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REVIEW 4 major objections 7 minor 49 references

MatExPre: A matrix exponential preconditioner for the high-frequency Helmholtz equation

T0 review · 4 major / 7 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read MatExPre approximates the inverse Helmholtz operator by a truncated matrix-exponential series and, with a 1/ω complex shift, solves high-frequency Helmholtz systems with total sparse matrix-vector work that grows only linearly in frequency.

desk verdict A clean and reproducible algebraic Helmholtz preconditioner with honest numerics; the O(omega) complexity is empirical, not proven, but the paper deserves serious review. read the letter →

arxiv 2506.03617 v1 pith:VIN4M2AY submitted 2025-06-04 math.NA cs.NA

Shubin Fu , Qing Huo Liu , Qiwei Zhan , Eric T. Chung , Changqing Ye This is my paper
classification math.NAcs.NA MSC 65F0865F1065F6065N2235J05
keywords Helmholtzequationmatrixexponentialpreconditionerhigh-frequencywavesSchrödinger-liketime-domainsolverGMRESandFGMREScomplexshiftspectralgapacousticwavemodeling
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

The paper introduces MatExPre, a preconditioner for the discretized high-frequency Helmholtz equation built entirely from sparse matrix-vector products. It starts from a Schrödinger-like time-domain reformulation and turns the resulting fixed-point iteration into an algebraic matrix-exponential identity for the inverse operator. The key estimate places the eigenvalues of the preconditioned matrix inside a small disk around 1, so nested Krylov solvers converge quickly. Numerical experiments on 2D and 3D homogeneous, lens, and benchmark seismic models report outer FGMRES iterations of 7–14 and inner iterations growing linearly with frequency, implying total work O(ω) at fixed points per wavelength.

What carries the argument

The load-bearing object is the matrix-exponential identity $A^{{-1}}$ = -it(Σ_{n=0}^∞ ψ_0(itA)^n)ψ_1(itA), truncated to N+1 terms as P_N. The ψ_l functions are the standard exponential-integrator entire functions defined by ψ_0(z)=e^z and ψ_{l+1}(z)=(ψ_l(z)-1/l!)/z; they turn the time integral of exp(iτA) into a single matrix-function application. P_N shares eigenvectors with A, and the eigenvalues of P_N A are 1 - exp(i(N+1)tλ_k), so convergence is controlled by the imaginary spectral gap λ* = min_k Im(λ_k). The inner GMRES solve of (I-exp(itA))z = r accelerates the geometric series, the outer FGMRES handles the preconditioner's variability, and the complex shift A_s = A + iω²sD is the tuning mechanism that keeps the gap effective at high frequency.

What would settle it

Compute all eigenvalues of a 2D or 3D PML Helmholtz matrix at several increasing frequencies and check whether min_k Im(λ_k) stays bounded away from zero; if it decays like 1/ω or touches zero, the radius exp(-(N+1)tλ*) stops shrinking and the inner solve stalls. Alternatively, run the reported fixed-ppw experiments at freq = 640 or 1280 and test whether total inner GMRES iterations still grow linearly in ω rather than quadratically.

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Extended reading notes

Core claim

The central claim is that the inverse of a PML-damped Helmholtz matrix A can be approximated by the truncated ψ-function series P_N = -it(Σ_{n=0}^N ψ_0(itA)^n)ψ_1(itA), and that this approximation is good enough to precondition GMRES at high frequency. The identity comes from writing the time integral of a matrix exponential as tψ_1(itA) and inverting I - exp(itA) by its geometric series. For any eigenvalue λ of A with Im(λ) ≥ λ* > 0, the eigenvalues of the preconditioned matrix P_N A lie in a disk centered at 1 with radius exp(-(N+1)tλ*), which explains why the inner fixed-point equation converges. The paper conjectures λ* > 0 for the damped Helmholtz operator and supports it with a non-negativity proof plus 1D eigenvalue scans, then demonstrates that a complex shift A + iω²sD enlarges the effective gap and stabilizes the outer solver. The resulting algorithm requires only sparse matrix-vector products and achieves, in the reported tests, total work linear in ω.

Load-bearing premise

Everything rests on the spectral gap λ* = min_k Im(λ_k) being strictly positive for the damped or PML-truncated Helmholtz operator; the paper proves only non-negativity, conjectures λ* > 0 from the Sommerfeld radiation condition and 1D scans, and every convergence rate is exponential in -λ*.

