{"as_of":"2026-08-13T06:15:00Z","caps":{"database_statements":6,"inbound":100,"outbound":100},"context_digest":"sha256:65ec043cf7df9c9bd46a890c8337faa58fda614d57ed610b32cbf4f6224d2496","coverage":[{"denominator":51,"lane":"reference_resolution","note":"Typed states for the displayed outbound observations.","records_observed":51,"source":"paper_references, paper_reference_links","source_observed_at":"2026-08-05T11:56:28.623464Z","state":"measured"},{"denominator":53,"lane":"standing_notices","note":"One-hop event checks from named stored sources.","records_observed":53,"source":"scholarly_work_events, retraction_status_cache","source_observed_at":"2026-08-12T06:34:41.77262+00:00","state":"measured"},{"denominator":2,"lane":"inbound_itemization","note":"Pith citing papers itemized under the disclosed page cap.","records_observed":2,"source":"paper_references, paper_reference_links","source_observed_at":"2026-06-28T21:13:13.357878Z","state":"measured"},{"denominator":1,"lane":"external_citation_measurements","note":"A source-named dated measurement, never combined with another source.","records_observed":0,"source":"arxiv_reference","source_observed_at":"2026-07-01T20:26:12.532490Z","state":"measured"}],"external_citation_measurements":[],"inbound":[{"citation":{"cited_paper":{"arxiv_id":"2509.02131","last_updated":"2025-09-02T09:26:08Z","snapshot_observed_at":"2026-08-13T04:10:03.444517Z","submitted_at":"2025-09-02T09:26:08Z","title":"Achieving wavenumber robustness in domain decomposition for heterogeneous Helmholtz equation: an overview of spectral coarse spaces","version":1},"cited_work":{"arxiv_id":"2509.02131","doi":null,"metadata_source":"arxiv_reference","pith_arxiv_id":"2509.02131","snapshot_observed_at":"2026-07-01T20:26:12.532490Z","title":"Dolean, M","venue":null,"work_id":"dbefb096-ff1c-48e4-95d9-2efc52640ea3","year":2025},"citing_paper":{"arxiv_id":"2403.18378","last_updated":"2026-02-06T06:11:32Z","snapshot_observed_at":"2026-08-02T08:31:12.743945Z","submitted_at":"2024-03-27T09:17:38Z","title":"Can Symmetric Positive Definite (SPD) coarse spaces perform well for indefinite Helmholtz problems?","version":5},"reference_index":11,"source":"pdf_text","source_observed_at":"2026-05-24T03:46:26.393347Z"},"links":{"cited_paper":"/paper/2509.02131","citing_paper":"/paper/2403.18378"},"observation_digest":"sha256:d47913cad3cd78e7867b4b2e05bd0b297dd27b0be07a9c0a56593867b62a155d","observation_id":"157e6f9f-de0d-4069-b5df-be650551439d","resolution":{"observed_at":"2026-05-24T03:48:53.078105Z","resolver_source":"arxiv_id","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-12T06:34:41.77262+00:00","source":"crossref"},{"observed_at":"2026-08-12T06:34:36.333875+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":{"arxiv_id":"2509.02131","last_updated":"2025-09-02T09:26:08Z","snapshot_observed_at":"2026-08-13T04:10:03.444517Z","submitted_at":"2025-09-02T09:26:08Z","title":"Achieving wavenumber robustness in domain decomposition for heterogeneous Helmholtz equation: an overview of spectral coarse spaces","version":1},"cited_work":{"arxiv_id":"2509.02131","doi":null,"metadata_source":"arxiv_reference","pith_arxiv_id":"2509.02131","snapshot_observed_at":"2026-07-01T20:26:12.532490Z","title":"Dolean, M","venue":null,"work_id":"dbefb096-ff1c-48e4-95d9-2efc52640ea3","year":2025},"citing_paper":{"arxiv_id":"2605.31552","last_updated":"2026-05-29T17:15:56Z","snapshot_observed_at":"2026-08-05T05:01:10.857322Z","submitted_at":"2026-05-29T17:15:56Z","title":"Spectral coarse spaces based on indefinite operators: the $H_k$-GenEO method","version":1},"reference_index":11,"source":"pdf_text","source_observed_at":"2026-06-28T21:13:13.357878Z"},"links":{"cited_paper":"/paper/2509.02131","citing_paper":"/paper/2605.31552"},"observation_digest":"sha256:5290a08f776f657fb49c5ceb32d4dafbb9983eb18bc60fc19c793812ba79e300","observation_id":"ebd31bad-2f70-4868-a35e-4e9449a751ea","resolution":{"observed_at":"2026-07-01T20:26:12.534470Z","resolver_source":"arxiv_id","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-12T06:34:41.77262+00:00","source":"crossref"},{"observed_at":"2026-08-12T06:34:36.333875+00:00","source":"retraction_watch"}],"state":"measured"}}],"links":{"evidence":"/evidence","html":"/paper/2509.02131/citation-record","integrity":"/paper/2509.02131/integrity","json":"/paper/2509.02131/citation-record.json","paper":"/paper/2509.02131"},"outbound":[{"citation":{"cited_paper":{"arxiv_id":"2401.03915","last_updated":"2024-01-08T14:23:56Z","snapshot_observed_at":"2026-08-13T04:47:18.920969Z","submitted_at":"2024-01-08T14:23:56Z","title":"A Robust Two-Level Schwarz Preconditioner For Sparse Matrices","version":1},"cited_work":{"arxiv_id":"2401.03915","doi":null,"metadata_source":"pith","pith_arxiv_id":"2401.03915","snapshot_observed_at":"2026-08-05T11:56:31.993865Z","title":"A Robust Two-Level Schwarz Preconditioner For Sparse Matrices","venue":"math.NA","work_id":"34bd7933-ae9d-4d5f-b53c-27a2bf7d6c74","year":2024},"citing_paper":{"arxiv_id":"2509.02131","last_updated":"2025-09-02T09:26:08Z","snapshot_observed_at":"2026-08-13T04:10:03.444517Z","submitted_at":"2025-09-02T09:26:08Z","title":"Achieving wavenumber robustness in domain decomposition for heterogeneous Helmholtz equation: an overview of spectral coarse spaces","version":1},"reference_index":1,"source":"pdf_text","source_observed_at":"2026-08-05T11:56:23.726678Z"},"links":{"cited_paper":"/paper/2401.03915","citing_paper":"/paper/2509.02131"},"observation_digest":"sha256:dea4344a52bb278eec2bd45566e151becbc1ea6063b0d6fb2280293634734edc","observation_id":"c31f5775-8437-4e7d-9c6d-cd7dac80c574","resolution":{"observed_at":"2026-08-05T11:56:31.997211Z","resolver_source":"local_arxiv","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-12T06:34:41.77262+00:00","source":"crossref"},{"observed_at":"2026-08-12T06:34:36.333875+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-05T11:56:32.146153Z","title":"A fully asynchronous multifrontal solver using dis- tributed dynamic scheduling","venue":null,"work_id":"8ffa9a0c-df4d-4cc7-9b9a-56be038e5701","year":2001},"citing_paper":{"arxiv_id":"2509.02131","last_updated":"2025-09-02T09:26:08Z","snapshot_observed_at":"2026-08-13T04:10:03.444517Z","submitted_at":"2025-09-02T09:26:08Z","title":"Achieving wavenumber robustness in domain decomposition for heterogeneous Helmholtz equation: an overview of spectral coarse spaces","version":1},"reference_index":2,"source":"pdf_text","source_observed_at":"2026-08-05T11:56:23.820539Z"},"links":{"citing_paper":"/paper/2509.02131"},"observation_digest":"sha256:3e0edef5b1f32a2b2086c31add71aa83745c6c1cc8b53f83ccdb92db994826d1","observation_id":"f26d1575-5538-40c3-aaa8-e86b1244d87f","resolution":{"observed_at":"2026-08-05T11:56:32.149030Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-12T06:34:41.77262+00:00","source":"crossref"},{"observed_at":"2026-08-12T06:34:36.333875+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-05T11:56:32.137830Z","title":"Is the pollution effect of the FEM avoidable for the Helmholtz equation considering high wave numbers?","venue":null,"work_id":"6c07f1b8-ad51-4a1f-ad1e-7b3327e7c04a","year":1997},"citing_paper":{"arxiv_id":"2509.02131","last_updated":"2025-09-02T09:26:08Z","snapshot_observed_at":"2026-08-13T04:10:03.444517Z","submitted_at":"2025-09-02T09:26:08Z","title":"Achieving wavenumber robustness in domain decomposition for heterogeneous Helmholtz equation: an overview of spectral coarse