{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2020:ISHPKRWZMVXVCQHPJ2LROBKADV","short_pith_number":"pith:ISHPKRWZ","canonical_record":{"source":{"id":"2006.08750","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2020-06-15T20:43:30Z","cross_cats_sorted":["cs.NA"],"title_canon_sha256":"2a0d431b582fc8137c22ee748615b1c94852338c1fe4c826f9410b610f2d9beb","abstract_canon_sha256":"38d0f740839642be3063c51306b6da173a5ecc5227650d5856e529def54d6b79"},"schema_version":"1.0"},"canonical_sha256":"448ef546d9656f5140ef4e971705401d4dbd9f34570934a9d97bec4cd5bc0032","source":{"kind":"arxiv","id":"2006.08750","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2006.08750","created_at":"2026-07-05T01:11:01Z"},{"alias_kind":"arxiv_version","alias_value":"2006.08750v2","created_at":"2026-07-05T01:11:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2006.08750","created_at":"2026-07-05T01:11:01Z"},{"alias_kind":"pith_short_12","alias_value":"ISHPKRWZMVXV","created_at":"2026-07-05T01:11:01Z"},{"alias_kind":"pith_short_16","alias_value":"ISHPKRWZMVXVCQHP","created_at":"2026-07-05T01:11:01Z"},{"alias_kind":"pith_short_8","alias_value":"ISHPKRWZ","created_at":"2026-07-05T01:11:01Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2020:ISHPKRWZMVXVCQHPJ2LROBKADV","target":"record","payload":{"canonical_record":{"source":{"id":"2006.08750","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2020-06-15T20:43:30Z","cross_cats_sorted":["cs.NA"],"title_canon_sha256":"2a0d431b582fc8137c22ee748615b1c94852338c1fe4c826f9410b610f2d9beb","abstract_canon_sha256":"38d0f740839642be3063c51306b6da173a5ecc5227650d5856e529def54d6b79"},"schema_version":"1.0"},"canonical_sha256":"448ef546d9656f5140ef4e971705401d4dbd9f34570934a9d97bec4cd5bc0032","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T01:11:01.071913Z","signature_b64":"CcztPeYoeFWJiMsHHMJlDV9ChWUaYMHE1wJn4C9731csXnAMGrw5UbR9M3vWrcTMbCqPQ1O1v73rRCBkRYq+Aw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"448ef546d9656f5140ef4e971705401d4dbd9f34570934a9d97bec4cd5bc0032","last_reissued_at":"2026-07-05T01:11:01.071568Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T01:11:01.071568Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2006.08750","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T01:11:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"d6QbPx8OQNIbG5Faoy5LARBnEreqECxQaS6MQ7MjHiW0TmOH21AMTUvxT2eGNWJ1UlZ29or7U1kSxhdnxtuIDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-22T00:40:20.781693Z"},"content_sha256":"1cb57514cb71e89e9523f40bc207cd23b04b50f342b6a9f86a45eb4d8e658cbf","schema_version":"1.0","event_id":"sha256:1cb57514cb71e89e9523f40bc207cd23b04b50f342b6a9f86a45eb4d8e658cbf"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2020:ISHPKRWZMVXVCQHPJ2LROBKADV","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A two-level shifted Laplace preconditioner for Helmholtz problems: Field-of-values analysis and wavenumber-independent convergence","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.NA"],"primary_cat":"math.NA","authors_text":"Luis Garc\\'ia Ramos, Reinhard Nabben","submitted_at":"2020-06-15T20:43:30Z","abstract_excerpt":"One of the main tools for solving linear systems arising from the discretization of the Helmholtz equation is the shifted Laplace preconditioner, which results from the discretization of a perturbed Helmholtz problem $-\\Delta u - (k^2 + i \\varepsilon )u = f$ where $0 \\neq \\varepsilon \\in \\mathbb{R}$ is an absorption parameter. In this work we revisit the idea of combining the shifted Laplace preconditioner with two-level deflation and apply it to Helmholtz problems discretized with linear finite elements. We use the convergence theory of GMRES based on the field of values to prove that GMRES a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2006.08750","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/2006.08750/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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T01:11:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"WLdYFUGJNqwm7CvMtPCY25Xgh06quLVvMl/5EK7Gmgo6/3q6Wei+0INfDL+IByYwrYEXASdgFsqGkW29/bCrDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-22T00:40:20.782032Z"},"content_sha256":"2c542b3b7585fd52305e43f4faca833243ae7ee7b66e9fe5e42b7afadf2b9375","schema_version":"1.0","event_id":"sha256:2c542b3b7585fd52305e43f4faca833243ae7ee7b66e9fe5e42b7afadf2b9375"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ISHPKRWZMVXVCQHPJ2LROBKADV/bundle.json","state_url