Editorial extensions

If this is right

  • With s = 1/ω and t ∝ 1/ω², outer FGMRES iterations stay nearly constant across frequency (7–14 in the reported tests), while inner GMRES iterations grow roughly linearly, so total SpMV work is O(ω).
  • The preconditioner needs no matrix factorization and no sequential sweeping pass, so its parallel scalability is set by SpMV and Krylov inner products; reported strong-scaling efficiency is near 1 at freq = 40 and around 80% at freq = 80, with weak scaling near 80%.
  • The same construction handles PML, first-order absorbing, and sound-hard boundary conditions without modification, as shown on 2D and 3D variable-velocity models including benchmark seismic data.
  • Eigenvalue clustering gives a direct design rule: choose t and N so (N+1)tλ* is large enough that exp(-(N+1)tλ*) is small, and set the complex shift s to balance outer versus inner iteration counts.
  • Relaxing the inner tolerance to about 0.08 leaves outer iteration counts unchanged while cutting inner work, which is useful at large scale.

Reading between the lines

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

  • The paper's O(ω) lower-bound argument via wave-support spreading is heuristic; a rigorous complexity statement would need to bound inner GMRES convergence in terms of eigenvector conditioning, which the paper leaves open. If that conditioning degrades with model complexity, the method's advantage on strongly heterogeneous media would shrink.
  • The radius estimate exp(-(N+1)tλ*) with t ≈ O(ω^{-2}) would naively predict poor approximation unless N grows with ω; the observed linear scaling therefore likely owes as much to GMRES's clustering behavior on non-normal matrices as to the disk radius, a mechanism the paper does not isolate.
  • A natural test of the spectral-gap conjecture is to use a few Krylov iterations on the PML matrix to estimate λ* on the fly and choose t adaptively; this could remove the hand-tuned t ∝ 1/ω² rule.
  • If λ* can be shown bounded below for PML-truncated acoustic operators, the same Schrödinger-exponential construction may extend to time-harmonic Maxwell and elasticity, where the operator is also non-self-adjoint after damping; the paper does not claim this.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 7 minor

Summary. The manuscript proposes MatExPre, a preconditioner for the discretized Helmholtz equation built from the identity A^{-1} = -it(Σ_{n=0}^∞ ψ_0(itA)^n)ψ_1(itA). The authors derive a Schrödinger-like time-domain fixed-point solver, connect it to exponential integrators, analyze the spectral clustering of the truncated polynomial P_N A, and introduce a complex shift A_s = A + iω²sD. Numerical experiments on 2D and 3D homogeneous models and on the SEG/EAGE salt, overthrust, and Marmousi-II benchmarks report 7–14 outer FGMRES iterations and inner GMRES counts that grow roughly linearly with frequency, leading to the claim that the number of SpMV operations grows linearly with frequency.

Significance. If the O(ω) SpMV claim holds robustly, MatExPre would be a useful contribution to high-frequency Helmholtz preconditioning because it avoids matrix factorizations and heavy sequential routines, uses only sparse matrix-vector products, and has a clean algebraic foundation. The identity (8) is parameter-free and elegant, Theorem 3.1 gives a transparent spectral clustering result under Im λ ≥ λ*>0, and the paper is honest about its conjectured strict spectral gap and about known GMRES limitations on non-normal matrices. The source code is publicly available, which strengthens reproducibility. However, the central complexity claim currently rests on numerical extrapolation rather than on a theorem controlling GMRES/FGMRES on non-normal systems, and the reported evidence for linear growth is not uniform across all tables.