spaces","version":1},"reference_index":3,"source":"pdf_text","source_observed_at":"2026-08-05T11:56:23.896070Z"},"links":{"citing_paper":"/paper/2509.02131"},"observation_digest":"sha256:6045cb7b0a884871ae2c3fcb0fc32e68f83c7507a4c5e2b99c2d3769eaaa72a3","observation_id":"e0850ee1-2c60-4a6d-8486-fce09182330b","resolution":{"observed_at":"2026-08-05T11:56:32.140708Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-12T06:34:41.77262+00:00","source":"crossref"},{"observed_at":"2026-08-12T06:34:36.333875+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":"10.1002/nme.6581","metadata_source":"openalex","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-10T05:30:23.456663Z","title":"An automatic perfectly matched layer for acoustic finite element simulations in convex domains of general shape","venue":"International Journal for Numerical Methods in Engineering","work_id":"1913e0d5-71ca-4184-98f5-0d344a791414","year":2021},"citing_paper":{"arxiv_id":"2509.02131","last_updated":"2025-09-02T09:26:08Z","snapshot_observed_at":"2026-08-13T04:10:03.444517Z","submitted_at":"2025-09-02T09:26:08Z","title":"Achieving wavenumber robustness in domain decomposition for heterogeneous Helmholtz equation: an overview of spectral coarse spaces","version":1},"reference_index":4,"source":"pdf_text","source_observed_at":"2026-08-05T11:56:23.975446Z"},"links":{"citing_paper":"/paper/2509.02131"},"observation_digest":"sha256:899bad996c09bbe34e7259649eefaf5c564becf3af62f5a007825da5251b8cee","observation_id":"6ace3d74-ebd2-4ba3-b07a-7415bc989228","resolution":{"observed_at":"2026-08-05T11:56:30.413324Z","resolver_source":"doi","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-12T06:34:41.77262+00:00","source":"crossref"},{"observed_at":"2026-08-12T06:34:36.333875+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":"10.1016/j.crma.2004.10.006","metadata_source":"openalex","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-10T05:30:23.456663Z","title":"An exact bounded PML for the Helmholtz equation","venue":"Comptes Rendus Mathématique","work_id":"052539ab-619b-42e0-894f-e721d424f1ca","year":2004},"citing_paper":{"arxiv_id":"2509.02131","last_updated":"2025-09-02T09:26:08Z","snapshot_observed_at":"2026-08-13T04:10:03.444517Z","submitted_at":"2025-09-02T09:26:08Z","title":"Achieving wavenumber robustness in domain decomposition for heterogeneous Helmholtz equation: an overview of spectral coarse spaces","version":1},"reference_index":5,"source":"pdf_text","source_observed_at":"2026-08-05T11:56:24.053507Z"},"links":{"citing_paper":"/paper/2509.02131"},"observation_digest":"sha256:ca749196ee1e540522d09be38bf30484a1c3de022132e024a79d0ee12b19476f","observation_id":"ebfda9fb-54aa-4c0c-90e8-da378d4c4bd8","resolution":{"observed_at":"2026-08-05T11:56:30.223398Z","resolver_source":"doi","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-12T06:34:41.77262+00:00","source":"crossref"},{"observed_at":"2026-08-12T06:34:36.333875+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-05T11:56:24.123539Z","title":"Domain decomposition precon- ditioning for the high-frequency time-harmonic Maxwell equations with absorption","venue":null,"work_id":null,"year":2019},"citing_paper":{"arxiv_id":"2509.02131","last_updated":"2025-09-02T09:26:08Z","snapshot_observed_at":"2026-08-13T04:10:03.444517Z","submitted_at":"2025-09-02T09:26:08Z","title":"Achieving wavenumber robustness in domain decomposition for heterogeneous Helmholtz equation: an overview of spectral coarse spaces","version":1},"reference_index":6,"source":"pdf_text","source_observed_at":"2026-08-05T11:56:24.123539Z"},"links":{"citing_paper":"/paper/2509.02131"},"observation_digest":"sha256:b8bda9bfe26f0239fdb66ecb6dadf0d57f535c4115154e23353d69d9fc76d0c4","observation_id":"ec77fbfe-a9eb-43f0-ba66-bbd3886c8bcb","resolution":{"observed_at":"2026-08-05T11:56:24.123539Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-05T11:56:24.229784Z","title":"On the Dirichlet-to-Neumann coarse space for solving the Helmholtz prob- lem using domain decomposition","venue":null,"work_id":null,"year":2019},"citing_paper":{"arxiv_id":"2509.02131","last_updated":"2025-09-02T09:26:08Z","snapshot_observed_at":"2026-08-13T04:10:03.444517Z","submitted_at":"2025-09-02T09:26:08Z","title":"Achieving wavenumber robustness in domain decomposition for heterogeneous Helmholtz equation: an overview of spectral coarse spaces","version":1},"reference_index":7,"source":"pdf_text","source_observed_at":"2026-08-05T11:56:24.229784Z"},"links":{"citing_paper":"/paper/2509.02131"},"observation_digest":"sha256:89b5670088bc64479edb630108c96d8239fb7470b5e19e6e2a3131f96c9db193","observation_id":"e142372b-010d-408f-9714-b62e31d78eb5","resolution":{"observed_at":"2026-08-05T11:56:24.229784Z","resolver_source":null,"status":"malformed_identifier"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-05T11:56:32.128996Z","title":"Can DtN and GenEO Coarse Spaces Be Sufficiently Robust for Heteroge- neous Helmholtz Problems?","venue":null,"work_id":"2b112246-dc94-41c4-ada8-ff709db1fa2b","year":2022},"citing_paper":{"arxiv_id":"2509.02131","last_updated":"2025-09-02T09:26:08Z","snapshot_observed_at":"2026-08-13T04:10:03.444517Z","submitted_at":"2025-09-02T09:26:08Z","title":"Achieving wavenumber robustness in domain decomposition for heterogeneous Helmholtz equation: an overview of spectral coarse spaces","version":1},"reference_index":8,"source":"pdf_text","source_observed_at":"2026-08-05T11:56:24.309780Z"},"links":{"citing_paper":"/paper/2509.02131"},"observation_digest":"sha256:d338b2054601fe5508b1c3d1ed29f4ecfa7a4c9a011c024847dbf26bc0ac26b3","observation_id":"eedd1e18-3630-404a-b8c8-a8ad737cfbc6","resolution":{"observed_at":"2026-08-05T11:56:32.131924Z","resolver_source":"raw_fallback","status":"malformed_identifier"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-12T06:34:41.77262+00:00","source":"crossref"},{"observed_at":"2026-08-12T06:34:36.333875+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-05T11:56:24.382590Z","title":"Overlapping Schwarz methods with Ge- nEO coarse spaces for indefinite and nonself-adjoint problems","venue":null,"work_id":null,"year":2022},"citing_paper":{"arxiv_id":"2509.02131","last_updated":"2025-09-02T09:26:08Z","snapshot_observed_at":"2026-08-13T04:10:03.444517Z","submitted_at":"2025-09-02T09:26:08Z","title":"Achieving wavenumber robustness in domain decomposition for heterogeneous Helmholtz equation: an overview of spectral coarse spaces","version":1},"reference_index":9,"source":"pdf_text","source_observed_at":"2026-08-05T11:56:24.382590Z"},"links":{"citing_paper":"/paper/2509.02131"},"observation_digest":"sha256:86feda03c3190bc157817f5e7b04a2c215a83cff2023c95763be1ba896664a7a","observation_id":"465509bd-2dc1-4a84-9b0f-30cb4b06f437","resolution":{"observed_at":"2026-08-05T11:56:24.382590Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-05T11:56:24.429343Z","title":"GenEO coarse spaces for heterogeneous indefinite elliptic problems","venue":null,"work_id":null,"year":2022},"citing_paper":{"arxiv_id":"2509.02131","last_updated":"2025-09-02T09:26:08Z","snapshot_observed_at":"2026-08-13T04:10:03.444517Z","submitted_at":"2025-09-02T09:26:08Z","title":"Achieving wavenumber robustness in domain decomposition for heterogeneous Helmholtz equation: an overview of spectral coarse spaces","version":1},"reference_index":10,"source":"pdf_text","source_observed_at":"2026-08-05T11:56:24.429343Z"},"links":{"citing_paper":"/paper/2509.02131"},"observation_digest":"sha256:55a4d89a58ab5a3ca06ff6e3f1eec94f1bf8fb55138ae728b2d64453c594c4d0","observation_id":"69f2a01b-4ae5-4d1e-9cf3-e67f0dcc17b9","resolution":{"observed_at":"2026-08-05T11:56:24.429343Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-05T11:56:24.528029Z","title":"AcomparisonofcoarsespacesforHelmholtzproblems in the high frequency regime","venue":null,"work_id":null,"year":2021},"citing_paper":{"arxiv_id":"2509.02131","last_updated":"2025-09-02T09:26:08Z","snapshot_observed_at":"2026-08-13T04:10:03.444517Z","submitted_at":"2025-09-02T09:26:08Z","title":"Achieving