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Achieving wavenumber robustness in domain decomposition for heterogeneous Helmholtz equation: an overview of spectral coarse spaces

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Pith's one-line read This paper claims that harmonic and extended-harmonic spectral coarse spaces give the best balance of GMRES efficiency and coarse-space size for high-frequency heterogeneous Helmholtz solvers, while DtN and Hk-GenEO spaces break down in the

desk verdict Solid benchmark with a credible ranking of Helmholtz coarse spaces, but the 3D 'breakdown' of DtN and Hk-GenEO is a fixed-budget artifact, not an established failure. read the letter →

arxiv 2509.02131 v1 pith:VYONSJVP submitted 2025-09-02 math.NA cs.NA

Victorita Dolean , Mark Fry , Matthias Langer , Emile Parolin , Pierre-Henri Tournier This is my paper · ORCID
classification math.NAcs.NA MSC 65N5565N3565F10
keywords HelmholtzequationdomaindecompositioncoarsespaceGenEOharmonicDtNwavenumberrobustnesstwo-levelpreconditioner
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 compares four families of spectral coarse spaces for two-level overlapping Schwarz solvers for the heterogeneous Helmholtz equation, asking which construction delivers wavenumber robustness at acceptable coarse-space cost. Its central claim is that harmonic and extended-harmonic coarse spaces offer the best trade-off: they keep GMRES iteration counts low, even under weak scaling and in high-frequency three-dimensional heterogeneous settings, while using comparatively compact coarse spaces. The paper also reports that the established DtN and Hk-GenEO coarse spaces remain competitive in two dimensions and at moderate problem sizes, but fail to converge in its three-dimensional benchmarks. If correct, the practical consequence is a clear ordering: for challenging 3D problems, prefer extended-harmonic or harmonic coarse spaces, and reserve DtN and Hk-GenEO for cheaper 2D or moderate-scale settings.

What carries the argument

The central object is the spectral coarse space: a set of global basis functions assembled by partition of unity from eigenvectors of local eigenproblems. The decisive difference between families is which local problem is diagonalized and which vectors are admitted. DtN diagonalizes an interface Dirichlet-to-Neumann map and extends the eigenfunctions by a Helmholtz extension. GenEO-type methods diagonalize volumetric operators; Hk-GenEO uses a positive-definite surrogate operator close to the indefinite Helmholtz operator. Harmonic and extended-harmonic spaces diagonalize self-adjoint, coercive eigenproblems whose eigenfunctions are constrained to be local Helmholtz-harmonic fields (solution

What would settle it

Rerun the two 3D benchmarks where DtN and Hk-GenEO break down—the COBRA cavity at k=360 m^-1 with 2916 subdomains and the GO_3D_OBS test with 1024 subdomains—allowing 400 or 1000 eigenvectors per subdomain and up to 1000 GMRES iterations. If either method then reaches the 1e-6 residual tolerance with moderate iteration counts, the reported 3D breakdown is an artifact of the 200-dimensional budget rather than a genuine robustness failure.

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Extended reading notes

Core claim

The discovery is comparative and empirical. Within a common two-level ORAS (optimized restricted additive Schwarz) preconditioner and a non-restarted GMRES solver, the paper tests DtN (Dirichlet-to-Neumann), GenEO-type (Generalized Eigenproblems in the Overlap, specifically Hk-GenEO), harmonic, and extended-harmonic coarse spaces on problems from a homogeneous square to the COBRA cavity and the GO_3D_OBS crustal geomodel. Every two-level method beats the one-level baseline by a wide margin, but the balance of robustness and coarse-space size shifts with dimensionality: harmonic and extended-harmonic spaces give the lowest iteration counts for a given coarse-space size and keep those counts u

Load-bearing premise

The load-bearing premise is that failing to converge within 200 eigenvectors per subdomain and 200 GMRES iterations counts as genuine loss of robustness; the paper reports no experiments with larger budgets that would rule out a budget-limited failure.

Editorial extensions

If this is right

  • For high-frequency 3D heterogeneous Helmholtz problems, practitioners should build two-level ORAS preconditioners with extended-harmonic or harmonic coarse spaces rather than DtN or Hk-GenEO spaces.
  • Two-level coarse correction is not optional: in these tests every spectral coarse space dramatically outperforms the one-level ORAS baseline as frequency or domain size grows.
  • Wavenumber robustness is achievable with compact coarse spaces whose dimension tracks the number of subdomain-interface unknowns, provided the selected modes are genuinely Helmholtz-harmonic.
  • Working eigenvalue thresholds are roughly 5–10 for extended-harmonic, about 10 for harmonic, 500–1000 for DtN, and 0.8–1 for Hk-GenEO.
  • In 3D, DtN and Hk-GenEO should be used with caution on large domains: the paper reports breakdown on the COBRA cavity and GO_3D_OBS benchmarks.

