Paper Citation Record · LEDGER

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

As of 13 August 2026, this Paper Citation Record lists 51 of 51 outbound references and 2 inbound Pith citation observations for arXiv:2509.02131.

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pith.paper-citation-record.v1
2509.02131 v1

Coverage vector

measured 51 of 51 reference resolution

Typed states for the displayed outbound observations.

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measured 2 of 2 inbound itemization

Pith citing papers itemized under the disclosed page cap.

Source: paper_references, paper_reference_links, observed 2026-06-28T21:13:13.357878Z

measured 0 of 1 external citation measurements

A source-named dated measurement, never combined with another source.

Source: arxiv_reference, observed 2026-07-01T20:26:12.532490Z

Reference resolution

51 of 51 outbound references displayed

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External citation measurements

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Outbound references

Observation c31f5775-8437-4e7d-9c6d-cd7dac80c574 · outbound

This paper cites A Robust Two-Level Schwarz Preconditioner For Sparse Matrices.

Achieving wavenumber robustness in domain decomposition for heterogeneous Helmholtz equation: an overview of spectral coarse spaces A Robust Two-Level Schwarz Preconditioner For Sparse Matrices

Reference 1

Resolution: verified exact; local_arxiv, observed 2026-08-05T11:56:31.997211Z

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Observation f26d1575-5538-40c3-aaa8-e86b1244d87f · outbound

This paper cites A fully asynchronous multifrontal solver using dis- tributed dynamic scheduling.

Achieving wavenumber robustness in domain decomposition for heterogeneous Helmholtz equation: an overview of spectral coarse spaces A fully asynchronous multifrontal solver using dis- tributed dynamic scheduling

Reference 2

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Observation e0850ee1-2c60-4a6d-8486-fce09182330b · outbound

This paper cites Is the pollution effect of the FEM avoidable for the Helmholtz equation considering high wave numbers?.

Achieving wavenumber robustness in domain decomposition for heterogeneous Helmholtz equation: an overview of spectral coarse spaces Is the pollution effect of the FEM avoidable for the Helmholtz equation considering high wave numbers?

Reference 3

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Observation 6ace3d74-ebd2-4ba3-b07a-7415bc989228 · outbound

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

Achieving wavenumber robustness in domain decomposition for heterogeneous Helmholtz equation: an overview of spectral coarse spaces An automatic perfectly matched layer for acoustic finite element simulations in convex domains of general shape

Reference 4

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Observation ebfda9fb-54aa-4c0c-90e8-da378d4c4bd8 · outbound

This paper cites An exact bounded PML for the Helmholtz equation.

Achieving wavenumber robustness in domain decomposition for heterogeneous Helmholtz equation: an overview of spectral coarse spaces An exact bounded PML for the Helmholtz equation

Reference 5

Resolution: verified exact; doi, observed 2026-08-05T11:56:30.223398Z

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Observation ec77fbfe-a9eb-43f0-ba66-bbd3886c8bcb · outbound

This paper cites Domain decomposition precon- ditioning for the high-frequency time-harmonic Maxwell equations with absorption.

Achieving wavenumber robustness in domain decomposition for heterogeneous Helmholtz equation: an overview of spectral coarse spaces Domain decomposition precon- ditioning for the high-frequency time-harmonic Maxwell equations with absorption

Reference 6

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Observation e142372b-010d-408f-9714-b62e31d78eb5 · outbound

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

Achieving wavenumber robustness in domain decomposition for heterogeneous Helmholtz equation: an overview of spectral coarse spaces On the Dirichlet-to-Neumann coarse space for solving the Helmholtz prob- lem using domain decomposition

Reference 7

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Observation eedd1e18-3630-404a-b8c8-a8ad737cfbc6 · outbound

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

Achieving wavenumber robustness in domain decomposition for heterogeneous Helmholtz equation: an overview of spectral coarse spaces Can DtN and GenEO Coarse Spaces Be Sufficiently Robust for Heteroge- neous Helmholtz Problems?

Reference 8

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Observation 465509bd-2dc1-4a84-9b0f-30cb4b06f437 · outbound

This paper cites Overlapping Schwarz methods with Ge- nEO coarse spaces for indefinite and nonself-adjoint problems.

Achieving wavenumber robustness in domain decomposition for heterogeneous Helmholtz equation: an overview of spectral coarse spaces Overlapping Schwarz methods with Ge- nEO coarse spaces for indefinite and nonself-adjoint problems

Reference 9

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Observation 69f2a01b-4ae5-4d1e-9cf3-e67f0dcc17b9 · outbound

This paper cites GenEO coarse spaces for heterogeneous indefinite elliptic problems.