wavenumber robustness in domain decomposition for heterogeneous Helmholtz equation: an overview of spectral coarse spaces","version":1},"reference_index":11,"source":"pdf_text","source_observed_at":"2026-08-05T11:56:24.528029Z"},"links":{"citing_paper":"/paper/2509.02131"},"observation_digest":"sha256:2d4c77d8a2c350111db3ee70b5b01cd6cdf98fd81bf6ea510d9f4b3fc9884efd","observation_id":"3ee30b98-85e4-40ea-9827-f2e94bf3522f","resolution":{"observed_at":"2026-08-05T11:56:24.528029Z","resolver_source":null,"status":"malformed_identifier"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-05T11:56:32.120416Z","title":"Wave-ray multigrid method for standing wave equations","venue":null,"work_id":"34699b2d-5f10-4d19-9127-7adee4cbf97b","year":1997},"citing_paper":{"arxiv_id":"2509.02131","last_updated":"2025-09-02T09:26:08Z","snapshot_observed_at":"2026-08-13T04:10:03.444517Z","submitted_at":"2025-09-02T09:26:08Z","title":"Achieving wavenumber robustness in domain decomposition for heterogeneous Helmholtz equation: an overview of spectral coarse spaces","version":1},"reference_index":12,"source":"pdf_text","source_observed_at":"2026-08-05T11:56:24.584039Z"},"links":{"citing_paper":"/paper/2509.02131"},"observation_digest":"sha256:fd8df6a9abd842ae8637d70141aa0756a4473a26927352bb5b20f61356258add","observation_id":"f2e58b1c-3081-4243-aeee-43cc22c9b5aa","resolution":{"observed_at":"2026-08-05T11:56:32.123377Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-12T06:34:41.77262+00:00","source":"crossref"},{"observed_at":"2026-08-12T06:34:36.333875+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":{"arxiv_id":"2406.06283","last_updated":"2024-06-10T14:06:46Z","snapshot_observed_at":"2026-08-12T23:46:28.214385Z","submitted_at":"2024-06-10T14:06:46Z","title":"Schwarz preconditioner with $H_k$-GenEO coarse space for the indefinite Helmholtz problem","version":1},"cited_work":{"arxiv_id":"2406.06283","doi":null,"metadata_source":"pith","pith_arxiv_id":"2406.06283","snapshot_observed_at":"2026-08-05T11:56:31.980804Z","title":"Schwarz preconditioner with $H_k$-GenEO coarse space for the indefinite Helmholtz problem","venue":"math.NA","work_id":"ad9928f1-570b-479c-8fbd-a73a562613e4","year":2024},"citing_paper":{"arxiv_id":"2509.02131","last_updated":"2025-09-02T09:26:08Z","snapshot_observed_at":"2026-08-13T04:10:03.444517Z","submitted_at":"2025-09-02T09:26:08Z","title":"Achieving wavenumber robustness in domain decomposition for heterogeneous Helmholtz equation: an overview of spectral coarse spaces","version":1},"reference_index":13,"source":"pdf_text","source_observed_at":"2026-08-05T11:56:24.660740Z"},"links":{"cited_paper":"/paper/2406.06283","citing_paper":"/paper/2509.02131"},"observation_digest":"sha256:a9d3bdfbb7bc7d1ca8994b85f4f4d6b860dc1cb52b35140f36fb18453cfe4b46","observation_id":"aac061ef-2279-451c-a7a2-2c57fbc32d49","resolution":{"observed_at":"2026-08-05T11:56:31.984664Z","resolver_source":"local_arxiv","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-12T06:34:41.77262+00:00","source":"crossref"},{"observed_at":"2026-08-12T06:34:36.333875+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-05T11:56:24.770701Z","title":"A coarse space for heterogeneous Helmholtz problems based on the Dirichlet-to-Neumann operator","venue":null,"work_id":null,"year":2014},"citing_paper":{"arxiv_id":"2509.02131","last_updated":"2025-09-02T09:26:08Z","snapshot_observed_at":"2026-08-13T04:10:03.444517Z","submitted_at":"2025-09-02T09:26:08Z","title":"Achieving wavenumber robustness in domain decomposition for heterogeneous Helmholtz equation: an overview of spectral coarse spaces","version":1},"reference_index":14,"source":"pdf_text","source_observed_at":"2026-08-05T11:56:24.770701Z"},"links":{"citing_paper":"/paper/2509.02131"},"observation_digest":"sha256:f0c280428e7031c79af0ce44493bbd760a9bf9fcf26a94ca05567078a87394db","observation_id":"834322d3-88f8-4c4a-8788-57a4bf362b92","resolution":{"observed_at":"2026-08-05T11:56:24.770701Z","resolver_source":null,"status":"malformed_identifier"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":"10.1190/segam2020-3427414.1","metadata_source":"openalex","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-10T05:30:23.456663Z","title":"Large-scale frequency-domain seismic wave modeling on h- adaptive tetrahedral meshes with iterative solver and multi-level domain-decomposition preconditioners","venue":null,"work_id":"029911ba-756e-451f-b1b0-a5a0d496f605","year":2020},"citing_paper":{"arxiv_id":"2509.02131","last_updated":"2025-09-02T09:26:08Z","snapshot_observed_at":"2026-08-13T04:10:03.444517Z","submitted_at":"2025-09-02T09:26:08Z","title":"Achieving wavenumber robustness in domain decomposition for heterogeneous Helmholtz equation: an overview of spectral coarse spaces","version":1},"reference_index":15,"source":"pdf_text","source_observed_at":"2026-08-05T11:56:24.883392Z"},"links":{"citing_paper":"/paper/2509.02131"},"observation_digest":"sha256:7979cbd05dec11f33e0b8c2e291c562c0284d928ee7fc960f3e3941b745fafb5","observation_id":"8fed09e7-899b-4224-963e-75ba3ab3fa84","resolution":{"observed_at":"2026-08-05T11:56:30.027136Z","resolver_source":"doi","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-12T06:34:41.77262+00:00","source":"crossref"},{"observed_at":"2026-08-12T06:34:36.333875+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":{"arxiv_id":"2403.18378","last_updated":"2026-02-06T06:11:32Z","snapshot_observed_at":"2026-08-02T08:31:12.743945Z","submitted_at":"2024-03-27T09:17:38Z","title":"Can Symmetric Positive Definite (SPD) coarse spaces perform well for indefinite Helmholtz problems?","version":5},"cited_work":{"arxiv_id":"2403.18378","doi":null,"metadata_source":"pith","pith_arxiv_id":"2403.18378","snapshot_observed_at":"2026-08-05T11:56:31.937358Z","title":"Can Symmetric Positive Definite (SPD) coarse spaces perform well for indefinite Helmholtz problems?","venue":"math.NA","work_id":"fa4361ac-52d3-4656-951c-dea3e1ce9307","year":2024},"citing_paper":{"arxiv_id":"2509.02131","last_updated":"2025-09-02T09:26:08Z","snapshot_observed_at":"2026-08-13T04:10:03.444517Z","submitted_at":"2025-09-02T09:26:08Z","title":"Achieving wavenumber robustness in domain decomposition for heterogeneous Helmholtz equation: an overview of spectral coarse spaces","version":1},"reference_index":16,"source":"pdf_text","source_observed_at":"2026-08-05T11:56:24.972798Z"},"links":{"cited_paper":"/paper/2403.18378","citing_paper":"/paper/2509.02131"},"observation_digest":"sha256:fc24dcdc2a793568a574688db4b9e29d6d4185a1332ded293c41fa131574c9a1","observation_id":"caccd204-91d1-4d0a-9a5c-2a269b28ce96","resolution":{"observed_at":"2026-08-05T11:56:31.971424Z","resolver_source":"local_arxiv","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-12T06:34:41.77262+00:00","source":"crossref"},{"observed_at":"2026-08-12T06:34:36.333875+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-05T11:56:25.061888Z","title":"Effective transmission conditions for domain decomposition methods applied to the time-harmonic curl-curl Maxwell’s equations","venue":null,"work_id":null,"year":2015},"citing_paper":{"arxiv_id":"2509.02131","last_updated":"2025-09-02T09:26:08Z","snapshot_observed_at":"2026-08-13T04:10:03.444517Z","submitted_at":"2025-09-02T09:26:08Z","title":"Achieving wavenumber robustness in domain decomposition for heterogeneous Helmholtz equation: an overview of spectral coarse