major comments (4)
  1. [Section 3.4, Algorithm 1, and the Conclusion] The claim that the total number of SpMV operations grows linearly with frequency is not established by the presented theory. Theorem 3.1 analyzes the fixed polynomial P_N, but Algorithm 1 solves (I−exp(itA_s))w = r with inner GMRES and wraps this in outer FGMRES; the total SpMV count also includes the SLEPc evaluations of ψ1(itA_s) and exp(itA_s) inside each inner iteration. The lower-bound argument in Section 3.4 is only a lower bound, so it cannot justify an upper bound. The tables report iteration counts, not total SpMV counts, and the outer count and the per-iteration matrix-function cost are never combined into a single complexity measure. Please either provide a theorem or a careful accounting for the total SpMV count, or state the linear-growth claim explicitly as empirical.
  2. [Section 3.1 and inequality (12)] The theory does not control non-normality. After noting, with reference [30], that GMRES convergence cannot be inferred from eigenvalues alone, the only quantitative bridge is inequality (12), which contains cond(X). No control of cond(X) is given for the PML-discretized Helmholtz matrices, and no bound is provided for the outer FGMRES iteration count on A A_s^{-1} in the regime s = 1/ω, where [26] predicts possible deterioration for small shifts. The numerical linear-growth results in Tables 3, 5, 7, 8, and 9 could therefore be specific to the test problems. Please add a discussion or analysis of non-normality, or at least a systematic numerical study of cond(X) and the total SpMV counts.
  3. [Section 2.1 and Theorems 2.2/2.3] The convergence theorems rest on a finite spectral decomposition of the source g and on the conjectured strict inequality λ*>0; Proposition 2.1 only proves Im λ ≥ 0, and the manuscript explicitly labels λ*>0 as a conjecture. As written, this theory does not directly cover Algorithm 1, which uses the shifted operator A_s. Since Im λ(A_s) ≥ ω²s λ_min(D) > 0, the shift removes the need for the conjecture for the implemented method. Please state this explicitly and give the corresponding version of Theorem 3.1 for A_s, including the resulting scaling of t λ*_s, so that the link between the theory and the observed O(ω) inner iteration counts is explicit and not left to the reader.
  4. [Tables 2 and 8; Section 4.1.2] The numerical evidence for linear growth is mixed. In Table 2 (s = 0), the first-FGMRES inner iterations are 15, 25, 43, 108 for freq = 40, 80, 160, 320; the ratio over an 8-fold frequency increase is 7.2, and the increase from 160 to 320 is 108/43 ≈ 2.5 for a 2-fold frequency increase. In Table 8, the overthrust model shows inner iterations 8.3 at freq = 40 and 24.8 at freq = 80, a threefold increase for a twofold frequency increase, which is superlinear. Please report total SpMV counts, including matrix-function evaluations, and either fit the scaling or qualify the linear-growth conclusion accordingly.
minor comments (7)
  1. [Proposition 2.1 proof] The displayed identity appears to miss a complex conjugation: taking \bar{Φ}/c² as the test function gives the stated result, whereas the written ∫ Φ∆Φ dx is not equal to −∫|∇Φ|² dx for complex Φ. The conclusion is correct after conjugation, but the proof should be corrected.
  2. [Theorems 2.2 and 2.3] The word 'convergences' should be 'converges' in both theorem statements.
  3. [Section 2.2] The estimate ||(ω²I+H)^{-1}|| ≈ O(ω^{-1}) is described as a postulate; this is a wavenumber-explicit regularity estimate and should be cited or proven rather than presented as an ad-hoc assumption.
  4. [Tables 6 and 7] The column alignment in Tables 6 and 7 is difficult to read because frequency, DoF count, outer iterations, and inner iterations are packed together without clear spacing; please reformat these tables.
  5. [Section 4.1.2] The phrase 'increases almost linearly in freq' for Table 2 should be quantified, since some rows (for example, the 320-row values 108, 76, 95, 110) do not follow a clean linear trend.
  6. [Figure 6] The left panel plots two time series with different scales but a single 'time (s)' axis label; consider separate panels or a legend that clarifies the two scales.
  7. [Abstract] The abstract states 'rigorous connection', while the body repeatedly relies on conjectures and formal spectral decompositions; consider softening 'rigorous' to 'algebraic' or 'exact at the discrete level'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the MatExPre derivation rests on an exact matrix-exponential identity, the O(omega) complexity claim is empirical, and the lambda* > 0 gap is an openly stated conjecture rather than a fitted input.

