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arxiv: 2606.10132 · v1 · pith:SFPFCYIK · submitted 2026-06-08 · math.NA · cs.NA

WING: A Simple Windowed Nonorthogonalized Initial Guess Procedure for Repeated Matrix Solves

David Wells , Matthew G. Knepley , Boyce E. Griffith This is my paper

Reviewed by Pith T0 review T1 audit T2 compute T3 formal T4 kernel 2026-06-27 15:20 UTCgrok-4.3pith:SFPFCYIKrecord.jsonopen to challenge →

classification math.NA cs.NA
keywords initial guessKrylov methodslinear systemspseudoinversewindowed methodsnumerical linear algebrarepeated solvesnonorthogonalization
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The pith

WING lowers the cost of initial guesses for repeated linear solves by skipping orthogonalization and using a pseudoinverse on prior solutions.

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

The paper introduces the WING algorithm to generate useful starting vectors for Krylov methods that solve many linear systems sharing the same matrix but with similar right-hand sides. It modifies Fischer's second algorithm to drop the orthogonalization step entirely and instead solve the possibly singular normal equations via a pseudoinverse. The resulting initial guess still reduces iteration counts, particularly when only coarse relative tolerances are required. The method is shown on fluid-structure interaction, mantle convection, and earthquake problems. A reader would care because repeated solves appear throughout large-scale simulations, and cheaper initial guesses directly cut total runtime.

Core claim

The WING algorithm forms an initial guess from a window of previous solutions without orthogonalizing them, by solving the normal equations with a pseudoinverse. This modification reduces the expense of constructing the guess while preserving enough accuracy to lower the number of Krylov iterations needed for convergence to a given tolerance, as demonstrated across the tested applications.

What carries the argument

The WING procedure, a windowed nonorthogonalized initial guess obtained by pseudoinverse solution of the normal equations.

If this is right

  • Fewer total Krylov iterations are required to reach the target tolerance for each solve in a sequence.
  • The savings are largest when the requested tolerance is coarse rather than tight.
  • The same matrix appears repeatedly across fluid-structure interaction, mantle convection, and earthquake simulations.
  • Forming the initial guess itself becomes cheaper by omitting the orthogonalization step.

Where Pith is reading between the lines

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

  • Choosing a suitable window size may allow the pseudoinverse step to approximate the effect of orthogonalization closely enough for many practical tolerances.
  • The approach could be paired with existing preconditioners without changing the outer solver loop.
  • Similar windowed pseudoinverse ideas might apply to other iterative methods that reuse information across related right-hand sides.

Load-bearing premise

The non-orthogonalized pseudoinverse solution of the normal equations from a window of prior solutions still yields an initial guess that meaningfully reduces Krylov iterations compared to simpler alternatives.

What would settle it

A set of runs on the fluid-structure, mantle, or earthquake test problems in which WING produces no reduction in Krylov iteration count relative to a zero or previous-solution guess at the reported coarse tolerances.

Figures

Figures reproduced from arXiv: 2606.10132 by Boyce E. Griffith, David Wells, Matthew G. Knepley.