spaces","version":1},"reference_index":17,"source":"pdf_text","source_observed_at":"2026-08-05T11:56:25.061888Z"},"links":{"citing_paper":"/paper/2509.02131"},"observation_digest":"sha256:edcb7bf90c22f1f42b603c8518a7d526ead50554ddfba93364b3bd99cdbea810","observation_id":"89d62bf4-68bf-4d9b-a131-a6fc27aadf06","resolution":{"observed_at":"2026-08-05T11:56:25.061888Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-05T11:56:25.127034Z","title":"Dolean, P","venue":null,"work_id":null,"year":2015},"citing_paper":{"arxiv_id":"2509.02131","last_updated":"2025-09-02T09:26:08Z","snapshot_observed_at":"2026-08-13T04:10:03.444517Z","submitted_at":"2025-09-02T09:26:08Z","title":"Achieving wavenumber robustness in domain decomposition for heterogeneous Helmholtz equation: an overview of spectral coarse spaces","version":1},"reference_index":18,"source":"pdf_text","source_observed_at":"2026-08-05T11:56:25.127034Z"},"links":{"citing_paper":"/paper/2509.02131"},"observation_digest":"sha256:23cdad9cf7fe648899e709fc4fc4204e74194fc5e1c1d9ba8a1fd4b504c1ed95","observation_id":"9a4dad2a-ec88-4555-a451-450a30db8f58","resolution":{"observed_at":"2026-08-05T11:56:25.127034Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-05T11:56:32.111761Z","title":"Iterative frequency-domain seismic wave solvers based on multi-level domain-decomposition preconditioners","venue":null,"work_id":"6d21f9e3-5f4c-408a-ae11-5c318680ac75","year":2020},"citing_paper":{"arxiv_id":"2509.02131","last_updated":"2025-09-02T09:26:08Z","snapshot_observed_at":"2026-08-13T04:10:03.444517Z","submitted_at":"2025-09-02T09:26:08Z","title":"Achieving wavenumber robustness in domain decomposition for heterogeneous Helmholtz equation: an overview of spectral coarse spaces","version":1},"reference_index":19,"source":"pdf_text","source_observed_at":"2026-08-05T11:56:25.247162Z"},"links":{"citing_paper":"/paper/2509.02131"},"observation_digest":"sha256:20c76d6b7def1cfd8f86a870a66fc12e40ece0a7b4a5fb627b5761adb7588788","observation_id":"da5a9d2f-05d2-4818-a813-d1208c69f1ed","resolution":{"observed_at":"2026-08-05T11:56:32.114801Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-12T06:34:41.77262+00:00","source":"crossref"},{"observed_at":"2026-08-12T06:34:36.333875+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-05T11:56:32.103264Z","title":"2020, pp","venue":null,"work_id":"d2f5bd14-325f-4aaf-b5e9-faf17979b7ab","year":2020},"citing_paper":{"arxiv_id":"2509.02131","last_updated":"2025-09-02T09:26:08Z","snapshot_observed_at":"2026-08-13T04:10:03.444517Z","submitted_at":"2025-09-02T09:26:08Z","title":"Achieving wavenumber robustness in domain decomposition for heterogeneous Helmholtz equation: an overview of spectral coarse spaces","version":1},"reference_index":20,"source":"pdf_text","source_observed_at":"2026-08-05T11:56:25.310179Z"},"links":{"citing_paper":"/paper/2509.02131"},"observation_digest":"sha256:ffd1aa7ceb2440adfc08fedab03510b1f4158a1e7d279c1bc68c7e5b437d2487","observation_id":"62e85a67-eeac-4517-b98d-6aa78dbec6c5","resolution":{"observed_at":"2026-08-05T11:56:32.106024Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-12T06:34:41.77262+00:00","source":"crossref"},{"observed_at":"2026-08-12T06:34:36.333875+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":"10.2478/cmam-2012-0027","metadata_source":"openalex","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-10T05:30:23.456663Z","title":"Analysis of a two-level Schwarz method with coarse spaces based on local Dirichlet-to-Neumann maps","venue":"Computational Methods in Applied Mathematics","work_id":"ad01f626-1ed5-4ece-8979-185ee55adb8a","year":2012},"citing_paper":{"arxiv_id":"2509.02131","last_updated":"2025-09-02T09:26:08Z","snapshot_observed_at":"2026-08-13T04:10:03.444517Z","submitted_at":"2025-09-02T09:26:08Z","title":"Achieving wavenumber robustness in domain decomposition for heterogeneous Helmholtz equation: an overview of spectral coarse spaces","version":1},"reference_index":21,"source":"pdf_text","source_observed_at":"2026-08-05T11:56:25.402757Z"},"links":{"citing_paper":"/paper/2509.02131"},"observation_digest":"sha256:ebd31303087e0db1374ec13d08a6d2f48149a84152b1b5ed228f53000a96c8e5","observation_id":"df18c7bd-46cb-4096-8bff-2a7d6284844d","resolution":{"observed_at":"2026-08-05T11:56:29.895451Z","resolver_source":"doi","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-12T06:34:41.77262+00:00","source":"crossref"},{"observed_at":"2026-08-12T06:34:36.333875+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":"10.1007/978-3-642-22061-6","metadata_source":"openalex","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-10T05:30:23.456663Z","title":"Why it is difficult to solve Helmholtz problems with classical iterative methods","venue":"Lecture notes in computational science and engineering","work_id":"a11e2212-a6f7-40d1-8dbe-961cffc9c80c","year":2012},"citing_paper":{"arxiv_id":"2509.02131","last_updated":"2025-09-02T09:26:08Z","snapshot_observed_at":"2026-08-13T04:10:03.444517Z","submitted_at":"2025-09-02T09:26:08Z","title":"Achieving wavenumber robustness in domain decomposition for heterogeneous Helmholtz equation: an overview of spectral coarse spaces","version":1},"reference_index":22,"source":"pdf_text","source_observed_at":"2026-08-05T11:56:25.557876Z"},"links":{"citing_paper":"/paper/2509.02131"},"observation_digest":"sha256:b77ba99ed007d05bdcb4e6961fe117810dddc035e51479b923a4bec07dcb0796","observation_id":"1022715c-ae25-4ce4-8982-0d4fcb8201d8","resolution":{"observed_at":"2026-08-05T11:56:29.734772Z","resolver_source":"doi","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-12T06:34:41.77262+00:00","source":"crossref"},{"observed_at":"2026-08-12T06:34:36.333875+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":"10.1142/s0218396x05002761","metadata_source":"openalex","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-10T05:30:23.456663Z","title":"FETI-DPH: a dual-primal domain decomposition method for acoustic scattering","venue":"Journal of Computational Acoustics","work_id":"220c9af4-75cf-495f-97de-576a0b71099a","year":2005},"citing_paper":{"arxiv_id":"2509.02131","last_updated":"2025-09-02T09:26:08Z","snapshot_observed_at":"2026-08-13T04:10:03.444517Z","submitted_at":"2025-09-02T09:26:08Z","title":"Achieving wavenumber robustness in domain decomposition for heterogeneous Helmholtz equation: an overview of spectral coarse spaces","version":1},"reference_index":23,"source":"pdf_text","source_observed_at":"2026-08-05T11:56:25.704062Z"},"links":{"citing_paper":"/paper/2509.02131"},"observation_digest":"sha256:906fea52a0eddc73ed4eb951595d1dc3a02be83de41e747ae6cd26d05dac7034","observation_id":"0378ff97-6526-47f0-a2e3-18b16cfaf66b","resolution":{"observed_at":"2026-08-05T11:56:29.541543Z","resolver_source":"doi","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-12T06:34:41.77262+00:00","source":"crossref"},{"observed_at":"2026-08-12T06:34:36.333875+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-05T11:56:32.094720Z","title":"A two-level domain decomposition method for the iterative solution of high frequency exterior Helmholtz problems","venue":null,"work_id":"b53da6c5-4b67-44aa-b0c8-20c84141a291","year":2000},"citing_paper":{"arxiv_id":"2509.02131","last_updated":"2025-09-02T09:26:08Z","snapshot_observed_at":"2026-08-13T04:10:03.444517Z","submitted_at":"2025-09-02T09:26:08Z","title":"Achieving wavenumber robustness in domain decomposition for heterogeneous Helmholtz equation: an overview of spectral coarse spaces","version":1},"reference_index":24,"source":"pdf_text","source_observed_at":"2026-08-05T11:56:25.856219Z"},"links":{"citing_paper":"/paper/2509.02131"},"observation_digest":"sha256:0aeffc8aeb13f53e78486e5f299d83f67f91728c3f2d2fa7d80e04ca6abfddf9","observation_id":"cd0c2c80-32bc-4b44-9adb-520d2c8f919d","resolution":{"observed_at":"2026-08-05T11:56:32.097758Z","resolver_source":"raw_fallback","status":"malformed_identifier"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-12T06:34:41.77262+00:00","source":"crossref"},{"observed_at":"2026-08-12T06:34:36.333875+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":"10.1002/1097-0207(20000930)49:3","metadata_source":"doi_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-05T11:56:29.299358Z","title":"Global-basis