full rationale

The central preconditioner comes from the exact identity (8)-(9): A^{-1} = -it(Sum_{n=0}^infty psi_0(itA)^n) psi_1(itA), which is an algebraic consequence of the variation-of-constants formula for the Schrodinger-like equation; truncating and multiplying by A gives P_N A = I - exp(i(N+1)tA), so Theorem 3.1 restates the definition rather than assuming its conclusion. Algorithm 1 applies the shifted operator A_s = A + i omega^2 s D of (13); the choices t = O(omega^{-2}) and s = O(omega^{-1}) are justified by external references [1] and [26] and by the spectral-radius estimate 1 - O(omega^{-1}), so the observed roughly linear inner-GMRES growth is a reported consequence of those parameter scalings, not a fitted value renamed as a prediction. The strict gap lambda* > 0 is explicitly introduced as a conjecture in Section 2.1 and is used only as a hypothesis in Theorems 2.2 and 3.1; a conditional theorem with an acknowledged open assumption is not circular. The paper also flags, rather than hides, the non-normal GMRES gap: equation (12) contains the eigenvector condition number cond(X), and the text cites [30] and states that GMRES convergence cannot be inferred from eigenvalues. The only author-derived citation, [37] by co-author Q. H. Liu, is used for background and quadrature technique ('as in [37]') and is not load-bearing for the algebraic derivation or for the numerical benchmarks, which are external SEG/EAGE, overthrust, and Marmousi-II models. Overall, no equation reduces to its own input and no prediction is forced by construction.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The central claim is supported by a mix of exact algebraic identities and unproven spectral assumptions. The method's efficiency depends on user-chosen parameters t, s, and fprtol, which are tuned experimentally. No new physical entities are introduced.

free parameters (4)
  • Time length t = t = 0.4/freq^2 (also 0.8, 1.2, 1.6 variants; 0.1/freq^2 for lens models)
    Controls the Krylov iteration count for evaluating exp(iτA)v and the contraction factor exp(-λ*T). Set by hand to balance inner GMRES cost and preconditioner quality; the specific constant 0.4 is chosen after the iteration count study in Table 1.
  • Complex shift parameter s = s = 1/freq (also 2/freq, 3/freq, 4/freq tested)
    Shifts the operator A_s = A + iω²sD used inside the preconditioner to enlarge Im(λ) and speed up inner GMRES. Chosen as O(ω^{-1}) based on Gander-Graham-Spence theory; exact constant tuned by experiments.
  • Inner fixed-point tolerance fprtol = 0.01 to 0.08 (0.08 adopted for 3D runs)
    Stops the inner GMRES solve of (I - exp(itA_s))w = r. Increased tolerance reduces inner iterations with minimal effect on outer FGMRES; values selected experimentally (Table 5).
  • PML strength C_pml = C_pml = 20
    Controls the absorbing strength of the PML and influences the eigenvalue imaginary part λ*. Fixed following practical thumb rules, but Section 3.3 shows the minimal imaginary part depends on it.
assumptions (4)
  • domain assumption Strict spectral gap conjecture: λ* = min_k Im(λ_k) > 0 for the complex absorbing potential or PML operator H.
    Invoked in Theorem 2.2 to guarantee contraction exp(-λ*T) and in Theorem 3.1 for eigenvalue clustering. Only Im(λ) ≥ 0 is proven (Proposition 2.1); strict positivity is conjectured from Sommerfeld condition and numerical evidence (Section 2.1, Section 3.3).
  • ad hoc to paper The source term g admits a finite spectral decomposition in eigenfunctions of H.
    Theorems 2.2 and 2.3 prove convergence by reducing to a finite number of eigenmodes; realistic delta-function sources do not satisfy this, so the proof does not cover the numerical experiments. Stated as 'Suppose that the source term g admits a finite spectral decomposition'.
  • domain assumption The discretized PML Helmholtz matrix A is diagonalizable with eigenvalues having positive imaginary parts.
    Used in Section 3.1 to derive the identity A^{-1} = -i(I - exp(itA))^{-1} ∫ exp(iτA)dτ and the truncation P_N. Diagonalizability is assumed, not proven for PML-discretized matrices.
  • domain assumption Wavenumber-explicit estimate ||(ω²I+H)^{-1}||_{L2→L2} ≈ O(ω^{-1}) holds for the PML or complex potential operator.
    Postulated in Section 2.2 to justify choosing T = O(ω^{-1}); the paper says 'we can postulate' without proof. Underlies the claimed complexity scaling.

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Cite this review

Pith. "Pith review of MatExPre: A matrix exponential preconditioner for the high-frequency Helmholtz equation." pith.science (2026). https://pith.science/paper/VIN4M2AY

@misc{pith2026250603617,
  author       = {Pith},
  title        = {Pith review of: MatExPre: A matrix exponential preconditioner for the high-frequency Helmholtz equation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VIN4M2AY}},
  note         = {Machine review of arXiv:2506.03617}
}
read the original abstract

In this article, we present a new preconditioner, MatExPre, for the high-frequency Helmholtz equation by leveraging the properties of matrix exponentials. Our approach begins by reformulating the Helmholtz equation into a Schr\"{o}dinger-like equation and constructing a time-domain solver based on a fixed-point iteration. We then establish a rigorous connection between the time-domain solver and matrix exponential integrators, which enables us to derive algebraic preconditioners that rely solely on sparse matrix-vector products. Spectral analysis and a detailed numerical implementation strategy, including performance improvements achieved through complex shifting, are discussed. Finally, numerical experiments on 2D and large-scale 3D homogeneous and inhomogeneous models, including benchmark seismic examples, substantiate the effectiveness and scalability of the proposed methods.