Reading between the lines

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

  • Because the paper's 3D 'breakdown' verdicts are made under a fixed budget of 200 eigenvectors per subdomain and 200 GMRES iterations, an immediate extension is to rerun those benchmarks with larger budgets; convergence there would demote the failure from structural to budget-limited.
  • The fact that the robust harmonic spaces stay roughly interface-sized suggests the information needed for wavenumber robustness lives on subdomain interfaces, so a hybrid that combines DtN-style boundary extraction with Helmholtz-harmonic projection might produce even cheaper robust spaces.
  • If coarse-space solve cost dominates each GMRES iteration, total wall-clock time in 2D could still favor DtN despite slightly higher iteration counts; the paper leaves setup and solve timing for future work, so the runtime ranking is still open.
  • The fixed first-level preconditioner, solver, meshes, and threshold sweeps make this comparison directly reusable: any new coarse space can be ranked on the same coarse-space-size-versus-iteration curves without new theory.
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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

2 major / 5 minor

Summary. The manuscript presents a unified numerical comparison of four spectral coarse spaces—DtN, Hk-GenEO, harmonic, and extended-harmonic—used in two-level ORAS preconditioners for the heterogeneous Helmholtz equation. The study is conducted in the FreeFEM/ffddm framework across 2D square, medical-imaging, COBRA-cavity, and GO_3D_OBS benchmarks, and reports iteration counts, coarse-space sizes, and scaling behavior. The central claim is that extended-harmonic coarse spaces offer the best trade-off between solver efficiency and coarse-space size, while DtN and Hk-GenEO, though competitive in 2D, break down in large 3D problems.

Significance. If the comparison is accepted, this is a practically valuable benchmark study for a community that lacks definitive guidance among spectral coarse spaces for high-frequency heterogeneous Helmholtz problems. The paper benefits from a single common implementation, large-scale 3D tests, a clear comparison framework, and a publicly available example script. However, the strength of the conclusions depends on the fairness and completeness of the numerical protocol, especially for the 3D negative verdicts.

major comments (2)
  1. [§5.3, Tables 6–7; §5.4, Table 7; §6, Table 8] The 3D 'breakdown' verdicts for DtN and Hk-GenEO rest on runs with a fixed per-subdomain budget of exactly 200 coarse vectors (CSs=200 in every row) and GMRES stopped at 200 iterations (entries '>200'). The manuscript does not test whether these methods would converge with, say, 400 or 1000 vectors per subdomain, even though the reported n∂Ωs ≈ 5000 makes such budgets feasible. 'Breakdown within a fixed budget' is not equivalent to 'inherently not robust'; the paper conflates the two. Since Table 8 and Section 6 carry these verdicts, this is a load-bearing gap: either experiments with larger budgets are needed, or the claims must be reformulated as budget-limited observations.
  2. [§5.1.3; also §5.2–§5.3] The headline tables report 'the parameters with the minimum number of iterations obtained for each method.' This means every method is evaluated at its individually tuned best. The resulting iteration counts and coarse-space sizes are therefore not produced under a common resource constraint. The threshold-sweep figures provide useful sensitivity information, but the 'best trade-off' conclusion in Section 6 would be more solid if the cross-method comparison were also made at a fixed budget or fixed threshold, rather than only at per-method optima.
minor comments (5)
  1. [§5.1.3, Tables 2–7] The column header 'CSs' is defined only in the text of §5.1.3. Each table caption should repeat the definition, because the repeated 'It CS CS s' blocks are otherwise hard to parse.
  2. [§5.2] The typo 'Matèrn' for the point-process name should be corrected to 'Matérn' or 'Matern'.
  3. [Table 8] The symbols ✓, ✓✓, and ✓/✗ are not quantitatively defined. State the criteria (e.g., convergence within a given iteration budget, relative coarse-space size) so the table is reproducible rather than subjective.
  4. [§4.3.2, Eq. (4.18)] The extended-harmonic eigenproblem is dense and introduced quickly. A short paragraph interpreting the action of Rs ˇRT_s − bA^{-1}_s Rs ˇRT_s ˇAs on a candidate vector would help the reader understand why the construction yields Helmholtz-harmonic coarse functions.
  5. [§6] The paper explicitly leaves setup and GMRES run times out of scope, yet the conclusion uses the word 'trade-off' repeatedly. A reader cannot fully judge practical trade-offs without at least an indication of setup costs; this should be acknowledged more prominently in the abstract or conclusions.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper is an empirical comparison, and its conclusions are drawn from its own experiments rather than from definitional or self-citational loops.