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

Reference 10

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Observation 3ee30b98-85e4-40ea-9827-f2e94bf3522f · outbound

This paper cites AcomparisonofcoarsespacesforHelmholtzproblems in the high frequency regime.

Achieving wavenumber robustness in domain decomposition for heterogeneous Helmholtz equation: an overview of spectral coarse spaces AcomparisonofcoarsespacesforHelmholtzproblems in the high frequency regime

Reference 11

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Observation f2e58b1c-3081-4243-aeee-43cc22c9b5aa · outbound

This paper cites Wave-ray multigrid method for standing wave equations.

Achieving wavenumber robustness in domain decomposition for heterogeneous Helmholtz equation: an overview of spectral coarse spaces Wave-ray multigrid method for standing wave equations

Reference 12

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Observation aac061ef-2279-451c-a7a2-2c57fbc32d49 · outbound

This paper cites Schwarz preconditioner with $H_k$-GenEO coarse space for the indefinite Helmholtz problem.

Achieving wavenumber robustness in domain decomposition for heterogeneous Helmholtz equation: an overview of spectral coarse spaces Schwarz preconditioner with $H_k$-GenEO coarse space for the indefinite Helmholtz problem

Reference 13

Resolution: verified exact; local_arxiv, observed 2026-08-05T11:56:31.984664Z

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Observation 834322d3-88f8-4c4a-8788-57a4bf362b92 · outbound

This paper cites A coarse space for heterogeneous Helmholtz problems based on the Dirichlet-to-Neumann operator.

Achieving wavenumber robustness in domain decomposition for heterogeneous Helmholtz equation: an overview of spectral coarse spaces A coarse space for heterogeneous Helmholtz problems based on the Dirichlet-to-Neumann operator

Reference 14

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Observation 8fed09e7-899b-4224-963e-75ba3ab3fa84 · outbound

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

Achieving wavenumber robustness in domain decomposition for heterogeneous Helmholtz equation: an overview of spectral coarse spaces Large-scale frequency-domain seismic wave modeling on h- adaptive tetrahedral meshes with iterative solver and multi-level domain-decomposition preconditioners

Reference 15

Resolution: verified exact; doi, observed 2026-08-05T11:56:30.027136Z

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Observation caccd204-91d1-4d0a-9a5c-2a269b28ce96 · outbound

This paper cites Can Symmetric Positive Definite (SPD) coarse spaces perform well for indefinite Helmholtz problems?.

Achieving wavenumber robustness in domain decomposition for heterogeneous Helmholtz equation: an overview of spectral coarse spaces Can Symmetric Positive Definite (SPD) coarse spaces perform well for indefinite Helmholtz problems?

Reference 16

Resolution: verified exact; local_arxiv, observed 2026-08-05T11:56:31.971424Z

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Observation 89d62bf4-68bf-4d9b-a131-a6fc27aadf06 · outbound

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

Achieving wavenumber robustness in domain decomposition for heterogeneous Helmholtz equation: an overview of spectral coarse spaces Effective transmission conditions for domain decomposition methods applied to the time-harmonic curl-curl Maxwell’s equations

Reference 17

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Observation 9a4dad2a-ec88-4555-a451-450a30db8f58 · outbound

This paper cites Dolean, P.

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

Reference 18

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Observation da5a9d2f-05d2-4818-a813-d1208c69f1ed · outbound

This paper cites Iterative frequency-domain seismic wave solvers based on multi-level domain-decomposition preconditioners.

Achieving wavenumber robustness in domain decomposition for heterogeneous Helmholtz equation: an overview of spectral coarse spaces Iterative frequency-domain seismic wave solvers based on multi-level domain-decomposition preconditioners

Reference 19

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Observation 62e85a67-eeac-4517-b98d-6aa78dbec6c5 · outbound

This paper cites 2020, pp.

Achieving wavenumber robustness in domain decomposition for heterogeneous Helmholtz equation: an overview of spectral coarse spaces 2020, pp

Reference 20

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Observation df18c7bd-46cb-4096-8bff-2a7d6284844d · outbound

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

Achieving wavenumber robustness in domain decomposition for heterogeneous Helmholtz equation: an overview of spectral coarse spaces Analysis of a two-level Schwarz method with coarse spaces based on local Dirichlet-to-Neumann maps

Reference 21

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Observation 1022715c-ae25-4ce4-8982-0d4fcb8201d8 · outbound

This paper cites Why it is difficult to solve Helmholtz problems with classical iterative methods.

Achieving wavenumber robustness in domain decomposition for heterogeneous Helmholtz equation: an overview of spectral coarse spaces Why it is difficult to solve Helmholtz problems with classical iterative methods

Reference 22

Resolution: verified exact; doi, observed 2026-08-05T11:56:29.734772Z

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Observation 0378ff97-6526-47f0-a2e3-18b16cfaf66b · outbound

This paper cites FETI-DPH: a dual-primal domain decomposition method for acoustic scattering.