Figures

Figures reproduced from arXiv: 2506.03617 by the authors.

Figure 1
Figure 1. Eigenvalues of the 1D Helmholtz operator with the PML method. The standard three-point FD [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. The minimal imaginary part of eigenvalues min [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. The minimal imaginary part of eigenvalues min [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figures from the paper (7 more)
Figure 4
Figure 4. Figure 4: The solution magnitude from the first four outer FGMRES iterations for the 2D homogeneous [PITH_FULL_IMAGE:figures/full_fig_p014_4.png]
Figure 5
Figure 5. Figure 5: Magnitude of the solutions for the 3D homogeneous Helmholtz equation at frequencies [PITH_FULL_IMAGE:figures/full_fig_p015_5.png]
Figure 6
Figure 6. Figure 6: Strong scalability results for the 3D homogeneous Helmholtz equation with [PITH_FULL_IMAGE:figures/full_fig_p017_6.png]
Figure 7
Figure 7. Figure 7: Weak scalability results for the 3D homogeneous Helmholtz equation. The plot shows the time [PITH_FULL_IMAGE:figures/full_fig_p017_7.png]
Figure 8
Figure 8. Figure 8: (a) The scaled velocity field of the SEG/EAGE salt model with the PML layer enclosed. The real [PITH_FULL_IMAGE:figures/full_fig_p018_8.png]
Figure 9
Figure 9. Figure 9: (a) The scaled velocity field of the SEG/EAGE overthrust model with the PML layer enclosed. The [PITH_FULL_IMAGE:figures/full_fig_p019_9.png]
Figure 10
Figure 10. Figure 10: The P-wave velocity field of the Marmousi-II model. The real part of the solution for the [PITH_FULL_IMAGE:figures/full_fig_p020_10.png]

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Works this paper leans on

49 extracted references · 23 canonical work pages

  1. [37]

    Luo and Q

    S. Luo and Q. H. Liu , A fixed-point iteration method for high frequency Helmholtz equations , Journal of Scientific Computing, 93 (2022), https://doi.org/10.1007/s10915-022-02039-8

    work page doi:10.1007/s10915-022-02039-8 2022

  2. [30]

    Greenbaum, V

    A. Greenbaum, V. Pt\' a k, and Z. Strako s , Any nonincreasing convergence curve is possible for GMRES , SIAM Journal on Matrix Analysis and Applications, 17 (1996), pp. 465--469, https://doi.org/10.1137/s0895479894275030

    work page doi:10.1137/s0895479894275030 1996

  3. [26]

    M. J. Gander, I. G. Graham, and E. A. Spence , Applying GMRES to the Helmholtz equation with shifted Laplacian preconditioning: what is the largest shift for which wavenumber-independent convergence is guaranteed? , Numerische Mathematik, 131 (2015), pp. 567--614, https://doi.org/10.1007/s00211-015-0700-2

    doi:10.1007/s00211-015-0700-2 2015

  4. [1]

    A. H. Al-Mohy and N. J. Higham , Computing the action of the matrix exponential, with an application to exponential integrators , SIAM Journal on Scientific Computing, 33 (2011), pp. 488--511, https://doi.org/10.1137/100788860

    doi:10.1137/100788860 2011

  5. [2]

    Aminzadeh, J

    F. Aminzadeh, J. Brac, and T. Kunz , 3-D salt and overthrust models , no. 1 in SEG/EAGE 3-D salt and overthrust models, Society of Exploration Geophysicists, US, 1997

    work page 1997

  6. [3]

    Appel\" o , F

    D. Appel\" o , F. Garcia, and O. Runborg , WaveHoltz : Iterative solution of the Helmholtz equation via the wave equation , SIAM Journal on Scientific Computing, 42 (2020), pp. A1950--A1983, https://doi.org/10.1137/19m1299062

    work page doi:10.1137/19m1299062 2020

  7. [4]