two-level method for indefinite systems. I. Convergence studies","venue":null,"work_id":"312349d5-b603-4227-b942-0cd3770b8ad6","year":2000},"citing_paper":{"arxiv_id":"2509.02131","last_updated":"2025-09-02T09:26:08Z","snapshot_observed_at":"2026-08-13T04:10:03.444517Z","submitted_at":"2025-09-02T09:26:08Z","title":"Achieving wavenumber robustness in domain decomposition for heterogeneous Helmholtz equation: an overview of spectral coarse spaces","version":1},"reference_index":25,"source":"pdf_text","source_observed_at":"2026-08-05T11:56:25.985987Z"},"links":{"citing_paper":"/paper/2509.02131"},"observation_digest":"sha256:b8edba5d97d4b848470b14692076b97096d25ce56f2e3190af0926ea0e70f451","observation_id":"89880f8c-d3a6-49df-a29a-03bedc0a817d","resolution":{"observed_at":"2026-08-05T11:56:29.404983Z","resolver_source":"doi","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-12T06:34:41.77262+00:00","source":"crossref"},{"observed_at":"2026-08-12T06:34:36.333875+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-05T11:56:32.085925Z","title":"Thehp-FEM applied to the Helmholtz equation with PML truncation does not suffer from the pollution effect","venue":null,"work_id":"acff3d8d-62dc-4df9-bb0d-d5c7cd76acbc","year":2024},"citing_paper":{"arxiv_id":"2509.02131","last_updated":"2025-09-02T09:26:08Z","snapshot_observed_at":"2026-08-13T04:10:03.444517Z","submitted_at":"2025-09-02T09:26:08Z","title":"Achieving wavenumber robustness in domain decomposition for heterogeneous Helmholtz equation: an overview of spectral coarse spaces","version":1},"reference_index":26,"source":"pdf_text","source_observed_at":"2026-08-05T11:56:26.094011Z"},"links":{"citing_paper":"/paper/2509.02131"},"observation_digest":"sha256:0c3f6bee32ae3e1e37d4a701a2ff509eeaa86fed7c250839ca04880ab897f3a8","observation_id":"816959f8-aa06-41ac-b134-c9c63592ec4e","resolution":{"observed_at":"2026-08-05T11:56:32.088950Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-12T06:34:41.77262+00:00","source":"crossref"},{"observed_at":"2026-08-12T06:34:36.333875+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-05T11:56:26.549777Z","title":null,"venue":null,"work_id":null,"year":2019},"citing_paper":{"arxiv_id":"2509.02131","last_updated":"2025-09-02T09:26:08Z","snapshot_observed_at":"2026-08-13T04:10:03.444517Z","submitted_at":"2025-09-02T09:26:08Z","title":"Achieving wavenumber robustness in domain decomposition for heterogeneous Helmholtz equation: an overview of spectral coarse spaces","version":1},"reference_index":27,"source":"pdf_text","source_observed_at":"2026-08-05T11:56:26.549777Z"},"links":{"citing_paper":"/paper/2509.02131"},"observation_digest":"sha256:6710b345b38ec6a019794d3b7f61ce5c6c9fb7a8bdba1db31be24233e02ea7d7","observation_id":"cd6c485d-49c5-4bba-889a-a97a367e4187","resolution":{"observed_at":"2026-08-05T11:56:26.549777Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":"2501.11060","doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-05T11:56:31.708469Z","title":"Galkowski and E","venue":null,"work_id":"a241f007-ada0-40ab-89da-45c806f4b2cf","year":2025},"citing_paper":{"arxiv_id":"2509.02131","last_updated":"2025-09-02T09:26:08Z","snapshot_observed_at":"2026-08-13T04:10:03.444517Z","submitted_at":"2025-09-02T09:26:08Z","title":"Achieving wavenumber robustness in domain decomposition for heterogeneous Helmholtz equation: an overview of spectral coarse spaces","version":1},"reference_index":28,"source":"pdf_text","source_observed_at":"2026-08-05T11:56:26.371939Z"},"links":{"citing_paper":"/paper/2509.02131"},"observation_digest":"sha256:b3570bdb3baf29134b03d02e58edde858f57f19ea65f41b8f38144994e3b981e","observation_id":"c75d8ff2-f6c9-44e2-ab9f-a51d24246d70","resolution":{"observed_at":"2026-08-05T11:56:31.790485Z","resolver_source":"raw_fallback","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-12T06:34:41.77262+00:00","source":"crossref"},{"observed_at":"2026-08-12T06:34:36.333875+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":{"arxiv_id":"2505.07566","last_updated":"2025-05-12T13:49:24Z","snapshot_observed_at":"2026-08-11T01:25:41.737535Z","submitted_at":"2025-05-12T13:49:24Z","title":"Probing the speckle to estimate the effective speed of sound, a first step towards quantitative ultrasound imaging","version":1},"cited_work":{"arxiv_id":"2505.07566","doi":null,"metadata_source":"pith","pith_arxiv_id":"2505.07566","snapshot_observed_at":"2026-08-05T11:56:31.177762Z","title":"Probing the speckle to estimate the effective speed of sound, a first step towards quantitative ultrasound imaging","venue":"math.AP","work_id":"314fcd81-feab-4bd1-ba85-aaa624e06177","year":2025},"citing_paper":{"arxiv_id":"2509.02131","last_updated":"2025-09-02T09:26:08Z","snapshot_observed_at":"2026-08-13T04:10:03.444517Z","submitted_at":"2025-09-02T09:26:08Z","title":"Achieving wavenumber robustness in domain decomposition for heterogeneous Helmholtz equation: an overview of spectral coarse spaces","version":1},"reference_index":29,"source":"pdf_text","source_observed_at":"2026-08-05T11:56:26.743829Z"},"links":{"cited_paper":"/paper/2505.07566","citing_paper":"/paper/2509.02131"},"observation_digest":"sha256:73b3cdbd19d13af9d6b0cd28070ce9603a58c62d75ae286c6fffad9ebabbf929","observation_id":"3acae57a-9968-4827-ae42-3faa47be8eef","resolution":{"observed_at":"2026-08-05T11:56:31.276973Z","resolver_source":"local_arxiv","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-12T06:34:41.77262+00:00","source":"crossref"},{"observed_at":"2026-08-12T06:34:36.333875+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":"2309.07777","doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-05T11:56:31.445605Z","title":"Garnier, L","venue":null,"work_id":"8fe1821c-a56b-4cf6-80e7-0e069c5dc50a","year":2023},"citing_paper":{"arxiv_id":"2509.02131","last_updated":"2025-09-02T09:26:08Z","snapshot_observed_at":"2026-08-13T04:10:03.444517Z","submitted_at":"2025-09-02T09:26:08Z","title":"Achieving wavenumber robustness in domain decomposition for heterogeneous Helmholtz equation: an overview of spectral coarse spaces","version":1},"reference_index":30,"source":"pdf_text","source_observed_at":"2026-08-05T11:56:26.676262Z"},"links":{"citing_paper":"/paper/2509.02131"},"observation_digest":"sha256:369bccc4355927fcd0697147f760dce6bda6153fb33bfb9d948df5bef0abebaf","observation_id":"253cf2dd-9803-401f-b741-d6370ebb75d4","resolution":{"observed_at":"2026-08-05T11:56:31.506883Z","resolver_source":"raw_fallback","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-12T06:34:41.77262+00:00","source":"crossref"},{"observed_at":"2026-08-12T06:34:36.333875+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-05T11:56:32.077144Z","title":"GO_3D_OBS: the multi-parameter benchmark geomodel for seismic imaging method assessment and next-generation 3D survey design (version 1.0)","venue":null,"work_id":"62a6c125-0a95-4bbd-ab03-f05240981105","year":2021},"citing_paper":{"arxiv_id":"2509.02131","last_updated":"2025-09-02T09:26:08Z","snapshot_observed_at":"2026-08-13T04:10:03.444517Z","submitted_at":"2025-09-02T09:26:08Z","title":"Achieving wavenumber robustness in domain decomposition for heterogeneous Helmholtz equation: an overview of spectral coarse