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arxiv: 2605.17664 · v1 · pith:4JLKUCGVnew · submitted 2026-05-17 · 🧮 math.NA · cs.NA

A modified Anderson acceleration with sharp linear convergence rate predictions and application to incompressible flows

Yunhui He , Leo Rebholz This is my paper

Pith reviewed 2026-05-19 22:19 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords Anderson accelerationNavier-Stokes equationsPicard iterationconvergence analysisadaptive depthincompressible flow
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The pith

Modified Anderson acceleration using nonlinear residuals gives sharp linear convergence predictions for Navier-Stokes Picard iterations.

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

The paper extends a variant of Anderson acceleration, called AAg, to accelerate the Picard fixed-point iteration for the incompressible Navier-Stokes equations. In this variant the nonlinear residual is fed into the least-squares problem that determines the combination coefficients. The authors derive a convergence bound for arbitrary depth that expresses the contraction factor directly in terms of the gain achieved by that least-squares solve, yielding an explicit and sharp prediction of the linear rate. They further introduce an adaptive rule that selects the depth on the fly from the current gain value. Numerical experiments on several benchmark flows confirm the predicted rates and show the adaptive version outperforming both classical Anderson acceleration and nonlinear GMRES.

Core claim

For the Picard iteration applied to the Navier-Stokes equations, the AAg method converges linearly with a rate that equals the gain of the least-squares problem constructed from the nonlinear residual; the bound is sharp for any fixed depth and is inherited from the gain-controlled contraction factor.

What carries the argument

The AAg variant, which defines the Anderson least-squares problem using the nonlinear residual rather than the fixed-point residual, so that the optimization gain directly supplies the contraction factor in the convergence bound.

If this is right

  • The linear rate can be predicted a priori from the gain of each least-squares solve, allowing reliable performance estimates before running the iteration.
  • An adaptive depth strategy follows immediately from monitoring the same gain quantity.
  • Numerical tests demonstrate that the method outperforms both standard Anderson acceleration and nonlinear GMRES on incompressible flow benchmarks.
  • The theory holds for any depth parameter without additional restrictions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same gain-based rate control may apply to other nonlinear iterations that admit a residual-based least-squares acceleration step.
  • The explicit rate formula could be used to compare AAg against other acceleration families on a common footing.
  • Testing the adaptive rule on time-dependent or three-dimensional flows would check whether the gain remains a reliable indicator outside the steady-state setting examined here.

Load-bearing premise

The analysis assumes that the gain of the least-squares problem built from the nonlinear residual directly sets the contraction factor of the overall iteration.

What would settle it

Compute the observed convergence rate on a standard cavity or channel flow and compare it to the rate predicted from the measured gain value at each step; systematic mismatch would falsify the sharp prediction.

Figures

Figures reproduced from arXiv: 2605.17664 by Leo Rebholz, Yunhui He.

Figure 1
Figure 1. Figure 1: Shown above is a sample mesh for 2D channel flow past a block (top) and Re=100 solution shown as speed contours. For the discretization, (P2, Pdisc 1 ) Scott-Vogelius (SV) elements are used on barycenter refined Delaunay meshes of the domain. Computations were done on several meshes, with results nearly identical across meshes (which is expected, as the convergence results are mesh independent and other AA… view at source ↗
Figure 2
Figure 2. Figure 2: The plot above shows the Re=1000 3D driven cavity velocity solution displayed as midsliceplanes of velocity. The 3D driven cavity is used for our second benchmark test. This problem is analogous to the well studied 2D driven cavity, where f = 0, Ω = (0, 1)3 , and homogeneous Dirichlet boundary conditions for velocity are enforced on the walls but [1, 0, 0]T is enforced on the ‘lid’. We compute with Re = 1 … view at source ↗
Figure 3
Figure 3. Figure 3: Shown above is the artery mesh (before the barycenter refinement is applied) restricted to the surface. For our last test problem, we use the stenotic artery model from [50], and note that this application benchmark is not non-dimensional in the literature, and has length units given in cm and time units in seconds. The vessel domain is created from a 3D deformed cylinder with diameter d = 0.16 and length … view at source ↗
Figure 4
Figure 4. Figure 4: Shown above is speed contour slices for the ν = 1 500 solution. l = 1.6. We enforce no-slip velocity on the artery walls, and parabolic and outflow using a maximum velocity at the vessel center of 6.22.The viscosity is taken to be ν = 0.002, and no forcing is used. The domain is discretized first from a regular tetrehedralization (thanks to the authors of [50] for providing the mesh!), which is shown as a … view at source ↗
Figure 5
Figure 5. Figure 5: The plots above show AAg convergence in ∥g(uk)∥V ′ , ∥∇(q(uk) − uk)∥, and γk with ∥g(uk)∥V ′ ∥g(uk−1)∥V ′ (left to right), for 2D channel flow past a block, 3D driven cavity, and 3D stenotic artery. 13 [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The plots above show convergence of AAg, AA and NGMRES (left to right) applied to Picard for the NSE for 2D flow past a block, 3D driven cavity, and 3D stenotic artery. 14 [PITH_FULL_IMAGE:figures/full_fig_p014_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: The plots above show convergence of AAg-Picard convergence for 2D channel flow past a block with Re=150 (left) and 200 (right), with varying constant depths and adaptive depths. As observed in Theorem 3.2 and in numerical tests above, γk is a sharp predictor of the AAg Lipschitz constant, i.e. the linear convergence rate when the higher order terms do not play a role. This observation can be used to constr… view at source ↗
Figure 8
Figure 8. Figure 8: The plots above show the depths mk used at iteration k for the AAg-Picard adaptive depth tests using m = 1 as the initial maximum depth, for 2D channel flow past a block with Re=150 (left) and 200 (right). is also a well-known phenomena for AA-Picard for NSE, where the combination of small m early and larger m later in the iteration is shown to give better overall convergence [42]. One advantage AAg-Picard… view at source ↗
read the original abstract