    Arnold, S

    A. Arnold, S. Geevers, I. Perugia, and D. Ponomarev , On the limiting amplitude principle for the wave equation with variable coefficients , Communications in Partial Differential Equations, 49 (2024), pp. 333--380, https://doi.org/10.1080/03605302.2024.2341070

    arXiv 2024

  8. [5]

    Auchmuty, E

    G. Auchmuty, E. Dean, R. Glowinski, and S. C. Zhang , Control methods for the numerical computation of periodic solutions of autonomous differential equations , in Control Problems for Systems Described by Partial Differential Equations and Applications, I. Lasiecka and R. Triggiani, eds., Berlin, Heidelberg, 1987, Springer, pp. 64--89

    work page 1987

Show all 49 references
  1. [6]

    Babu s ka and S

    I. Babu s ka and S. A. Sauter , Is the pollution effect of the FEM avoidable for the Helmholtz equation considering high wave numbers? , SIAM Journal on Numerical Analysis, 34 (1997), pp. 2392--2423, https://doi.org/10.1137/s0036142994269186

    1997 doi

  2. [7]

    Benzi and G

    M. Benzi and G. H. Golub , A preconditioner for generalized saddle point problems , SIAM Journal on Matrix Analysis and Applications, 26 (2004), pp. 20--41, https://doi.org/10.1137/s0895479802417106

    2004 doi

  3. [8]

    Benzi, G

    M. Benzi, G. H. Golub, and J. Liesen , Numerical solution of saddle point problems , Acta Numerica, 14 (2005), pp. 1--137, https://doi.org/10.1017/s0962492904000212

    2005 doi

  4. [9]

    Berenger , A perfectly matched layer for the absorption of electromagnetic waves , Journal of Computational Physics, 114 (1994), pp

    J.-P. Berenger , A perfectly matched layer for the absorption of electromagnetic waves , Journal of Computational Physics, 114 (1994), pp. 185--200, https://doi.org/10.1006/jcph.1994.1159

    1994

  5. [10]

    M. O. Bristeau, R. Glowinski, and J. P\' e riaux , Controllability methods for the computation of time-periodic solutions; application to scattering , Journal of Computational Physics, 147 (1998), pp. 265--292, https://doi.org/10.1006/jcph.1998.6044

    1998

  6. [11]

    Calandra, S

    H. Calandra, S. Gratton, X. Pinel, and X. Vasseur , An improved two-grid preconditioner for the solution of three-dimensional Helmholtz problems in heterogeneous media , Numerical Linear Algebra with Applications, 20 (2012), pp. 663--688, https://doi.org/10.1002/nla.1860

    2012 doi

  7. [12]

    Carson, J

    E. Carson, J. Liesen, and Z. Strako s , Towards understanding CG and GMRES through examples , Linear Algebra and its Applications, 692 (2024), pp. 241--291, https://doi.org/10.1016/j.laa.2024.04.003

    2024 doi

  8. [13]

    Chen and X

    Z. Chen and X. Xiang , A source transfer domain decomposition method for Helmholtz equations in unbounded domain , SIAM Journal on Numerical Analysis, 51 (2013), pp. 2331--2356, https://doi.org/10.1137/130917144

    2013 doi

  9. [14]

    W. C. Chew and Q. H. Liu , Perfectly matched layers for elastodynamics: A new absorbing boundary condition , Journal of Computational Acoustics, 04 (1996), pp. 341--359, https://doi.org/10.1142/s0218396x96000118

    1996 doi

  10. [15]

    Cummings and X

    P. Cummings and X. Feng , Sharp regularity coefficient estimates for complex-valued acoustic and elastic Helmholtz equations , Mathematical Models and Methods in Applied Sciences, 16 (2006), pp. 139--160, https://doi.org/10.1142/s021820250600108x

    2006 doi

  11. [16]

    Despr\' e s , M\' e thodes de d\' e composition de domaine pour les probl\` e ms de propagation d'ondes en r\' e gime harmonique

    B. Despr\' e s , M\' e thodes de d\' e composition de domaine pour les probl\` e ms de propagation d'ondes en r\' e gime harmonique. Le th\' e or\` e me de Borg pour l'\' e quation de Hill vectorielle , PhD thesis, Universit\' e Paris IX Dauphine, 1991, http://www.theses.fr/19...