":"https://pith.science/pith/ISHPKRWZMVXVCQHPJ2LROBKADV/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ISHPKRWZMVXVCQHPJ2LROBKADV/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-08-22T00:40:20Z","links":{"resolver":"https://pith.science/pith/ISHPKRWZMVXVCQHPJ2LROBKADV","bundle":"https://pith.science/pith/ISHPKRWZMVXVCQHPJ2LROBKADV/bundle.json","state":"https://pith.science/pith/ISHPKRWZMVXVCQHPJ2LROBKADV/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ISHPKRWZMVXVCQHPJ2LROBKADV/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2020:ISHPKRWZMVXVCQHPJ2LROBKADV","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"38d0f740839642be3063c51306b6da173a5ecc5227650d5856e529def54d6b79","cross_cats_sorted":["cs.NA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2020-06-15T20:43:30Z","title_canon_sha256":"2a0d431b582fc8137c22ee748615b1c94852338c1fe4c826f9410b610f2d9beb"},"schema_version":"1.0","source":{"id":"2006.08750","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2006.08750","created_at":"2026-07-05T01:11:01Z"},{"alias_kind":"arxiv_version","alias_value":"2006.08750v2","created_at":"2026-07-05T01:11:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2006.08750","created_at":"2026-07-05T01:11:01Z"},{"alias_kind":"pith_short_12","alias_value":"ISHPKRWZMVXV","created_at":"2026-07-05T01:11:01Z"},{"alias_kind":"pith_short_16","alias_value":"ISHPKRWZMVXVCQHP","created_at":"2026-07-05T01:11:01Z"},{"alias_kind":"pith_short_8","alias_value":"ISHPKRWZ","created_at":"2026-07-05T01:11:01Z"}],"graph_snapshots":[{"event_id":"sha256:2c542b3b7585fd52305e43f4faca833243ae7ee7b66e9fe5e42b7afadf2b9375","target":"graph","created_at":"2026-07-05T01:11:01Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2006.08750/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"One of the main tools for solving linear systems arising from the discretization of the Helmholtz equation is the shifted Laplace preconditioner, which results from the discretization of a perturbed Helmholtz problem $-\\Delta u - (k^2 + i \\varepsilon )u = f$ where $0 \\neq \\varepsilon \\in \\mathbb{R}$ is an absorption parameter. In this work we revisit the idea of combining the shifted Laplace preconditioner with two-level deflation and apply it to Helmholtz problems discretized with linear finite elements. We use the convergence theory of GMRES based on the field of values to prove that GMRES a","authors_text":"Luis Garc\\'ia Ramos, Reinhard Nabben","cross_cats":["cs.NA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2020-06-15T20:43:30Z","title":"A two-level shifted Laplace preconditioner for Helmholtz problems: Field-of-values analysis and wavenumber-independent convergence"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2006.08750","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:1cb57514cb71e89e9523f40bc207cd23b04b50f342b6a9f86a45eb4d8e658cbf","target":"record","created_at":"2026-07-05T01:11:01Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"38d0f740839642be3063c51306b6da173a5ecc5227650d5856e529def54d6b79","cross_cats_sorted":["cs.NA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2020-06-15T20:43:30Z","title_canon_sha256":"2a0d431b582fc8137c22ee748615b1c94852338c1fe4c826f9410b610f2d9beb"},"schema_version":"1.0","source":{"id":"2006.08750","kind":"arxiv","version":2}},"canonical_sha256":"448ef546d9656f5140ef4e971705401d4dbd9f34570934a9d97bec4cd5bc0032","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"448ef546d9656f5140ef4e971705401d4dbd9f34570934a9d97bec4cd5bc0032","first_computed_at":"2026-07-05T01:11:01.071568Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T01:11:01.071568Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"CcztPeYoeFWJiMsHHMJlDV9ChWUaYMHE1wJn4C9731csXnAMGrw5UbR9M3vWrcTMbCqPQ1O1v73rRCBkRYq+Aw==","signature_status":"signed_v1","signed_at":"2026-07-05T01:11:01.071913Z","signed_message":"canonical_sha256_bytes"},"source_id":"2006.08750","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1cb57514cb71e89e9523f40bc207cd23b04b50f342b6a9f86a45eb4d8e658cbf","sha256:2c542b3b7585fd52305e43f4faca833243ae7ee7b66e9fe5e42b7afadf2b9375"],"state_sha256":"81e5b951b6cc75eb8dca9889699f8d7d740fb2bbae43f262d25770e21dd20b27"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"BaoZBwW3IzPwgSrTRlR3fx2HLRU5LdrA3SlqOjRCiEd2uchGrs1F5eE/EPa4TMDs6namRc7F9lsh+7i/naijBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-22T00:40:20.785791Z","bundle_sha256":"57b985fdf23293ed566a91f7e85c8fc99883492156d323f5a29382d226595ce9"}}