full rationale

The paper is a numerical overview, not a derivation: it implements four coarse-space families in a common FreeFEM/ffddm framework and compares them on benchmark problems. The main claim—that extended-harmonic coarse spaces offer the best trade-off between solver efficiency and coarse space size—is supported by the reported GMRES iteration counts and coarse-space sizes (Tables 2–7, Figures 2–18), not by an equation that presupposes the conclusion. There is no step in which an object is defined in terms of the quantity it is said to predict, and no fitted parameter is renamed as a prediction. The eigenvalue thresholds are tuned per test case ('The results reported in the table correspond to the parameters with the minimum number of iterations obtained for each method', Section 5.1.3), but this is best-case benchmarking rather than a circular fit: the measured iteration counts are genuine outputs of the solver on the same problems, and the comparison is explicitly framed as an assessment of each method under favorable tuning. The paper does rely on citations to prior work, including [43] (co-authored by Parolin) for the extended-harmonic space and [42,38] for harmonic spaces, but the comparative verdict is obtained from the paper's own numerical experiments, so the self-citations are not load-bearing. The 3D negative verdicts for DtN and Hk-GenEO are limited by the fixed computational budget of 200 coarse vectors per subdomain and 200 GMRES iterations (Tables 6–7: all entries show CSs = 200 and '>200' for non-convergence; Section 5.3 and 5.4 call this 'breakdown'). This is an experimental limitation and a possible overstatement—'breakdown within a fixed budget' is not necessarily 'inherently not robust'—but it is not a circularity: the conclusion does not reduce by definition to the budget, and the paper does not claim to have tested larger budgets. Overall, the central content is self-contained empirical comparison, so the circularity score is 0.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

The comparison relies on no new theory; the central ranking is empirical. The main unpaid inputs are implementation fidelity to prior method papers, benchmark-specific tuning parameters (thresholds, caps, discretizations, overlap), and the representativeness of the chosen geometries.

free parameters (5)
  • Eigenvalue threshold tau per coarse-space family = DtN 500-1000; harmonic ~10; extended-harmonic 5-10; Hk-GenEO 0.8-1
    Each method's threshold is tuned per experiment to minimize GMRES iterations ('parameters with the minimum number of iterations', Section 5.1.3); the reported 'sensible choices' are empirical fits to the benchmarks, not theory-driven.
  • Per-subdomain coarse-space budget cap = 200 eigenvectors per subdomain in 3D tests (Tables 6-7)
    COBRA and GO_3D_OBS runs cap the local coarse space at 200 vectors, which bounds DtN and Hk-GenEO; the 'breakdown' verdict depends on this budget.
  • GMRES maximum iterations = 200 for 3D tables
    Methods exceeding 200 iterations are recorded as >200 and judged to have failed; the choice of ceiling affects which methods are labeled non-robust.
  • Mesh resolution (points per wavelength) = 10 (square), 22 (imaging), 8 (COBRA), 4 (GO_3D_OBS)
    Applied to control pollution; not fitted to the conclusion, but the ranking of methods may be resolution-dependent.
  • Overlap width and partition of unity = Minimal overlap (2 cell layers), default steep P1 partition of unity
    The default choices are used for all headline tables; Figure 12 shows overlap and partition-of-unity smoothness change iteration counts, so the comparison is tied to these defaults.
assumptions (5)
  • domain assumption The continuous Helmholtz boundary value problem (2.1) is well posed and the chosen Robin/PML truncations faithfully approximate the Sommerfeld radiation condition.
    Used throughout Section 2 and in all benchmark setups; the imaging test uses a PML from [25,5] and the square/cobra use Robin boundaries.
  • domain assumption P1/P2 finite element discretization with fixed points-per-wavelength controls the pollution effect sufficiently.
    Section 2 states kh bounded or P2 elements relax constraints; experiments fix nppwl between 4 and 22, assuming the discrete problem accurately represents the continuous one.
  • domain assumption Each spectral coarse space is implemented faithfully as specified in its source paper ([14], [9], [13], [38], [42], [43]) and the ffddm framework is correct.
    The paper does not re-derive the methods; the comparison is only as valid as the implementations of the cited constructions within FreeFEM.
  • standard math GMRES with right preconditioning at tolerance 1e-6 is a reliable proxy for solver efficiency.
    Used in all experiments; iteration count is the standard comparison metric in the literature, though it omits per-iteration costs.
  • domain assumption The 3D benchmark problems (COBRA cavity, GO_3D_OBS crustal model) are representative of realistic mid-to-high-frequency Helmholtz challenges.
    The generality of conclusions such as 'DtN breaks down in 3D' rests on these two geometries being representative of difficult 3D wave problems.