Achieving wavenumber robustness in domain decomposition for heterogeneous Helmholtz equation: an overview of spectral coarse spaces FETI-DPH: a dual-primal domain decomposition method for acoustic scattering

Reference 23

Resolution: verified exact; doi, observed 2026-08-05T11:56:29.541543Z

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Observation cd0c2c80-32bc-4b44-9adb-520d2c8f919d · outbound

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

Achieving wavenumber robustness in domain decomposition for heterogeneous Helmholtz equation: an overview of spectral coarse spaces A two-level domain decomposition method for the iterative solution of high frequency exterior Helmholtz problems

Reference 24

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Observation 89880f8c-d3a6-49df-a29a-03bedc0a817d · outbound

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

Achieving wavenumber robustness in domain decomposition for heterogeneous Helmholtz equation: an overview of spectral coarse spaces Global-basis two-level method for indefinite systems. I. Convergence studies

Reference 25

Resolution: verified exact; doi, observed 2026-08-05T11:56:29.404983Z

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

Achieving wavenumber robustness in domain decomposition for heterogeneous Helmholtz equation: an overview of spectral coarse spaces Thehp-FEM applied to the Helmholtz equation with PML truncation does not suffer from the pollution effect

Reference 26

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Observation cd6c485d-49c5-4bba-889a-a97a367e4187 · outbound

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

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This paper cites Galkowski and E.

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

Reference 28

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Observation 3acae57a-9968-4827-ae42-3faa47be8eef · outbound

This paper cites Probing the speckle to estimate the effective speed of sound, a first step towards quantitative ultrasound imaging.

Achieving wavenumber robustness in domain decomposition for heterogeneous Helmholtz equation: an overview of spectral coarse spaces Probing the speckle to estimate the effective speed of sound, a first step towards quantitative ultrasound imaging

Reference 29

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Observation 253cf2dd-9803-401f-b741-d6370ebb75d4 · outbound

This paper cites Garnier, L.

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

Reference 30

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

Achieving wavenumber robustness in domain decomposition for heterogeneous Helmholtz equation: an overview of spectral coarse spaces GO_3D_OBS: the multi-parameter benchmark geomodel for seismic imaging method assessment and next-generation 3D survey design (version 1.0)

Reference 31

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This paper cites Domain decomposition preconditioners for high-order discretiza- tions of the heterogeneous Helmholtz equation.

Achieving wavenumber robustness in domain decomposition for heterogeneous Helmholtz equation: an overview of spectral coarse spaces Domain decomposition preconditioners for high-order discretiza- tions of the heterogeneous Helmholtz equation

Reference 32

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Observation e606c3df-af04-447c-8d31-2a3cf7d7a1a8 · outbound

This paper cites Recent Results on Domain Decomposition Preconditioning for the High-Frequency Helmholtz Equation Using Absorption.

Achieving wavenumber robustness in domain decomposition for heterogeneous Helmholtz equation: an overview of spectral coarse spaces Recent Results on Domain Decomposition Preconditioning for the High-Frequency Helmholtz Equation Using Absorption

Reference 33

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

Achieving wavenumber robustness in domain decomposition for heterogeneous Helmholtz equation: an overview of spectral coarse spaces Toward a robust workflow for deep crustal imaging by FWI of OBS data: the eastern Nankai Trough revisited

Reference 34

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Achieving wavenumber robustness in domain decomposition for heterogeneous Helmholtz equation: an overview of spectral coarse spaces An additive Schwarz method type theory for Lions’s algorithm and a symmetrized optimized restricted additive Schwarz method

Reference 35

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This paper cites Domain decomposition with local impedance conditions for the Helmholtz equation with absorption.

Achieving wavenumber robustness in domain decomposition for heterogeneous Helmholtz equation: an overview of spectral coarse spaces Domain decomposition with local impedance conditions for the Helmholtz equation with absorption

Reference 36

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This paper cites SLEPc:Ascalableandflexibletoolkitforthesolutionofeigenvalue problems.

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

Reference 37

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This paper cites Analytical and numerical studies of a finite element PML for the Helmholtz equation.

Achieving wavenumber robustness in domain decomposition for heterogeneous Helmholtz equation: an overview of spectral coarse spaces Analytical and numerical studies of a finite element PML for the Helmholtz equation

Reference 38

Resolution: verified exact; doi, observed 2026-08-05T11:56:28.853843Z

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

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Observation 20a7dcc1-5b7e-453d-8f3f-087d000fbaf9 · outbound

This paper cites Hu and Z.