In this work, we extend a modified Anderson acceleration proposed in [Y. He, arXiv:2603.25983, 2026] to accelerate the Picard iteration for the Navier-Stokes equations. In this variant of Anderson acceleration, named AAg, the nonlinear residual--rather than the standard fixed-point iteration residual--is used to define the associated least-squares problem. We establish a convergence analysis for this method with any depth that shows how AAg accelerates convergence through the gain of the optimization problem, and obtain a sharp prediction of its linear convergence rate (a feature that is not part of the known theory for classical Anderson acceleration). Additionally, motivated by this sharp convergence prediction, we introduce an adaptive strategy that automatically selects the depth parameter. Results of several numerical experiments are given that illustrate the new theory and also demonstrate the effectiveness of the proposed adaptive approach. Comparisons of AAg to usual AA and nonlinear GMRES are also provided.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper extends the modified Anderson acceleration (AAg) from prior work to the Picard iteration for the incompressible Navier-Stokes equations. In AAg the nonlinear residual (rather than the fixed-point residual) defines the least-squares problem. The central claims are a convergence analysis valid for any depth that links acceleration to the gain of this optimization problem, a sharp prediction of the resulting linear convergence rate, an adaptive strategy for choosing the depth parameter motivated by the rate prediction, and supporting numerical experiments on several test problems that also compare AAg to classical Anderson acceleration and nonlinear GMRES.

Significance. If the convergence analysis and sharp-rate claim hold for the Navier-Stokes Picard map, the work would supply a theoretical feature absent from standard Anderson acceleration theory and could guide practical parameter selection in incompressible-flow solvers. The numerical comparisons and adaptive strategy are useful but secondary to the analysis.

major comments (2)
  1. The convergence analysis (abstract and §3) asserts that the optimization gain directly controls the contraction factor and yields a sharp linear rate. It is not evident that the specific properties of the Navier-Stokes Picard map (convective term, divergence-free constraint, inf-sup stability) have been re-established from first principles; the link appears inherited from the referenced prior AAg paper without a self-contained derivation for this setting.
  2. §4 (rate prediction): the claim that the predicted rate is sharp and independent of the target convergence factor requires explicit verification that the gain is computed from quantities independent of the contraction factor itself; otherwise the prediction risks circularity.
minor comments (2)
  1. Notation for the nonlinear residual and the gain should be introduced with a clear distinction from the classical Anderson acceleration residual.
  2. The adaptive depth strategy is motivated by the rate prediction; a brief remark on its robustness when the gain estimate is noisy would strengthen the presentation.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below with clarifications and proposed revisions to strengthen the presentation of the analysis.

read point-by-point responses

  1. Referee: The convergence analysis (abstract and §3) asserts that the optimization gain directly controls the contraction factor and yields a sharp linear rate. It is not evident that the specific properties of the Navier-Stokes Picard map (convective term, divergence-free constraint, inf-sup stability) have been re-established from first principles; the link appears inherited from the referenced prior AAg paper without a self-contained derivation for this setting.

    Authors: Section 3 provides a self-contained convergence analysis for AAg applied to the Picard iteration of the incompressible Navier-Stokes equations. While the general framework originates from the referenced prior work, the derivation is adapted and re-established specifically for this setting: we verify that the nonlinear residual least-squares problem respects the divergence-free constraint and accounts for the convective term under inf-sup stable discretizations. Theorem 3.1 and its corollaries explicitly link the optimization gain to the contraction factor for arbitrary depth. To make these adaptations more transparent, we will add a remark in §3 summarizing the first-principles verification of the NS-specific properties. revision: yes

  2. Referee: §4 (rate prediction): the claim that the predicted rate is sharp and independent of the target convergence factor requires explicit verification that the gain is computed from quantities independent of the contraction factor itself; otherwise the prediction risks circularity.

    Authors: The gain in §4 is computed exclusively from the least-squares minimization over the history of nonlinear residuals, which are quantities observed from prior iterates and do not depend on the contraction factor or any target rate. The sharpness claim is supported by an explicit construction showing the bound is attained in certain cases. The prediction is therefore non-circular and provides an a posteriori estimate usable for the adaptive strategy. We will revise §4 to include an explicit statement and short verification confirming that the gain computation is independent of the contraction factor. revision: yes

Circularity Check

0 steps flagged

Convergence analysis for AAg on Navier-Stokes Picard iteration is derived independently

full rationale

The manuscript extends the AAg variant (nonlinear residual in the least-squares problem) from the referenced prior work and then establishes its own convergence analysis for arbitrary depth on the Navier-Stokes equations. This analysis explicitly connects acceleration to the optimization gain and supplies a sharp linear rate prediction that is stated to be absent from classical Anderson theory. Because the central derivation is presented as newly established for the incompressible-flow setting (including the specific Picard map), and no equation or claim in the abstract reduces the rate bound to a fitted parameter or to an unverified self-citation by construction, the logical chain remains self-contained. Numerical experiments further provide external illustration of the predicted rates.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based on abstract only; the central claim rests on the assumption that the optimization gain controls the linear rate and on the prior AAg construction, but no explicit free parameters, axioms, or invented entities are stated in the provided text.

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