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Cite this review

Pith. "Pith review of Achieving wavenumber robustness in domain decomposition for heterogeneous Helmholtz equation: an overview of spectral coarse spaces." pith.science (2026). https://pith.science/paper/VYONSJVP

@misc{pith2026250902131,
  author       = {Pith},
  title        = {Pith review of: Achieving wavenumber robustness in domain decomposition for heterogeneous Helmholtz equation: an overview of spectral coarse spaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VYONSJVP}},
  note         = {Machine review of arXiv:2509.02131}
}
read the original abstract

Solving time-harmonic wave propagation problems in the frequency domain within heterogeneous media poses significant mathematical and computational challenges, particularly in the high-frequency regime. Among the available numerical approaches, domain decomposition methods are widely regarded as effective due to their suitability for parallel computing and their capacity to maintain robustness with respect to physical parameters, such as the wavenumber. These methods can achieve near-constant time-to-solution as the wavenumber increases, though often at the expense of a computationally intensive coarse correction step. This work focuses on identifying the best algorithms and numerical strategies for benchmark problems modelled by the Helmholtz equation. Specifically, we examine and compare several coarse spaces which are part of different families, e.g. GenEO (Generalised Eigenvalue Overlap) type coarse spaces and harmonic coarse spaces, that underpin two-level domain decomposition methods. By leveraging spectral information and multiscale approaches, we aim to provide a comprehensive overview of the strengths and weaknesses of these methods. Numerical experiments demonstrate that the effectiveness of these coarse spaces depends on the specific problem and numerical configuration, highlighting the trade-offs between computational cost, robustness, and practical applicability.

Figures

Figures reproduced from arXiv: 2509.02131 by the authors.

Figure 1
Figure 1. Schematic of the 2D wave guide model problem with example triangular mesh. Real part (left) and [PITH_FULL_IMAGE:figures/full_fig_p011_1.png] view at source ↗
Figure 2
Figure 2. Influence of the coarse space size (left) and threshold choice (right) on the iteration count in strong [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Influence of the subdomain diameter on the iteration count (left) and coarse space size (right) in strong [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figures from the paper (15 more)
Figure 4
Figure 4. Figure 4: Piecewise constant layer profiles for the wave speed [PITH_FULL_IMAGE:figures/full_fig_p013_4.png]
Figure 5
Figure 5. Figure 5: Influence of the coarse space size (left) and threshold choice (right) on the iteration count in strong [PITH_FULL_IMAGE:figures/full_fig_p013_5.png]
Figure 6
Figure 6. Figure 6: Influence of the subdomain diameter on the iteration count (left) and coarse space size (right) in strong [PITH_FULL_IMAGE:figures/full_fig_p014_6.png]
Figure 7
Figure 7. Figure 7: Values of the coefficient c 2 0m (left), real part of total field Re(u) (middle), and real part of scattered field Re(us) (right). Strong scaling test We first provide results for a strong scaling test, increasing the number of subdomains for a fixed problem size. The …
Figure 8
Figure 8. Figure 8: Influence of the coarse space size (left) and threshold choice (right) on the iteration count in strong [PITH_FULL_IMAGE:figures/full_fig_p017_8.png]
Figure 9
Figure 9. Figure 9: Influence of the subdomain diameter on the iteration count in strong scaling for the imaging test case. [PITH_FULL_IMAGE:figures/full_fig_p017_9.png]
Figure 10
Figure 10. Figure 10: Influence of the coarse space size (left) and threshold choice (right) on the iteration count in weak [PITH_FULL_IMAGE:figures/full_fig_p018_10.png]
Figure 11
Figure 11. Figure 11: Influence of the full domain diameter on the iteration count in weak scaling for the imaging test case. [PITH_FULL_IMAGE:figures/full_fig_p018_11.png]
Figure 12
Figure 12. Figure 12: Influence of the overlap and the partition of unity function on the iteration count for the imaging test [PITH_FULL_IMAGE:figures/full_fig_p019_12.png]
Figure 13
Figure 13. Figure 13: Partitioning of the cobra cavity domain into 2916 subdomains (left) and real part of the total field [PITH_FULL_IMAGE:figures/full_fig_p020_13.png]
Figure 14
Figure 14. Figure 14: Influence of the coarse space size (left) and threshold choice (right) on the iteration count in weak [PITH_FULL_IMAGE:figures/full_fig_p021_14.png]
Figure 15
Figure 15. Figure 15: Influence of the full domain diameter on the iteration count in weak scaling for the cobra cavity test [PITH_FULL_IMAGE:figures/full_fig_p021_15.png]
Figure 16
Figure 16. Figure 16: Target of the regional GO_3D_OBS model representing the crust of a subduction zone [ [PITH_FULL_IMAGE:figures/full_fig_p022_16.png]
Figure 17
Figure 17. Figure 17: Tetrahedral mesh of the GO_3D_OBS model adapted to the local wavelength. Note that the mesh in [PITH_FULL_IMAGE:figures/full_fig_p022_17.png]
Figure 18
Figure 18. Figure 18: Influence of the coarse space size (left) and threshold choice (right) on the iteration count in a strong [PITH_FULL_IMAGE:figures/full_fig_p023_18.png]