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

Reference 40

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Observation 2d039481-05f7-4bac-8f15-12b6768e6079 · outbound

This paper cites Scattering analysis of a large body with deep cavities.

Achieving wavenumber robustness in domain decomposition for heterogeneous Helmholtz equation: an overview of spectral coarse spaces Scattering analysis of a large body with deep cavities

Reference 41

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Observation bcb8a229-9d8d-494c-914d-6ff74b873186 · outbound

This paper cites Karypis and V.

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

Reference 42

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Observation 4865a5c0-5796-49a6-892f-87b8f2e9c9c1 · outbound

This paper cites Coarse spaces for non-symmetric two-level preconditioners based on local extended generalized eigenproblems.

Achieving wavenumber robustness in domain decomposition for heterogeneous Helmholtz equation: an overview of spectral coarse spaces Coarse spaces for non-symmetric two-level preconditioners based on local extended generalized eigenproblems

Reference 43

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Observation 9dec54ae-233d-4fe7-aab8-2d94d820545c · outbound

This paper cites Two-level Restricted Additive Schwarz preconditioner based on Multiscale Spectral Generalized FEM for Heterogeneous Helmholtz Problems.

Achieving wavenumber robustness in domain decomposition for heterogeneous Helmholtz equation: an overview of spectral coarse spaces Two-level Restricted Additive Schwarz preconditioner based on Multiscale Spectral Generalized FEM for Heterogeneous Helmholtz Problems

Reference 44

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Observation 71433be8-0db7-419c-bf5d-94ffa51b5b5d · outbound

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

Achieving wavenumber robustness in domain decomposition for heterogeneous Helmholtz equation: an overview of spectral coarse spaces Is 3D frequency-domain FWI of full- azimuth/long-offset OBN data feasible? The Gorgon data FWI case study

Reference 45

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Observation 58263bed-dc6a-4470-8cbc-3c1973eb4313 · outbound

This paper cites A coarse space construction based on local Dirichlet-to- Neumann maps.

Achieving wavenumber robustness in domain decomposition for heterogeneous Helmholtz equation: an overview of spectral coarse spaces A coarse space construction based on local Dirichlet-to- Neumann maps

Reference 46

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Observation f333c645-e0b4-440f-a674-492a3acbec7d · outbound

This paper cites Tournier, P.

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

Reference 47

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Observation 3e1f0312-9ef6-4e99-87a5-93dfca0f6ee6 · outbound

This paper cites Abstract robust coarse spaces for systems of PDEs via generalized eigenproblems in the overlaps.

Achieving wavenumber robustness in domain decomposition for heterogeneous Helmholtz equation: an overview of spectral coarse spaces Abstract robust coarse spaces for systems of PDEs via generalized eigenproblems in the overlaps

Reference 48

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Observation 1694bc09-553b-45c9-91af-f44f2688dfe6 · outbound

This paper cites A general approach for high order absorbing boundary conditions for the Helmholtz equation.

Achieving wavenumber robustness in domain decomposition for heterogeneous Helmholtz equation: an overview of spectral coarse spaces A general approach for high order absorbing boundary conditions for the Helmholtz equation

Reference 49

Resolution: verified exact; doi, observed 2026-08-05T11:56:28.735522Z

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Observation 307710ae-c672-4158-b26f-70401bee5080 · outbound

This paper cites Tournier, P.

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

Reference 50

Resolution: verified fuzzy; raw_fallback, observed 2026-08-05T11:56:32.006751Z

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This paper cites an unresolved cited work.

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

Reference 1816

Resolution: verified exact; doi, observed 2026-08-05T11:56:29.216793Z

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Pith citing papers

Observation 157e6f9f-de0d-4069-b5df-be650551439d · inbound

Can Symmetric Positive Definite (SPD) coarse spaces perform well for indefinite Helmholtz problems? cites this paper.

Can Symmetric Positive Definite (SPD) coarse spaces perform well for indefinite Helmholtz problems? Achieving wavenumber robustness in domain decomposition for heterogeneous Helmholtz equation: an overview of spectral coarse spaces

Reference 11

Resolution: verified exact; arxiv_id, observed 2026-05-24T03:48:53.078105Z

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Observation ebd31bad-2f70-4868-a35e-4e9449a751ea · inbound

Spectral coarse spaces based on indefinite operators: the $H_k$-GenEO method cites this paper.

Spectral coarse spaces based on indefinite operators: the $H_k$-GenEO method Achieving wavenumber robustness in domain decomposition for heterogeneous Helmholtz equation: an overview of spectral coarse spaces

Reference 11

Resolution: verified exact; arxiv_id, observed 2026-07-01T20:26:12.534470Z

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