    1991

  12. [17]

    Eiermann and O

    M. Eiermann and O. G. Ernst , A restarted Krylov subspace method for the evaluation of matrix functions , SIAM Journal on Numerical Analysis, 44 (2006), pp. 2481--2504, https://doi.org/10.1137/050633846

    2006 doi

  13. [18]

    Engquist and A

    B. Engquist and A. Majda , Radiation boundary conditions for acoustic and elastic wave calculations , Communications on Pure and Applied Mathematics, 32 (1979), pp. 313--357, https://doi.org/10.1002/cpa.3160320303

    1979 doi

  14. [19]

    Engquist and L

    B. Engquist and L. Ying , Sweeping preconditioner for the Helmholtz equation: Hierarchical matrix representation , Communications on Pure and Applied Mathematics, 64 (2011), pp. 697--735, https://doi.org/10.1002/cpa.20358

    2011 doi

  15. [20]

    Y. A. Erlangga , Advances in iterative methods and preconditioners for the Helmholtz equation , Archives of Computational Methods in Engineering, 15 (2007), pp. 37--66, https://doi.org/10.1007/s11831-007-9013-7

    2007 doi

  16. [21]

    Y. A. Erlangga, C. W. Oosterlee, and C. Vuik , A novel multigrid based preconditioner for heterogeneous helmholtz problems , SIAM Journal on Scientific Computing, 27 (2006), pp. 1471--1492, https://doi.org/10.1137/040615195

    2006 doi

  17. [22]

    Y. A. Erlangga, C. Vuik, and C. W. Oosterlee , On a class of preconditioners for solving the Helmholtz equation , Applied Numerical Mathematics, 50 (2004), pp. 409--425, https://doi.org/10.1016/j.apnum.2004.01.009

    2004 doi

  18. [23]

    O. G. Ernst and M. J. Gander , Why it is difficult to solve Helmholtz problems with classical iterative methods , in Numerical Analysis of Multiscale Problems, I. G. Graham, T. Y. Hou, O. Lakkis, and R. Scheichl, eds., Springer, Berlin, Heidelberg, 2012, ch. 10, pp. 325--363, ...

    2012 doi

  19. [24]

    Feng and H

    X. Feng and H. Wu , Discontinuous Galerkin methods for the Helmholtz equation with large wave number , SIAM Journal on Numerical Analysis, 47 (2009), pp. 2872--2896, https://doi.org/10.1137/080737538

    2009 doi

  20. [25]

    Galkowski and E

    J. Galkowski and E. A. Spence , Does the Helmholtz boundary element method suffer from the pollution effect? , SIAM Review, 65 (2023), pp. 806--828, https://doi.org/10.1137/22m1474199

    2023 doi

  21. [27]

    M. J. Gander, F. Magoul\` e s, and F. Nataf , Optimized Schwarz methods without overlap for the Helmholtz equation , SIAM Journal on Scientific Computing, 24 (2002), pp. 38--60, https://doi.org/10.1137/s1064827501387012

    2002 doi

  22. [28]

    M. J. Gander and H. Zhang , A class of iterative solvers for the Helmholtz equation: factorizations, sweeping preconditioners, source transfer, single layer potentials, polarized traces, and optimized Schwarz methods , SIAM Review, 61 (2019), pp. 3--76, https://doi.org/10.1137...

    2019 doi

  23. [29]

    M. J. Gander and H. Zhang , Schwarz methods by domain truncation , Acta Numerica, 31 (2022), pp. 1--134, https://doi.org/10.1017/s0962492922000034

    2022 doi

  24. [31]

    M. J. Grote and J. H. Tang , On controllability methods for the Helmholtz equation , Journal of Computational and Applied Mathematics, 358 (2019), pp. 306--326, https://doi.org/10.1016/j.cam.2019.03.016

    2019 doi

  25. [32]

    Hernandez, J

    V. Hernandez, J. E. Roman, and V. Vidal , SLEPc : A scalable and flexible toolkit for the solution of eigenvalue problems , Association for Computing Machinery. Transactions on Mathematical Software, 31 (2005), pp. 351--362, https://doi.org/10.1145/1089014.1089019

    2005

  26. [33]

    N. J. Higham and T. Mary , Mixed precision algorithms in numerical linear algebra , Acta Numerica, 31 (2022), pp. 347--414, https://doi.org/10.1017/s0962492922000022

    2022 doi

  27. [34]