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Works this paper leans on

51 extracted references · 37 canonical work pages

  1. [1]

    A Robust Two-Level Schwarz Preconditioner For Sparse Matrices

    H. Al Daas, P. Jolivet, F. Nataf, and P.-H. Tournier.A Robust Two-Level Schwarz Preconditioner For Sparse Matrices. 2024. arXiv:2401.03915

    work page Pith review arXiv 2024

  2. [2]

    A fully asynchronous multifrontal solver using dis- tributed dynamic scheduling

    P. Amestoy, I. Duff, J.-Y. L’Excellent, and J. Koster. “A fully asynchronous multifrontal solver using dis- tributed dynamic scheduling”. In:SIAM J. Matrix Anal. Appl.23.1 (2001), pp. 15–41.url: http://mumps. enseeiht.fr

    work page 2001

  3. [3]

    Is the pollution effect of the FEM avoidable for the Helmholtz equation considering high wave numbers?

    I. M. Babuska and S. A. Sauter. “Is the pollution effect of the FEM avoidable for the Helmholtz equation considering high wave numbers?” In:SIAM J. Numer. Anal.34.6 (1997), pp. 2392–2423. doi: 10 . 1137 / S0036142993255035

    work page 1997

  4. [4]

    An automatic perfectly matched layer for acoustic finite element simulations in convex domains of general shape

    H. Beriot and A. Modave. “An automatic perfectly matched layer for acoustic finite element simulations in convex domains of general shape”. In:International Journal for Numerical Methods in Engineering122 (2021), pp. 1239–1261.doi: 10.1002/nme.6581

    work page doi:10.1002/nme.6581 2021

  5. [5]

    An exact bounded PML for the Helmholtz equation

    A. Bermúdez, L. Hervella-Nieto, A. Prieto, and R. Rodríguez. “An exact bounded PML for the Helmholtz equation”. In:C. R. Math. Acad. Sci. Paris339.11 (2004), pp. 803–808.doi: 10.1016/j.crma.2004.10.006

    work page doi:10.1016/j.crma.2004.10.006 2004

  6. [6]

    Domain decomposition precon- ditioning for the high-frequency time-harmonic Maxwell equations with absorption

    M. Bonazzoli, V. Dolean, I. G. Graham, E. A. Spence, and P.-H. Tournier. “Domain decomposition precon- ditioning for the high-frequency time-harmonic Maxwell equations with absorption”. In:Math. Comp.88.320 (2019), pp. 2559–2604.doi: 10.1090/mcom/3447

    doi:10.1090/mcom/3447 2019

  7. [7]

    On the Dirichlet-to-Neumann coarse space for solving the Helmholtz prob- lem using domain decomposition

    N. Bootland and V. Dolean. “On the Dirichlet-to-Neumann coarse space for solving the Helmholtz prob- lem using domain decomposition”. In:Numerical mathematics and advanced applications—ENUMATH 2019. Vol. 139. Lect. Notes Comput. Sci. Eng. Springer, Cham, 2021, pp. 175–184.doi: 10.1007/978- 3- 030- 55874-1\_16

    doi:10.1007/978- 2019

  8. [8]

    Can DtN and GenEO Coarse Spaces Be Sufficiently Robust for Heteroge- neous Helmholtz Problems?

    N. Bootland and V. Dolean. “Can DtN and GenEO Coarse Spaces Be Sufficiently Robust for Heteroge- neous Helmholtz Problems?” In:Mathematical and Computational Applications27.3 (2022). doi: 10.3390/ mca27030035

    work page 2022

Show all 51 references
  1. [9]

    Overlapping Schwarz methods with Ge- nEO coarse spaces for indefinite and nonself-adjoint problems

    N. Bootland, V. Dolean, I. G. Graham, C. Ma, and R. Scheichl. “Overlapping Schwarz methods with Ge- nEO coarse spaces for indefinite and nonself-adjoint problems”. In:IMA J. Numer. Anal.43.4 (Aug. 2022), pp. 1899–1936. doi: 10.1093/imanum/drac036

    2022 doi

  2. [10]

    GenEO coarse spaces for heterogeneous indefinite elliptic problems

    N. Bootland, V. Dolean, I. G. Graham, C. Ma, and R. Scheichl. “GenEO coarse spaces for heterogeneous indefinite elliptic problems”. In:Domain decomposition methods in science and engineering XXVI. Vol. 145. Lect. Notes Comput. Sci. Eng. Springer, Cham, 2022, pp. 117–125.doi: 1...