spaces","version":1},"reference_index":31,"source":"pdf_text","source_observed_at":"2026-08-05T11:56:26.978576Z"},"links":{"citing_paper":"/paper/2509.02131"},"observation_digest":"sha256:a4900e9c9d7e63d22e3c23e785af5519175c9fa0a969d441164771bfe8fb4fb9","observation_id":"1ff26e5f-69b8-44da-b3b0-f530600093f2","resolution":{"observed_at":"2026-08-05T11:56:32.080191Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-12T06:34:41.77262+00:00","source":"crossref"},{"observed_at":"2026-08-12T06:34:36.333875+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":"10.1093/imanum/draa080","metadata_source":"openalex","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-10T05:30:23.456663Z","title":"Domain decomposition preconditioners for high-order discretiza- tions of the heterogeneous Helmholtz equation","venue":"IMA Journal of Numerical Analysis","work_id":"cea51b4e-6409-4116-b014-977fcbbfc452","year":2021},"citing_paper":{"arxiv_id":"2509.02131","last_updated":"2025-09-02T09:26:08Z","snapshot_observed_at":"2026-08-13T04:10:03.444517Z","submitted_at":"2025-09-02T09:26:08Z","title":"Achieving wavenumber robustness in domain decomposition for heterogeneous Helmholtz equation: an overview of spectral coarse spaces","version":1},"reference_index":32,"source":"pdf_text","source_observed_at":"2026-08-05T11:56:26.879584Z"},"links":{"citing_paper":"/paper/2509.02131"},"observation_digest":"sha256:3f2973c07ef05dca28bd6f9369920505afad69796d4bb7a83916fcc18eee2769","observation_id":"d34f0199-5c65-43bd-af79-6861ac15dfea","resolution":{"observed_at":"2026-08-05T11:56:29.089862Z","resolver_source":"doi","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-12T06:34:41.77262+00:00","source":"crossref"},{"observed_at":"2026-08-12T06:34:36.333875+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-05T11:56:27.266110Z","title":"Recent Results on Domain Decomposition Preconditioning for the High-Frequency Helmholtz Equation Using Absorption","venue":null,"work_id":null,"year":2017},"citing_paper":{"arxiv_id":"2509.02131","last_updated":"2025-09-02T09:26:08Z","snapshot_observed_at":"2026-08-13T04:10:03.444517Z","submitted_at":"2025-09-02T09:26:08Z","title":"Achieving wavenumber robustness in domain decomposition for heterogeneous Helmholtz equation: an overview of spectral coarse spaces","version":1},"reference_index":33,"source":"pdf_text","source_observed_at":"2026-08-05T11:56:27.266110Z"},"links":{"citing_paper":"/paper/2509.02131"},"observation_digest":"sha256:7f981af12c5ba75f74e351b49c741162e38f1ccb1b1c6fd04badf03affa23135","observation_id":"e606c3df-af04-447c-8d31-2a3cf7d7a1a8","resolution":{"observed_at":"2026-08-05T11:56:27.266110Z","resolver_source":null,"status":"malformed_identifier"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-05T11:56:32.068690Z","title":"Toward a robust workflow for deep crustal imaging by FWI of OBS data: the eastern Nankai Trough revisited","venue":null,"work_id":"51818ee5-3a4f-4259-b5e1-337bce5d055e","year":2017},"citing_paper":{"arxiv_id":"2509.02131","last_updated":"2025-09-02T09:26:08Z","snapshot_observed_at":"2026-08-13T04:10:03.444517Z","submitted_at":"2025-09-02T09:26:08Z","title":"Achieving wavenumber robustness in domain decomposition for heterogeneous Helmholtz equation: an overview of spectral coarse spaces","version":1},"reference_index":34,"source":"pdf_text","source_observed_at":"2026-08-05T11:56:27.152077Z"},"links":{"citing_paper":"/paper/2509.02131"},"observation_digest":"sha256:120132bede4de05436c46eff39b26a3193ef0de78a3ea23b058eda4cf621f3b0","observation_id":"5ae107ab-3058-4494-8731-2a379399d13c","resolution":{"observed_at":"2026-08-05T11:56:32.071762Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-12T06:34:41.77262+00:00","source":"crossref"},{"observed_at":"2026-08-12T06:34:36.333875+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":"10.1137/16m1060066","metadata_source":"openalex","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-10T05:30:23.456663Z","title":"An additive Schwarz method type theory for Lions’s algorithm and a symmetrized optimized restricted additive Schwarz method","venue":"SIAM Journal on Scientific Computing","work_id":"7c994cab-fa7d-4e03-91ac-e9004e2d8a1c","year":2017},"citing_paper":{"arxiv_id":"2509.02131","last_updated":"2025-09-02T09:26:08Z","snapshot_observed_at":"2026-08-13T04:10:03.444517Z","submitted_at":"2025-09-02T09:26:08Z","title":"Achieving wavenumber robustness in domain decomposition for heterogeneous Helmholtz equation: an overview of spectral coarse spaces","version":1},"reference_index":35,"source":"pdf_text","source_observed_at":"2026-08-05T11:56:27.497535Z"},"links":{"citing_paper":"/paper/2509.02131"},"observation_digest":"sha256:517c405909dfd180092244fc201150f11a79b169216dd9561ec60d2524818656","observation_id":"7c39a7d5-d264-4aa0-9a49-3a36c5ae6291","resolution":{"observed_at":"2026-08-05T11:56:28.987903Z","resolver_source":"doi","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-12T06:34:41.77262+00:00","source":"crossref"},{"observed_at":"2026-08-12T06:34:36.333875+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-05T11:56:32.059631Z","title":"Domain decomposition with local impedance conditions for the Helmholtz equation with absorption","venue":null,"work_id":"7c17ac44-b1ee-4fd2-a436-d79b742086c6","year":2020},"citing_paper":{"arxiv_id":"2509.02131","last_updated":"2025-09-02T09:26:08Z","snapshot_observed_at":"2026-08-13T04:10:03.444517Z","submitted_at":"2025-09-02T09:26:08Z","title":"Achieving wavenumber robustness in domain decomposition for heterogeneous Helmholtz equation: an overview of spectral coarse spaces","version":1},"reference_index":36,"source":"pdf_text","source_observed_at":"2026-08-05T11:56:27.384615Z"},"links":{"citing_paper":"/paper/2509.02131"},"observation_digest":"sha256:707d69e72655d6ef55c73609345ed0f83e024b4e885d61db66de1728bc1c2d52","observation_id":"418161e4-14c9-4003-987d-4210ca48949a","resolution":{"observed_at":"2026-08-05T11:56:32.062951Z","resolver_source":"raw_fallback","status":"malformed_identifier"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-12T06:34:41.77262+00:00","source":"crossref"},{"observed_at":"2026-08-12T06:34:36.333875+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-05T11:56:32.050746Z","title":"SLEPc:Ascalableandflexibletoolkitforthesolutionofeigenvalue problems","venue":null,"work_id":"2573ad16-ae16-4141-bd4a-77bbd36aaddb","year":2005},"citing_paper":{"arxiv_id":"2509.02131","last_updated":"2025-09-02T09:26:08Z","snapshot_observed_at":"2026-08-13T04:10:03.444517Z","submitted_at":"2025-09-02T09:26:08Z","title":"Achieving wavenumber robustness in domain decomposition for heterogeneous Helmholtz equation: an overview of spectral coarse spaces","version":1},"reference_index":37,"source":"pdf_text","source_observed_at":"2026-08-05T11:56:27.756206Z"},"links":{"citing_paper":"/paper/2509.02131"},"observation_digest":"sha256:ba2e334e38fc499171d23a8adbeecaaf6eb268080f74e5c1f0cf99b90a7c07ab","observation_id":"19014a6a-b89d-4da0-9388-ece4e6d53d30","resolution":{"observed_at":"2026-08-05T11:56:32.053642Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-12T06:34:41.77262+00:00","source":"crossref"},{"observed_at":"2026-08-12T06:34:36.333875+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":"10.1142/s0218396x00000096","metadata_source":"doi_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-05T11:56:28.816737Z","title":"Analytical and numerical studies of a finite element PML for the Helmholtz equation","venue":null,"work_id":"6c9c5c99-cae6-4e6c-bf1b-576e1f0d3491","year":2000},"citing_paper":{"arxiv_id":"2509.02131","last_updated":"2025-09-02T09:26:08Z","snapshot_observed_at":"2026-08-13T04:10:03.444517Z","submitted_at":"2025-09-02T09:26:08Z","title":"Achieving wavenumber robustness in domain decomposition for heterogeneous Helmholtz equation: an overview of spectral coarse