Figure 1
Figure 1. Figure 1: 𝑥, 𝑦, and 𝑧 components of the Cartesian grid velocity for the IBFE-4 example after 500 time steps. example from Section 5.2 of Griffith and Luo.13 This section only presents algorithmic details of this method relevant to the present study. A complete description of the discretization is available in Griffith and Luo.13 4.2.1 Benchmark Description In the IFED method, a structure’s velocity, which is represe… view at source ↗
Figure 2
Figure 2. Figure 2: Iteration counts for the IBFE-4 experiment for both finite element systems with a coarser relative tolerance. 0 2 4 6 8 10 Number of stored vectors 400 500 600 700 800 Total solver time, s velocity system solve time, tolerance = 10−8 (500 timesteps) Fischer 1 Fischer 2 WING POD 0 2 4 6 8 10 Number of stored vectors 400 500 600 700 800 900 1000 Total solver time, s force system solve time, tolerance = 10−8 … view at source ↗
Figure 3
Figure 3. Figure 3: Solver times for the IBFE-4 experiment for both finite element systems with a coarser relative tolerance. 9 [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Iteration counts for the IBFE-4 experiment for both finite element systems with a stricter relative tolerance. three stored vectors because they avoid the cost of frequent reorthogonalization. Indeed, the sharpest improvements across all cases come from using no initial guess to using just two stored vectors for both WING and POD. 4.2.3 Solver performance with a relative tolerance of 10−14 In this subsecti… view at source ↗
Figure 5
Figure 5. Figure 5: Solver times for the IBFE-4 experiment for both finite element systems with a stricter relative tolerance. 4.3 PyLith PyLith2, 4 is a finite element code with a primary focus on modeling interseismic and coseismic defor￾mation of Earth’s crust and upper mantle. PyLith supports 2D and 3D static, quasistatic (neglecting inertia), and dynamic (including inertia) formulations of the governing equations, which … view at source ↗
Figure 6
Figure 6. Figure 6: Boundary conditions and representative solution for the PyLith subduction problem. and Lagrange multipliers, and the strong form for the system of equations is s 𝑇 = (u 𝜆) 𝑇 (18) f(x, 𝑡) + ∇ · S(u) = 0 in Ω, (19) S · n = 𝜏(x, 𝑡) on Γ𝜏, (20) u = u0(x, 𝑡) on Γ𝑢, (21) u + − u − − d(x, 𝑡) = 0 on Γf , (22) S · n = −𝜆(x, 𝑡) on Γf +, (23) S · n = +𝜆(x, 𝑡) on Γf − , (24) in which 𝜏 is the traction and f is the bod… view at source ↗
Figure 7
Figure 7. Figure 7: Solver iteration numbers and times for the 3D subduction experiment for finite element system from PyLith. 4.4 Mantle Convection This example is based on the step-32 tutorial program which is included in the deal.II finite element library.5 deal.II is a parallel finite element library written in C++ with support for distributed linear algebra using PETSc and the HYPRE17 preconditioner library. A thorough d… view at source ↗
Figure 8
Figure 8. Figure 8: Temperature field for the mantle convection benchmark at the initial condition and after 200 time steps. The resulting linear system for velocity and pressure is the standard Stokes system  𝐴 𝐵𝑇 𝐵 0  𝑈 𝑃  =  𝐹𝑈 0  (29) in which 𝐴 is a discretization of the Laplacian and 𝐵 𝑇 is a discretization of the gradient. We solve this system with the FGMRES algorithm and a Schur complement preconditioner with a… view at source ↗
Figure 9
Figure 9. Figure 9: Iteration counts for the mantle convection benchmark. 0 2 4 6 8 10 Number of stored vectors 100 200 300 400 500 600 Total solver time, s Stokes system solve time, tolerance = 10−8 (154 timesteps) Fischer 1 WING POD 0 2 4 6 8 10 Number of stored vectors 1.8 2.0 2.2 2.4 2.6 Total solver time, s Temperature system solve time, tolerance = 10−12 (154 timesteps) Fischer 1 WING POD [PITH_FULL_IMAGE:figures/full_… view at source ↗
Figure 10
Figure 10. Figure 10: Solver times for the mantle convection benchmark. in which 𝑛 > 1 is the timestep number, 𝑘 is the timestep, and 𝜈𝛼 is a stabilization parameter discussed at length in Kronbichler et al.16 The resulting linear system, therefore, is a mass matrix plus the time step times a Laplace matrix. This is, in practice, sufficiently well-conditioned that we solve it with the conjugate gradient algorithm and a Jacobi … view at source ↗
read the original abstract

Many numerical methods require solution of a sequence of linear systems with the same matrix and similar right-hand sides. Krylov subspace methods are a common tool for solving such linear systems, and a carefully chosen initial guess for the solution can reduce the total number of iterations, and thereby the total computational cost, required for convergence to a specified numerical tolerance. This paper introduces the WING algorithm, a modification of Fischer's second algorithm, which lowers the cost of forming an acceptably close initial guess by skipping orthogonalization and solving the possibly singular normal equations with a pseudoinverse. We demonstrate the efficacy of the new algorithm, particularly for solving linear systems with coarse relative tolerances, with numerical benchmarks based on fluid-structure interaction, mantle convection, and earthquake models.