    2022 doi

  3. [11]

    AcomparisonofcoarsespacesforHelmholtzproblems in the high frequency regime

    N.Bootland,V.Dolean,P.Jolivet,andP.-H.Tournier.“AcomparisonofcoarsespacesforHelmholtzproblems in the high frequency regime”. In:Comput. Math. Appl.98 (2021), pp. 239–253. doi: 10.1016/j.camwa. 2021.07.011

    2021 doi

  4. [12]

    Wave-ray multigrid method for standing wave equations

    A. Brandt and I. Livshits. “Wave-ray multigrid method for standing wave equations”. In:Electron. Trans. Numer. Anal.6 (1997), pp. 162–181

    1997

  5. [13]

    Chaumont-Frelet, V

    T. Chaumont-Frelet, V. Dolean, M. Fry, I. G. Graham, and M. Langer. Spectral coarse spaces based on indefinite operators: theHk-GenEO method. 2024. arXiv:2406.06283

    2024 arXiv

  6. [14]

    A coarse space for heterogeneous Helmholtz problems based on the Dirichlet-to-Neumann operator

    L. Conen, V. Dolean, R. Krause, and F. Nataf. “A coarse space for heterogeneous Helmholtz problems based on the Dirichlet-to-Neumann operator”. In:J. Comput. Appl. Math.271 (2014), pp. 83–99.doi: 10.1016/j. cam.2014.03.031

    2014 doi

  7. [15]

    Large-scale frequency-domain seismic wave modeling on h- adaptive tetrahedral meshes with iterative solver and multi-level domain-decomposition preconditioners

    V. Dolean, P. Jolivet, P. Tournier, and S. Operto. “Large-scale frequency-domain seismic wave modeling on h- adaptive tetrahedral meshes with iterative solver and multi-level domain-decomposition preconditioners”. In: 90th Annual International Meeting, SEG Technical Program Ex...

    2020 doi

  8. [16]

    Dolean, M

    V. Dolean, M. Fry, and M. Langer.Improvements to the theoretical estimates of the Schwarz preconditioner with ∆-GenEO coarse space for the indefinite Helmholtz problem. 2024. arXiv:2403.18378

    2024 arXiv

  9. [17]

    Effective transmission conditions for domain decomposition methods applied to the time-harmonic curl-curl Maxwell’s equations

    V. Dolean, M. J. Gander, S. Lanteri, J.-F. Lee, and Z. Peng. “Effective transmission conditions for domain decomposition methods applied to the time-harmonic curl-curl Maxwell’s equations”. In:J. Comput. Phys. 280 (2015), pp. 232–247.doi: 10.1016/j.jcp.2014.09.024

    2015 doi

  10. [18]

    Dolean, P

    V. Dolean, P. Jolivet, and F. Nataf.An introduction to domain decomposition methods. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2015, pp. x+238.doi: 10.1137/1.9781611974065.ch1

    2015 doi

  11. [19]

    Iterative frequency-domain seismic wave solvers based on multi-level domain-decomposition preconditioners

    V. Dolean, P. Jolivet, P. Tournier, and S. Operto. “Iterative frequency-domain seismic wave solvers based on multi-level domain-decomposition preconditioners”. In:EAGE 2020 Annual Conference & Exhibition Online

    2020

  12. [20]

    2020, pp

    European Association of Geoscientists & Engineers. 2020, pp. 1–5

    2020

  13. [21]

    Analysis of a two-level Schwarz method with coarse spaces based on local Dirichlet-to-Neumann maps

    V. Dolean, F. Nataf, R. Scheichl, and N. Spillane. “Analysis of a two-level Schwarz method with coarse spaces based on local Dirichlet-to-Neumann maps”. In:Comput. Methods Appl. Math.12.4 (2012), pp. 391–414.doi: 10.2478/cmam-2012-0027

    2012 doi

  14. [22]

    Why it is difficult to solve Helmholtz problems with classical iterative methods

    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. Vol. 83. Lect. Notes Comput. Sci. Eng. Springer, Heidelberg, 2012, pp. 325–363.doi: 10.1007/978-3-642-22061-6\_10

    2012 doi

  15. [23]

