claim that 'geometric and asymptotic convergence factors align closely' is also not supported by any displayed numerical values in the text.

minor comments (5)

[Abstract and Section 2] The notation 'PPP2/P1' appears to be a rendering failure and should be written as $\mathbb{P}_2/\mathbb{P}_1$; the same issue occurs in the abstract and in Section 2.
[Section 4] The theorem statement contains grammatical and punctuation issues: 'Given nc and non-singular smoother M' should be 'Given nc and a non-singular smoother M', and the sentence beginning 'Consider the left and right generalized eigenvectors' should be integrated so that the hypotheses and conclusions are unambiguous.
[Eqs. (3.1) and (4.1)] Eq. (3.1) uses the damped smoother $I - \omega M^{-1}K$, while Eq. (4.1) and the theorem use $I - M^{-1}K$. The paper should clarify whether the damping parameter omega is absorbed into M, since Figure 5.2 varies omega and the theory is stated without omega.
[Section 5, Figure 5.1] The sentence 'Figure 5.1 implies that widely scattered eigenvalues can cause the iterative method to stagnate' is not a direct consequence of an eigenvalue plot; if the authors intend this as an observation, they should show corresponding iteration histories or residual curves.
[References] Reference [5] is listed as 'submitted'; if the PyAMG paper has appeared by the time of publication, the reference should be updated with journal and volume/page information.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: predicted convergence factor |1-λ_{nc+1}| is computed from the generalized eigenproblem with the actual K and M, not fitted to observed convergence.

full rationale

The paper's central predictive quantity is |1-λ_{nc+1}| from Eq. (4.2)/(4.4), quoting Theorem 5.1/Corollary 5.2 of [2]. This quantity is computed from the generalized eigenvalue problem K v = λ M v in Eq. (4.3) using the actual Stokes operator K and the actual additive Vanka relaxation operator M; it is not obtained by fitting observed iteration counts or residuals. The comparison in Section 5 (Figure 5.2) measures geometric and asymptotic convergence factors from independent V(1,0) SA-AMG runs and compares them to this eigenvalue-based number. The coarse-space dimension nc is taken from the SA-AMG interpolation operator (1095×191), not tuned to match the observed factor. Thus the 'prediction' is a function of the solver components, not of the observed convergence rates, and no equation in the paper defines the observed factor in terms of |1-λ_{nc+1}|. The heavy reliance on [2] and [19], both with overlapping authors, is a self-citation load, but [2] states explicit assumptions (K nonsingular, M nonsingular, M^{-1}K diagonalizable) that do not include the target numerical convergence factors, so under the review rules it counts as independent mathematical support rather than circularity. The unverified diagonalizability of M^{-1}K for the nonsymmetric additive Vanka operator, and the absence of an explicit minimax statement guaranteeing that the optimal-interpolation result is a lower bound for the non-optimal SA-AMG interpolation, are correctness/rigor caveats, not instances of circular reduction. No circular step is exhibited.

Assumptions & free parameters 2 free parameters · 3 assumptions · 0 invented entities

No new entities are introduced. The paper's predictive quantity |1 - λ_{nc+1}| is a function of the existing operator K, relaxation M, and chosen coarse-space size nc; these are inputs, not fitted parameters.

free parameters (2)

Vanka damping parameter ω = 0.30 to 0.84 (Table 5.1)
Damping parameter for the relaxation operator M; varied by hand to test sensitivity, not fitted to match the theoretical bound.
coarse-space size nc = 191 (standard) and 500 (aggressive) for the 1095-DoF test
Number of coarse degrees of freedom chosen by the SA-AMG hierarchy in Section 5; the predicted bound depends on nc.

assumptions (3)

standard math Generalized optimal AMG convergence theory from [2] (Theorem 5.1 and Corollary 5.3)
The paper relies on the theorem that ρ(E_TG(P♯)) = |1 - λ_{nc+1}| for optimal interpolation; this is not re-derived and is cited from prior work by overlapping authors.
domain assumption M^{-1}K is diagonalizable for both additive and multiplicative Vanka relaxation operators
The theory requires diagonalizability; the paper states the theorem's assumptions but does not verify them for the specific nonsymmetric additive Vanka operator (Section 5).
domain assumption The monolithic SA-AMG interpolation from [19] is fixed and independent of the eigenvalue analysis
The comparison treats the SA-AMG hierarchy as given; the theoretical bound applies to the optimal interpolation, and the paper implicitly assumes the actual interpolation is a valid non-optimal choice.