spaces","version":1},"reference_index":38,"source":"pdf_text","source_observed_at":"2026-08-05T11:56:27.636868Z"},"links":{"citing_paper":"/paper/2509.02131"},"observation_digest":"sha256:ee5caaf3121a39b9e6b20b5d75356224f47e97260565586605d3d0411560180a","observation_id":"e2e7abb4-2bf4-4a86-9846-b80222adc761","resolution":{"observed_at":"2026-08-05T11:56:28.853843Z","resolver_source":"doi","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-12T06:34:41.77262+00:00","source":"crossref"},{"observed_at":"2026-08-12T06:34:36.333875+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-05T11:56:32.032137Z","title":null,"venue":null,"work_id":"263a8b4c-d019-494e-ad0a-d4d0c18f2d1e","year":2015},"citing_paper":{"arxiv_id":"2509.02131","last_updated":"2025-09-02T09:26:08Z","snapshot_observed_at":"2026-08-13T04:10:03.444517Z","submitted_at":"2025-09-02T09:26:08Z","title":"Achieving wavenumber robustness in domain decomposition for heterogeneous Helmholtz equation: an overview of spectral coarse spaces","version":1},"reference_index":39,"source":"pdf_text","source_observed_at":"2026-08-05T11:56:27.927315Z"},"links":{"citing_paper":"/paper/2509.02131"},"observation_digest":"sha256:6802b285c0d5b06fe0aabdd211219b003a921ea47bd147b1d25016d1aab5e632","observation_id":"48a03c5c-e4bc-423a-90ee-f24f59511726","resolution":{"observed_at":"2026-08-05T11:56:32.034956Z","resolver_source":"raw_fallback","status":"unresolved"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-12T06:34:41.77262+00:00","source":"crossref"},{"observed_at":"2026-08-12T06:34:36.333875+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-05T11:56:32.041344Z","title":"Hu and Z","venue":null,"work_id":"d0286323-6be3-4bac-987a-2cee461cf87e","year":null},"citing_paper":{"arxiv_id":"2509.02131","last_updated":"2025-09-02T09:26:08Z","snapshot_observed_at":"2026-08-13T04:10:03.444517Z","submitted_at":"2025-09-02T09:26:08Z","title":"Achieving wavenumber robustness in domain decomposition for heterogeneous Helmholtz equation: an overview of spectral coarse spaces","version":1},"reference_index":40,"source":"pdf_text","source_observed_at":"2026-08-05T11:56:27.848025Z"},"links":{"citing_paper":"/paper/2509.02131"},"observation_digest":"sha256:bbb7743a96b5039d1f8e4ede70a6f601dc2d86b5d0e78eb98e11d6af216f7c2e","observation_id":"20a7dcc1-5b7e-453d-8f3f-087d000fbaf9","resolution":{"observed_at":"2026-08-05T11:56:32.044812Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-12T06:34:41.77262+00:00","source":"crossref"},{"observed_at":"2026-08-12T06:34:36.333875+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":"2003.81228","doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-05T11:56:30.887428Z","title":"Scattering analysis of a large body with deep cavities","venue":null,"work_id":"9464c7fc-b208-4e8b-8451-2bdbbbdcf3dd","year":2003},"citing_paper":{"arxiv_id":"2509.02131","last_updated":"2025-09-02T09:26:08Z","snapshot_observed_at":"2026-08-13T04:10:03.444517Z","submitted_at":"2025-09-02T09:26:08Z","title":"Achieving wavenumber robustness in domain decomposition for heterogeneous Helmholtz equation: an overview of spectral coarse spaces","version":1},"reference_index":41,"source":"pdf_text","source_observed_at":"2026-08-05T11:56:28.103741Z"},"links":{"citing_paper":"/paper/2509.02131"},"observation_digest":"sha256:2a52b899f51598d0338262c9b6cf283d495738dcacd297090e30ed7c2587a375","observation_id":"2d039481-05f7-4bac-8f15-12b6768e6079","resolution":{"observed_at":"2026-08-05T11:56:31.045761Z","resolver_source":"raw_fallback","status":"metadata_mismatch"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-12T06:34:41.77262+00:00","source":"crossref"},{"observed_at":"2026-08-12T06:34:36.333875+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-05T11:56:32.022713Z","title":"Karypis and V","venue":null,"work_id":"05d934c8-0034-425e-b15a-64b60c89317a","year":1998},"citing_paper":{"arxiv_id":"2509.02131","last_updated":"2025-09-02T09:26:08Z","snapshot_observed_at":"2026-08-13T04:10:03.444517Z","submitted_at":"2025-09-02T09:26:08Z","title":"Achieving wavenumber robustness in domain decomposition for heterogeneous Helmholtz equation: an overview of spectral coarse spaces","version":1},"reference_index":42,"source":"pdf_text","source_observed_at":"2026-08-05T11:56:27.997846Z"},"links":{"citing_paper":"/paper/2509.02131"},"observation_digest":"sha256:84b1f8890c2d42739a8ccbfb05a4138eee4b4746370b0f2eb3f99eae44779762","observation_id":"bcb8a229-9d8d-494c-914d-6ff74b873186","resolution":{"observed_at":"2026-08-05T11:56:32.025575Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-12T06:34:41.77262+00:00","source":"crossref"},{"observed_at":"2026-08-12T06:34:36.333875+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":{"arxiv_id":"2404.02758","last_updated":"2025-07-04T16:41:36Z","snapshot_observed_at":"2026-08-13T00:38:25.867229Z","submitted_at":"2024-04-03T14:03:39Z","title":"Coarse spaces for non-symmetric two-level preconditioners based on local extended generalized eigenproblems","version":3},"cited_work":{"arxiv_id":"2404.02758","doi":null,"metadata_source":"pith","pith_arxiv_id":"2404.02758","snapshot_observed_at":"2026-08-05T11:56:30.599029Z","title":"Coarse spaces for non-symmetric two-level preconditioners based on local extended generalized eigenproblems","venue":"math.NA","work_id":"171252ba-427c-4781-acab-e8211d7a7687","year":2024},"citing_paper":{"arxiv_id":"2509.02131","last_updated":"2025-09-02T09:26:08Z","snapshot_observed_at":"2026-08-13T04:10:03.444517Z","submitted_at":"2025-09-02T09:26:08Z","title":"Achieving wavenumber robustness in domain decomposition for heterogeneous Helmholtz equation: an overview of spectral coarse spaces","version":1},"reference_index":43,"source":"pdf_text","source_observed_at":"2026-08-05T11:56:28.217664Z"},"links":{"cited_paper":"/paper/2404.02758","citing_paper":"/paper/2509.02131"},"observation_digest":"sha256:57250bc9cfbea2fd76b3c20b7029592613dde96446daf43912b8c0f426126caa","observation_id":"4865a5c0-5796-49a6-892f-87b8f2e9c9c1","resolution":{"observed_at":"2026-08-05T11:56:30.685548Z","resolver_source":"local_arxiv","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-12T06:34:41.77262+00:00","source":"crossref"},{"observed_at":"2026-08-12T06:34:36.333875+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":{"arxiv_id":"2409.06533","last_updated":"2025-03-03T01:31:04Z","snapshot_observed_at":"2026-08-12T22:47:42.775436Z","submitted_at":"2024-09-10T14:09:12Z","title":"Two-level Restricted Additive Schwarz preconditioner based on Multiscale Spectral Generalized FEM for Heterogeneous Helmholtz Problems","version":2},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2409.06533","snapshot_observed_at":"2026-08-05T11:56:28.152271Z","title":null,"venue":null,"work_id":null,"year":2024},"citing_paper":{"arxiv_id":"2509.02131","last_updated":"2025-09-02T09:26:08Z","snapshot_observed_at":"2026-08-13T04:10:03.444517Z","submitted_at":"2025-09-02T09:26:08Z","title":"Achieving wavenumber robustness in domain decomposition for heterogeneous Helmholtz equation: an overview of spectral coarse spaces","version":1},"reference_index":44,"source":"pdf_text","source_observed_at":"2026-08-05T11:56:28.152271Z"},"links":{"cited_paper":"/paper/2409.06533","citing_paper":"/paper/2509.02131"},"observation_digest":"sha256:da0cdc85790b4a89017d5e232da22a5291565924ea42e0d4803b3694008ed7bb","observation_id":"9dec54ae-233d-4fe7-aab8-2d94d820545c","resolution":{"observed_at":"2026-08-05T11:56:28.152271Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-05T11:56:32.012906Z","title":"Is 3D frequency-domain FWI of full- azimuth/long-offset OBN data feasible? The Gorgon data FWI case