    FETI-DPH: a dual-primal domain decomposition method for acoustic scattering

    C. Farhat, P. Avery, R. Tezaur, and J. Li. “FETI-DPH: a dual-primal domain decomposition method for acoustic scattering”. In:J. Comput. Acoust.13.3 (2005), pp. 499–524.doi: 10.1142/S0218396X05002761

    2005 doi

  16. [24]

    A two-level domain decomposition method for the iterative solution of high frequency exterior Helmholtz problems

    C. Farhat, A. Macedo, and M. Lesoinne. “A two-level domain decomposition method for the iterative solution of high frequency exterior Helmholtz problems”. In:Numer. Math.85.2 (2000), pp. 283–308.doi: 10.1007/ PL00005389

    2000

  17. [25]

    Global-basis two-level method for indefinite systems. I. Convergence studies

    J. Fish and Y. Qu. “Global-basis two-level method for indefinite systems. I. Convergence studies”. In:Internat. J. Numer. Methods Engrg.49.3 (2000), pp. 439–460.doi: 10.1002/1097-0207(20000930)49:3<439::AID- NME981>3.0.CO;2-A

    2000 doi

  18. [26]

    Thehp-FEM applied to the Helmholtz equation with PML truncation does not suffer from the pollution effect

    J. Galkowski, D. Lafontaine, E. A. Spence, and J. Wunsch. “Thehp-FEM applied to the Helmholtz equation with PML truncation does not suffer from the pollution effect”. In:Commun. Math. Sci.22.7 (2024), pp. 1761–

    2024

  19. [27]

    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”. In: SIAM Rev.61.1 (2019), pp. 3–76.doi: 10.1137/16M109781X

    2019 doi

  20. [28]

    Galkowski and E

    J. Galkowski and E. A. Spence. Convergence theory for two-level hybrid Schwarz preconditioners for high- frequency Helmholtz problems. 2025. arXiv:2501.11060

    2025

  21. [29]

    Garnier, L

    J. Garnier, L. Giovangigli, Q. Goepfert, and P. Millien.Probing the speckle to estimate the effective speed of sound, a first step towards quantitative ultrasound imaging. 2025. arXiv:2505.07566

    2025 arXiv

  22. [30]

    Garnier, L

    J. Garnier, L. Giovangigli, Q. Goepfert, and P. Millien.Scattered wavefield in the stochastic homogenization regime. 2023. arXiv:2309.07777

    2023

  23. [31]

    GO_3D_OBS: the multi-parameter benchmark geomodel for seismic imaging method assessment and next-generation 3D survey design (version 1.0)

    A. Górszczyk and S. Operto. “GO_3D_OBS: the multi-parameter benchmark geomodel for seismic imaging method assessment and next-generation 3D survey design (version 1.0)”. In:Geoscientific Model Development 14.3 (2021), pp. 1773–1799. 25

    2021

  24. [32]

    Domain decomposition preconditioners for high-order discretiza- tions of the heterogeneous Helmholtz equation

    S. Gong, I. G. Graham, and E. A. Spence. “Domain decomposition preconditioners for high-order discretiza- tions of the heterogeneous Helmholtz equation”. In:IMA J. Numer. Anal.41.3 (2021), pp. 2139–2185.doi: 10.1093/imanum/draa080

    2021 doi

  25. [33]

    Recent Results on Domain Decomposition Preconditioning for the High-Frequency Helmholtz Equation Using Absorption

    I. G. Graham, E. A. Spence, and E. Vainikko. “Recent Results on Domain Decomposition Preconditioning for the High-Frequency Helmholtz Equation Using Absorption”. In:Lahaye D., Tang J., Vuik K. (eds) Modern Solvers for Helmholtz Problems. Geosystems Mathematics. Birkhäuser, Cha...

    2017 doi

  26. [34]

    Toward a robust workflow for deep crustal imaging by FWI of OBS data: the eastern Nankai Trough revisited

    A. Górszczyk, S. Operto, and M. Malinowski. “Toward a robust workflow for deep crustal imaging by FWI of OBS data: the eastern Nankai Trough revisited”. In:Journal of Geophysical Research: Solid Earth122.6 (2017), pp. 4601–4630

    2017

  27. [35]

    An additive Schwarz method type theory for Lions’s algorithm and a symmetrized optimized restricted additive Schwarz method

    R. Haferssas, P. Jolivet, and F. Nataf. “An additive Schwarz method type theory for Lions’s algorithm and a symmetrized optimized restricted additive Schwarz method”. In:SIAM J. Sci. Comput.39.4 (2017), A1345– A1365. doi: 10.1137/16M1060066

    2017 doi

  28. [36]

    Domain decomposition with local impedance conditions for the Helmholtz equation with absorption