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Pith. "Pith review of Generalized Optimal AMG Convergence Theory for Stokes Equations Using Smooth Aggregation and Vanka Relaxation Strategies." pith.science (2026). https://pith.science/paper/U3LO7ZDW

@misc{pith2026250106621,
  author       = {Pith},
  title        = {Pith review of: Generalized Optimal AMG Convergence Theory for Stokes Equations Using Smooth Aggregation and Vanka Relaxation Strategies},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/U3LO7ZDW}},
  note         = {Machine review of arXiv:2501.06621}
}

abstract

This paper discusses our recent generalized optimal algebraic multigrid (AMG) convergence theory applied to the steady-state Stokes equations discretized using Taylor-Hood elements ($\pmb{ \mathbb{P}}_2/\mathbb{P}_{1}$). The generalized theory is founded on matrix-induced orthogonality of the left and right eigenvectors of a generalized eigenvalue problem involving the system matrix and relaxation operator. This framework establishes a rigorous lower bound on the spectral radius of the two-grid error-propagation operator, enabling precise predictions of the convergence rate for symmetric indefinite problems, such as those arising from saddle-point systems. We apply this theory to the recently developed monolithic smooth aggregation AMG (SA-AMG) solver for Stokes, constructed using evolution-based strength of connection, standard aggregation, and smoothed prolongation. The performance of these solvers is evaluated using additive and multiplicative Vanka relaxation strategies. Additive Vanka relaxation constructs patches algebraically on each level, resulting in a nonsymmetric relaxation operator due to the partition of unity being applied on one side of the block-diagonal matrix. Although symmetry can be restored by eliminating the partition of unity, this compromises convergence. Alternatively, multiplicative Vanka relaxation updates velocity and pressure sequentially within each patch, propagating updates multiplicatively across the domain and effectively addressing velocity-pressure coupling, ensuring a symmetric relaxation. We demonstrate that the generalized optimal AMG theory consistently provides accurate lower bounds on the convergence rate for SA-AMG applied to Stokes equations. These findings suggest potential avenues for further enhancement in AMG solver design for saddle-point systems.

Figures

Figures reproduced from arXiv: 2501.06621 by the authors.

**Figure 5.1.** Spectrum of complex eigenvalues for different damping parameters. [PITH_FULL_IMAGE:figures/full_fig_p008_5_1.png] view at source ↗

**Figure 5.2.** Plot of the observed geometric and asymptotic convergence factors with [PITH_FULL_IMAGE:figures/full_fig_p009_5_2.png] view at source ↗

**Figure 5.3.** Convergence behavior of two Vanka relaxation methods, Additive Vanka [PITH_FULL_IMAGE:figures/full_fig_p009_5_3.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

20 extracted references · 17 canonical work pages

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [2]

A. Ali, J. Brannick, K. Kahl, O. A. Krzysik, J. B. Schroder, and B. S. Southworth, Generalized optimal AMG convergence theory for nonsymmetric and indefinite problems , arXiv preprint arXiv:2401.11146, (2024)

arXiv 2024

[2] [19]

Voronin, S

A. Voronin, S. MacLachlan, L. N. Olson, and R. Tuminaro, Monolithic algebraic multigrid preconditioners for the Stokes equations , arXiv preprint arXiv:2306.06795, (2023)

arXiv 2023

[3] [1]

J. H. Adler, T. R. Benson, and S. P. MacLachlan, Preconditioning a mass-conserving discontinuous Galerkin discretization of the Stokes equations , Numerical Linear Algebra with Applications, 24 (2017), p. e2047

work page 2017

[4] [3]

A. Ali, J. J. Brannick, K. Kahl, O. A. Krzysik, J. B. Schroder, and B. S. Southworth, Constrained local approximate ideal restriction for advection-diffusion problems , SIAM Journal on Scientific Computing, (2024), pp. S96–S122

work page 2024

[5] [4]

Bacq and Y

P.-L. Bacq and Y. Notay, A new semialgebraic two-grid method for Oseen problems , SIAM Journal on Scientific Computing, 45 (2022), pp. S226–S253

work page 2022

[6] [5]

N. Bell, L. N. Olson, J. Schroder, and B. S. Southworth, PyAMG: Algebraic Multigrid Solvers in Python , Journal of Open Source Software, (2022). submitted

work page 2022

[7] [6]

Benzi, G

M. Benzi, G. H. Golub, and J. Liesen, Numerical solution of saddle point problems , Acta numerica, 14 (2005), pp. 1–137

work page 2005

[8] [7]

Braess and R

D. Braess and R. Sarazin, An efficient smoother for the Stokes problem , Applied Numerical Mathematics, 23 (1997), pp. 3–19

work page 1997