{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2020:YPA4KB3PWFLYHURKWCIQG4EK5Z","short_pith_number":"pith:YPA4KB3P","schema_version":"1.0","canonical_sha256":"c3c1c5076fb15783d22ab09103708aee6e67ef0de0c6f67c9fc34308570dffe8","source":{"kind":"arxiv","id":"2005.13390","version":2},"attestation_state":"computed","paper":{"title":"Analysis of a Helmholtz preconditioning problem motivated by uncertainty quantification","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.NA"],"primary_cat":"math.NA","authors_text":"Euan A. Spence, Ivan G. Graham, Owen R. Pembery","submitted_at":"2020-05-27T14:42:25Z","abstract_excerpt":"This paper analyses the following question: let $\\mathbf{A}_j$, $j=1,2,$ be the Galerkin matrices corresponding to finite-element discretisations of the exterior Dirichlet problem for the heterogeneous Helmholtz equations $\\nabla\\cdot (A_j \\nabla u_j) + k^2 n_j u_j= -f$. How small must $\\|A_1 -A_2\\|_{L^q}$ and $\\|{n_1} - {n_2}\\|_{L^q}$ be (in terms of $k$-dependence) for GMRES applied to either $(\\mathbf{A}_1)^{-1}\\mathbf{A}_2$ or $\\mathbf{A}_2(\\mathbf{A}_1)^{-1}$ to converge in a $k$-independent number of iterations for arbitrarily large $k$? (In other words, for $\\mathbf{A}_1$ to be a good l"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2005.13390","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2020-05-27T14:42:25Z","cross_cats_sorted":["cs.NA"],"title_canon_sha256":"db40509c3379b378ea105ae99c3ab4a6b1e9c63f9d9ef79ccba24ac90d0e5491","abstract_canon_sha256":"8e671b39db05e4bff4faa4531438593273150e9034ea854fd485a9578600ee07"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T02:57:07.587773Z","signature_b64":"drwr/9Wl81zuCSZxf/jWoC6s95vGeBA6xSTrfhepWWaYdKknl+sK5sBTCxQ8gihHuh9JuxL8eYEEZE9JfIlAAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c3c1c5076fb15783d22ab09103708aee6e67ef0de0c6f67c9fc34308570dffe8","last_reissued_at":"2026-07-05T02:57:07.587285Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T02:57:07.587285Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Analysis of a Helmholtz preconditioning problem motivated by uncertainty quantification","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.NA"],"primary_cat":"math.NA","authors_text":"Euan A. Spence, Ivan G. Graham, Owen R. Pembery","submitted_at":"2020-05-27T14:42:25Z","abstract_excerpt":"This paper analyses the following question: let $\\mathbf{A}_j$, $j=1,2,$ be the Galerkin matrices corresponding to finite-element discretisations of the exterior Dirichlet problem for the heterogeneous Helmholtz equations $\\nabla\\cdot (A_j \\nabla u_j) + k^2 n_j u_j= -f$. How small must $\\|A_1 -A_2\\|_{L^q}$ and $\\|{n_1} - {n_2}\\|_{L^q}$ be (in terms of $k$-dependence) for GMRES applied to either $(\\mathbf{A}_1)^{-1}\\mathbf{A}_2$ or $\\mathbf{A}_2(\\mathbf{A}_1)^{-1}$ to converge in a $k$-independent number of iterations for arbitrarily large $k$? (In other words, for $\\mathbf{A}_1$ to be a good l"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2005.13390","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2005.13390/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"2005.13390","created_at":"2026-07-05T02:57:07.587347+00:00"},{"alias_kind":"arxiv_version","alias_value":"2005.13390v2","created_at":"2026-07-05T02:57:07.587347+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2005.13390","created_at":"2026-07-05T02:57:07.587347+00:00"},{"alias_kind":"pith_short_12","alias_value":"YPA4KB3PWFLY","created_at":"2026-07-05T02:57:07.587347+00:00"},{"alias_kind":"pith_short_16","alias_value":"YPA4KB3PWFLYHURK","created_at":"2026-07-05T02:57:07.587347+00:00"},{"alias_kind":"pith_short_8","alias_value":"YPA4KB3P","created_at":"2026-07-05T02:57:07.587347+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/YPA4KB3PWFLYHURKWCIQG4EK5Z","json":"https://pith.science/pith/YPA4KB3PWFLYHURKWCIQG4EK5Z.json","graph_json":"https://pith.science/api/pith-number/YPA4KB3PWFLYHURKWCIQG4EK5Z/graph.json","events_json":"https://pith.science/api/pith-number/YPA4KB3PWFLYHURKWCIQG4EK5Z/events.json","paper":"https://pith.science/paper/YPA4KB3P"},"agent_actions":{"view_html":"https://pith.science/pith/YPA4KB3PWFLYHURKWCIQG4EK5Z","download_json":"https://pith.science/pith/YPA4KB3PWFLYHURKWCIQG4EK5Z.json","view_paper":"https://pith.science/paper/YPA4KB3P","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2005.13390&json=true","fetch_graph":"https://pith.science/api/pith-number/YPA4KB3PWFLYHURKWCIQG4EK5Z/graph.json","fetch_events":"https://pith.science/api/pith-number/YPA4KB3PWFLYHURKWCIQG4EK5Z/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/YPA4KB3PWFLYHURKWCIQG4EK5Z/action/timestamp_anchor","attest_storage":"https://pith.science/pith/YPA4KB3PWFLYHURKWCIQG4EK5Z/action/storage_attestation","attest_author":"https://pith.science/pith/YPA4KB3PWFLYHURKWCIQG4EK5Z/action/author_attestation","sign_citation":"https://pith.science/pith/YPA4KB3PWFLYHURKWCIQG4EK5Z/action/citation_signature","submit_replication":"https://pith.science/pith/YPA4KB3PWFLYHURKWCIQG4EK5Z/action/replication_record"}},"created_at":"2026-07-05T02:57:07.587347+00:00","updated_at":"2026-07-05T02:57:07.587347+00:00"}