    I. G. Graham, E. A. Spence, and J. Zou. “Domain decomposition with local impedance conditions for the Helmholtz equation with absorption”. In:SIAM J. Numer. Anal.58.5 (2020), pp. 2515–2543.doi: 10.1137/ 19M1272512

    2020

  29. [37]

    SLEPc:Ascalableandflexibletoolkitforthesolutionofeigenvalue problems

    V.Hernandez,J.E.Roman,andV.Vidal.“SLEPc:Ascalableandflexibletoolkitforthesolutionofeigenvalue problems”. In:ACM Transactions on Mathematical Software31.3 (2005), pp. 351–362.url: https://slepc. upv.es

    2005

  30. [38]

    Analytical and numerical studies of a finite element PML for the Helmholtz equation

    I. Harari, M. Slavutin, and E. Turkel. “Analytical and numerical studies of a finite element PML for the Helmholtz equation”. In:J. Comput. Acoust.8 (2000), pp. 121–137.doi: 10.1142/S0218396X00000096

    2000 doi

  31. [39]

    J.-M. Jin. The Finite Element Method in Electromagnetics. John Wiley & Sons, 2015

    2015

  32. [40]

    Hu and Z

    Q. Hu and Z. Li. A novel coarse space applying to the weighted Schwarz method for Helmholtz equations

  33. [41]

    Scattering analysis of a large body with deep cavities

    J. Liu and J.-M. Jin. “Scattering analysis of a large body with deep cavities”. In:IEEE Transactions on Antennas and Propagation51.6 (2003), pp. 1157–1167.doi: 10.1109/TAP.2003.812280

    2003

  34. [42]

    Karypis and V

    G. Karypis and V. Kumar. A software package for partitioning unstructured graphs, partitioning meshes, and computing fill-reducing orderings of sparse matrices. Tech. rep. University of Minnesota, Department of Computer Science and Engineering, Army HPC Research Center, Minnea...

    1998

  35. [43]

    Nataf and E

    F. Nataf and E. Parolin.Coarse spaces for non-symmetric two-level preconditioners based on local generalized eigenproblems. 2024. arXiv:2404.02758

    2024 arXiv

  36. [44]

    C. Ma, C. Alber, and R. Scheichl.Two-level restricted additive Schwarz preconditioner based on multiscale spectral generalized FEM for heterogeneous Helmholtz problems. 2024. arXiv:2409.06533

    2024 arXiv

  37. [45]

    Is 3D frequency-domain FWI of full- azimuth/long-offset OBN data feasible? The Gorgon data FWI case study

    S. Operto, P. Amestoy, H. Aghamiry, S. Beller, A. Buttari, et al. “Is 3D frequency-domain FWI of full- azimuth/long-offset OBN data feasible? The Gorgon data FWI case study”. In: The Leading Edge 42.3 (2023), pp. 173–183

    2023

  38. [46]

    A coarse space construction based on local Dirichlet-to- Neumann maps

    F. Nataf, H. Xiang, V. Dolean, and N. Spillane. “A coarse space construction based on local Dirichlet-to- Neumann maps”. In:SIAM J. Sci. Comput.33.4 (2011), pp. 1623–1642.doi: 10.1137/100796376

    2011 doi

  39. [47]

    Tournier, P

    P.-H. Tournier, P. Jolivet, V. Dolean, H. S. Aghamiry, S. Operto, et al. “3D finite-difference and finite- element frequency-domain wave simulation with multilevel optimized additive Schwarz domain-decomposition preconditioner: a tool for full-waveform inversion of sparse node...

    2022 doi

  40. [48]

    Abstract robust coarse spaces for systems of PDEs via generalized eigenproblems in the overlaps

    N. Spillane, V. Dolean, P. Hauret, F. Nataf, C. Pechstein, et al. “Abstract robust coarse spaces for systems of PDEs via generalized eigenproblems in the overlaps”. In:Numer. Math. 126.4 (2014), pp. 741–770.doi: 10.1007/s00211-013-0576-y

    2014 doi

  41. [49]

    A general approach for high order absorbing boundary conditions for the Helmholtz equation

    A. Zarmi and E. Turkel. “A general approach for high order absorbing boundary conditions for the Helmholtz equation”. In:J. Comput. Phys.242 (2013), pp. 387–404.doi: 10.1016/j.jcp.2013.02.042. 26

    2013 doi

  42. [50]

    Tournier, P

    P.-H. Tournier, P. Jolivet, and F. Nataf. FFDDM: FreeFem Domain Decomposition Method. 2019. url: https://doc.freefem.org/documentation/ffddm/index.html

    2019

  43. [1816]

    doi: 10.4310/cms.240918021620

    doi

Pith tools

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