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

1 major / 0 minor

Summary. The paper introduces the WING algorithm, a modification of Fischer's second algorithm for generating initial guesses when solving sequences of linear systems with fixed matrix and similar right-hand sides. WING skips orthogonalization of a window of prior solutions and solves the (possibly singular) normal equations via pseudoinverse to reduce the cost of forming the guess. Numerical benchmarks on fluid-structure interaction, mantle convection, and earthquake models are used to demonstrate that the approach reduces Krylov iterations, particularly at coarse relative tolerances.

Significance. If the pseudoinverse-based non-orthogonalized window reliably produces initial guesses that reduce iteration counts without instability from correlated vectors, the method would provide a simple, parameter-free alternative to orthogonalized projection techniques for repeated solves. The direct definition, external benchmark evaluation, and focus on coarse-tolerance regimes are strengths that could make the procedure practically useful in applications such as time-dependent simulations.

major comments (1)
  1. [Abstract] Abstract: the central performance claim that WING yields 'meaningful' Krylov savings 'particularly for coarse relative tolerances' rests on the unexamined assumption that the pseudoinverse solution of the normal equations remains stable and superior to the trivial previous-solution guess when the window vectors are nearly linearly dependent (the motivating regime of similar right-hand sides). No conditioning analysis of the Gram matrix or controlled comparison under varying correlation strength is supplied, leaving the weakest assumption untested.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the review and address the single major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central performance claim that WING yields 'meaningful' Krylov savings 'particularly for coarse relative tolerances' rests on the unexamined assumption that the pseudoinverse solution of the normal equations remains stable and superior to the trivial previous-solution guess when the window vectors are nearly linearly dependent (the motivating regime of similar right-hand sides). No conditioning analysis of the Gram matrix or controlled comparison under varying correlation strength is supplied, leaving the weakest assumption untested.

    Authors: We acknowledge that the manuscript supplies neither a conditioning analysis of the Gram matrix nor a controlled synthetic comparison that varies correlation strength. The WING procedure computes the minimum-norm solution of the normal equations via the Moore-Penrose pseudoinverse precisely to remain defined when the window vectors become linearly dependent. The performance statements in the abstract rest on the three application benchmarks (fluid-structure interaction, mantle convection, earthquake models) in Sections 4.1--4.3, where the right-hand sides are temporally correlated by construction and the solution vectors therefore exhibit the dependence regime of interest. In those experiments WING reduces Krylov iterations relative to the previous-solution guess, with the reduction being largest at the coarser tolerances examined. While an explicit stability study would add insight, the reported numerical evidence is drawn directly from the motivating correlated regime. revision: no

Circularity Check

0 steps flagged

No circularity; algorithm defined directly and benchmarked externally

full rationale

The paper introduces WING as an explicit algorithmic modification of Fischer's second algorithm: it skips orthogonalization of the window and solves the (possibly singular) normal equations via pseudoinverse. This definition stands on its own without reducing to fitted parameters, self-referential definitions, or load-bearing self-citations. Efficacy is shown via numerical experiments on independent benchmark problems (fluid-structure interaction, mantle convection, earthquake models) rather than by construction from the inputs. No steps match any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard properties of Krylov methods and pseudoinverses for singular least-squares problems, with efficacy shown via numerical benchmarks on specific application models.

axioms (2)
  • standard math Properties of the Moore-Penrose pseudoinverse for solving possibly singular normal equations hold in the context of the windowed guess construction.
    Invoked when describing the solution of the normal equations without orthogonalization.
  • domain assumption A window of prior solutions provides a useful subspace for initial guess construction in the target applications.
    Underlying the windowed aspect of the algorithm.

pith-pipeline@v0.9.1-grok · 5662 in / 1291 out tokens · 29127 ms · 2026-06-27T15:20:50.202338+00:00 · methodology

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Reference graph

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