    Hochbruck and A

    M. Hochbruck and A. Ostermann , Exponential integrators , Acta Numerica, 19 (2010), pp. 209--286, https://doi.org/10.1017/s0962492910000048

    2010 doi

  28. [35]

    Ihlenburg and I

    F. Ihlenburg and I. Babu s ka , Finite element solution of the Helmholtz equation with high wave number part II : The h - p version of the FEM , SIAM Journal on Numerical Analysis, 34 (1997), pp. 315--358, https://doi.org/10.1137/s0036142994272337

    1997 doi

  29. [36]

    Lafontaine, E

    D. Lafontaine, E. A. Spence, and J. Wunsch , Wavenumber-explicit convergence of the hp- FEM for the full-space heterogeneous Helmholtz equation with smooth coefficients , Computers & Mathematics with Applications, 113 (2022), pp. 59--69, https://doi.org/10.1016/j.camwa.2022.03.007

    2022 doi

  30. [38]

    D. E. Manolopoulos , Derivation and reflection properties of a transmission-free absorbing potential , The Journal of Chemical Physics, 117 (2002), pp. 9552--9559, https://doi.org/10.1063/1.1517042

    2002 doi

  31. [39]

    G. S. Martin, R. Wiley, and K. J. Marfurt , Marmousi2: an elastic upgrade for Marmousi , The Leading Edge, 25 (2006), pp. 156--166, https://doi.org/10.1190/1.2172306

    2006 doi

  32. [40]

    J. M. Melenk and S. Sauter , Wavenumber explicit convergence analysis for Galerkin discretizations of the Helmholtz equation , SIAM Journal on Numerical Analysis, 49 (2011), pp. 1210--1243, https://doi.org/10.1137/090776202

    2011 doi

  33. [41]

    N\' e d\' e lec , Acoustic and electromagnetic equations , Springer, New York, 2001, https://doi.org/10.1007/978-1-4757-4393-7

    J.-C. N\' e d\' e lec , Acoustic and electromagnetic equations , Springer, New York, 2001, https://doi.org/10.1007/978-1-4757-4393-7

    2001 doi

  34. [42]

    Poulson, B

    J. Poulson, B. Engquist, S. Li, and L. Ying , A parallel sweeping preconditioner for heterogeneous 3D Helmholtz equations , SIAM Journal on Scientific Computing, 35 (2013), pp. C194--C212, https://doi.org/10.1137/120871985

    2013 doi

  35. [43]

    Saad , A flexible inner-outer preconditioned GMRES algorithm , SIAM Journal on Scientific Computing, 14 (1993), pp

    Y. Saad , A flexible inner-outer preconditioned GMRES algorithm , SIAM Journal on Scientific Computing, 14 (1993), pp. 461--469, https://doi.org/10.1137/0914028

    1993 doi

  36. [44]

    Saad and M

    Y. Saad and M. H. Schultz , GMRES : a generalized minimal residual algorithm for solving nonsymmetric linear systems , SIAM Journal on Scientific and Statistical Computing, 7 (1986), pp. 856--869, https://doi.org/10.1137/0907058

    1986 doi

  37. [45]

    C. C. Stolk , A time-domain preconditioner for the Helmholtz equation , SIAM Journal on Scientific Computing, 43 (2021), pp. A3469--A3502, https://doi.org/10.1137/20m1359997

    2021 doi

  38. [46]

    Trottenberg, C

    U. Trottenberg, C. W. Oosterlee, and A. Sch\" u ller , Multigrid , Elsevier, Amsterdam, 2007

    2007

  39. [47]

    Tsuji, J

    P. Tsuji, J. Poulson, B. Engquist, and L. Ying , Sweeping preconditioners for elastic wave propagation with spectral element methods , ESAIM. Mathematical Modelling and Numerical Analysis, 48 (2014), pp. 433--447, https://doi.org/10.1051/m2an/2013114

    2014

  40. [48]

    H. F. Walker and P. Ni , Anderson acceleration for fixed-point iterations , SIAM Journal on Numerical Analysis, 49 (2011), pp. 1715--1735, https://doi.org/10.1137/10078356x

    2011 doi

  41. [49]

    write newline

    " write newline "" before.all 'output.state := FUNCTION fin.entry add.period write newline FUNCTION new.block output.state before.all = 'skip after.block 'output.state := if FUNCTION not #0 #1 if FUNCTION and 'skip pop #0 if FUNCTION or pop #1 'skip if FUNCTION new.block.check...

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.