study","venue":null,"work_id":"73f6b2bd-ae92-4025-af2f-2713b6141789","year":2023},"citing_paper":{"arxiv_id":"2509.02131","last_updated":"2025-09-02T09:26:08Z","snapshot_observed_at":"2026-08-13T04:10:03.444517Z","submitted_at":"2025-09-02T09:26:08Z","title":"Achieving wavenumber robustness in domain decomposition for heterogeneous Helmholtz equation: an overview of spectral coarse spaces","version":1},"reference_index":45,"source":"pdf_text","source_observed_at":"2026-08-05T11:56:28.359572Z"},"links":{"citing_paper":"/paper/2509.02131"},"observation_digest":"sha256:08197859ea9733509ce1c3284a33216ff1e5d67a6ba4a9bdd3966c41971d7417","observation_id":"71433be8-0db7-419c-bf5d-94ffa51b5b5d","resolution":{"observed_at":"2026-08-05T11:56:32.016302Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-12T06:34:41.77262+00:00","source":"crossref"},{"observed_at":"2026-08-12T06:34:36.333875+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-05T11:56:28.293710Z","title":"A coarse space construction based on local Dirichlet-to- Neumann maps","venue":null,"work_id":null,"year":2011},"citing_paper":{"arxiv_id":"2509.02131","last_updated":"2025-09-02T09:26:08Z","snapshot_observed_at":"2026-08-13T04:10:03.444517Z","submitted_at":"2025-09-02T09:26:08Z","title":"Achieving wavenumber robustness in domain decomposition for heterogeneous Helmholtz equation: an overview of spectral coarse spaces","version":1},"reference_index":46,"source":"pdf_text","source_observed_at":"2026-08-05T11:56:28.293710Z"},"links":{"citing_paper":"/paper/2509.02131"},"observation_digest":"sha256:94687c23af9e458d265e1629906eac555cdd0287d98c00da7bc0e13e3fe362b3","observation_id":"58263bed-dc6a-4470-8cbc-3c1973eb4313","resolution":{"observed_at":"2026-08-05T11:56:28.293710Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-05T11:56:28.476934Z","title":"Tournier, P","venue":null,"work_id":null,"year":2022},"citing_paper":{"arxiv_id":"2509.02131","last_updated":"2025-09-02T09:26:08Z","snapshot_observed_at":"2026-08-13T04:10:03.444517Z","submitted_at":"2025-09-02T09:26:08Z","title":"Achieving wavenumber robustness in domain decomposition for heterogeneous Helmholtz equation: an overview of spectral coarse spaces","version":1},"reference_index":47,"source":"pdf_text","source_observed_at":"2026-08-05T11:56:28.476934Z"},"links":{"citing_paper":"/paper/2509.02131"},"observation_digest":"sha256:64844df94daac4811b8d2171d8e7c8ffc1a183f0c1d9be6423b08ca598cd4574","observation_id":"f333c645-e0b4-440f-a674-492a3acbec7d","resolution":{"observed_at":"2026-08-05T11:56:28.476934Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-05T11:56:28.418740Z","title":"Abstract robust coarse spaces for systems of PDEs via generalized eigenproblems in the overlaps","venue":null,"work_id":null,"year":2014},"citing_paper":{"arxiv_id":"2509.02131","last_updated":"2025-09-02T09:26:08Z","snapshot_observed_at":"2026-08-13T04:10:03.444517Z","submitted_at":"2025-09-02T09:26:08Z","title":"Achieving wavenumber robustness in domain decomposition for heterogeneous Helmholtz equation: an overview of spectral coarse spaces","version":1},"reference_index":48,"source":"pdf_text","source_observed_at":"2026-08-05T11:56:28.418740Z"},"links":{"citing_paper":"/paper/2509.02131"},"observation_digest":"sha256:cb1d2669cdcc4fe8055708c0b21f277bdf1fadc902b93484d51f2366cf054f5d","observation_id":"3e1f0312-9ef6-4e99-87a5-93dfca0f6ee6","resolution":{"observed_at":"2026-08-05T11:56:28.418740Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":"10.1016/j.jcp.2013.02.042","metadata_source":"openalex","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-10T05:30:23.456663Z","title":"A general approach for high order absorbing boundary conditions for the Helmholtz equation","venue":"Journal of Computational Physics","work_id":"6b5f05d8-6b45-4152-b9b6-fd32d6e4f31b","year":2013},"citing_paper":{"arxiv_id":"2509.02131","last_updated":"2025-09-02T09:26:08Z","snapshot_observed_at":"2026-08-13T04:10:03.444517Z","submitted_at":"2025-09-02T09:26:08Z","title":"Achieving wavenumber robustness in domain decomposition for heterogeneous Helmholtz equation: an overview of spectral coarse spaces","version":1},"reference_index":49,"source":"pdf_text","source_observed_at":"2026-08-05T11:56:28.623464Z"},"links":{"citing_paper":"/paper/2509.02131"},"observation_digest":"sha256:b3e920ff872dad83c58c5713a6d680fe7a53f1cb163de47b829977b76090c0cc","observation_id":"1694bc09-553b-45c9-91af-f44f2688dfe6","resolution":{"observed_at":"2026-08-05T11:56:28.735522Z","resolver_source":"doi","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-12T06:34:41.77262+00:00","source":"crossref"},{"observed_at":"2026-08-12T06:34:36.333875+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-05T11:56:32.003532Z","title":"Tournier, P","venue":null,"work_id":"63fe6fed-c0ad-4a68-8a8e-93a3490116dc","year":2019},"citing_paper":{"arxiv_id":"2509.02131","last_updated":"2025-09-02T09:26:08Z","snapshot_observed_at":"2026-08-13T04:10:03.444517Z","submitted_at":"2025-09-02T09:26:08Z","title":"Achieving wavenumber robustness in domain decomposition for heterogeneous Helmholtz equation: an overview of spectral coarse spaces","version":1},"reference_index":50,"source":"pdf_text","source_observed_at":"2026-08-05T11:56:28.550065Z"},"links":{"citing_paper":"/paper/2509.02131"},"observation_digest":"sha256:e50997112b9e02a233b902dc76fc4523de3b75b6f83731d98c4408eef38b0c3a","observation_id":"307710ae-c672-4158-b26f-70401bee5080","resolution":{"observed_at":"2026-08-05T11:56:32.006751Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-12T06:34:41.77262+00:00","source":"crossref"},{"observed_at":"2026-08-12T06:34:36.333875+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":"10.4310/cms.240918021620","metadata_source":"openalex","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-10T05:30:23.456663Z","title":null,"venue":"Communications in Mathematical Sciences","work_id":"0b4badbf-57c1-4e68-8f26-b91686117631","year":null},"citing_paper":{"arxiv_id":"2509.02131","last_updated":"2025-09-02T09:26:08Z","snapshot_observed_at":"2026-08-13T04:10:03.444517Z","submitted_at":"2025-09-02T09:26:08Z","title":"Achieving wavenumber robustness in domain decomposition for heterogeneous Helmholtz equation: an overview of spectral coarse spaces","version":1},"reference_index":1816,"source":"pdf_text","source_observed_at":"2026-08-05T11:56:26.233056Z"},"links":{"citing_paper":"/paper/2509.02131"},"observation_digest":"sha256:f673856e473efbf522688450419a5aff48e82f68fec4c99298e2b3281592670c","observation_id":"e3c62e0e-c772-4346-bc57-029a88c54411","resolution":{"observed_at":"2026-08-05T11:56:29.216793Z","resolver_source":"doi","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-12T06:34:41.77262+00:00","source":"crossref"},{"observed_at":"2026-08-12T06:34:36.333875+00:00","source":"retraction_watch"}],"state":"measured"}}],"paper":{"arxiv_id":"2509.02131","last_updated":"2025-09-02T09:26:08Z","latest_version":1,"primary_category":"math.NA","snapshot_observed_at":"2026-08-13T04:10:03.444517Z","submitted_at":"2025-09-02T09:26:08Z","title":"Achieving wavenumber robustness in domain decomposition for heterogeneous Helmholtz equation: an overview of spectral coarse spaces"},"reference_resolution":{"displayed":51,"state_counts":{"malformed_identifier":7,"metadata_mismatch":1,"parse_uncertain":0,"unresolved":11,"verified_exact":19,"verified_fuzzy":13},"total_outbound_references":51},"refusal":"A citation records a reference. It does not transfer a finding from one paper to another.","schema":"pith.paper-citation-record.v1","standing_sources":[{"observed_at":"2026-08-12T06:34:41.77262+00:00","source":"crossref"},{"observed_at":"2026-08-12T06:34:36.333875+00:00","source":"retraction_watch"}],"thesis":"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."}