{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:IXW663BKT6HGNWQKRLV2V33SRE","short_pith_number":"pith:IXW663BK","canonical_record":{"source":{"id":"1001.1362","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2010-01-08T21:13:53Z","cross_cats_sorted":["cs.NA"],"title_canon_sha256":"77b5dfd18ddd05e86509618d9379fab628bba370146e954d5a98eb49bfc5efb0","abstract_canon_sha256":"8d9602e8f7db98161139429d02fc347cac02ef9f6874ebef014164acdf15ee99"},"schema_version":"1.0"},"canonical_sha256":"45edef6c2a9f8e66da0a8aebaaef72892f38822dbfb6ac7c10561ba40a5b7c37","source":{"kind":"arxiv","id":"1001.1362","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1001.1362","created_at":"2026-06-03T23:06:32Z"},{"alias_kind":"arxiv_version","alias_value":"1001.1362v1","created_at":"2026-06-03T23:06:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1001.1362","created_at":"2026-06-03T23:06:32Z"},{"alias_kind":"pith_short_12","alias_value":"IXW663BKT6HG","created_at":"2026-06-03T23:06:32Z"},{"alias_kind":"pith_short_16","alias_value":"IXW663BKT6HGNWQK","created_at":"2026-06-03T23:06:32Z"},{"alias_kind":"pith_short_8","alias_value":"IXW663BK","created_at":"2026-06-03T23:06:32Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:IXW663BKT6HGNWQKRLV2V33SRE","target":"record","payload":{"canonical_record":{"source":{"id":"1001.1362","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2010-01-08T21:13:53Z","cross_cats_sorted":["cs.NA"],"title_canon_sha256":"77b5dfd18ddd05e86509618d9379fab628bba370146e954d5a98eb49bfc5efb0","abstract_canon_sha256":"8d9602e8f7db98161139429d02fc347cac02ef9f6874ebef014164acdf15ee99"},"schema_version":"1.0"},"canonical_sha256":"45edef6c2a9f8e66da0a8aebaaef72892f38822dbfb6ac7c10561ba40a5b7c37","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-03T23:06:32.341161Z","signature_b64":"2tkYwqSEOK05y2So5mAVh6p1I4hzvT3poEz0QlfwVPgivx/z3opz0EAVwEKyd7IbcbwSchAJQ3Qo2ZFtZhfJDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"45edef6c2a9f8e66da0a8aebaaef72892f38822dbfb6ac7c10561ba40a5b7c37","last_reissued_at":"2026-06-03T23:06:32.340759Z","signature_status":"signed_v1","first_computed_at":"2026-06-03T23:06:32.340759Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1001.1362","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-06-03T23:06:32Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"cwkFuYWn1gcWqKVZab3UMNWT34geFQLZdA64rfQ6NbIs7JmJih2kEAaEvpqYHbJaMFKichZ9WG+3fIlFjKJVBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T07:28:38.592827Z"},"content_sha256":"93a0a4ef1c496028d27fa1503ce3c2c199e3407c90c28cfc687bad058bfddda1","schema_version":"1.0","event_id":"sha256:93a0a4ef1c496028d27fa1503ce3c2c199e3407c90c28cfc687bad058bfddda1"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:IXW663BKT6HGNWQKRLV2V33SRE","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Schwarz Methods: To Symmetrize or Not to Symmetrize","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.NA"],"primary_cat":"math.NA","authors_text":"Michael Holst, Stefan Vandewalle","submitted_at":"2010-01-08T21:13:53Z","abstract_excerpt":"A preconditioning theory is presented which establishes sufficient conditions for multiplicative and additive Schwarz algorithms to yield self-adjoint positive definite preconditioners. It allows for the analysis and use of non-variational and non-convergent linear methods as preconditioners for conjugate gradient methods, and it is applied to domain decomposition and multigrid. It is illustrated why symmetrizing may be a bad idea for linear methods. It is conjectured that enforcing minimal symmetry achieves the best results when combined with conjugate gradient acceleration. Also, it is shown"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1001.1362","kind":"arxiv","version":1},"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/1001.1362/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":"2026-06-03T23:06:32Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"k52q8nd+iACrJebWN6LvDufOQxCGaS2sJkVByBSfOfkxBNbfT8rKValVfGRiKBj2bVe4GJEGLwXT/vJj88TjBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T07:28:38.593570Z"},"content_sha256":"0cc06c77f704df5e1a1d2f69bfa5a2f684b72760d45128290e336950ea76280b","schema_version":"1.0","event_id":"sha256:0cc06c77f704df5e1a1d2f69bfa5a2f684b72760d45128290e336950ea76280b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/IXW663BKT6HGNWQKRLV2V33SRE/bundle.json","state_url":"https://pith.science/pith/IXW663BKT6HGNWQKRLV2V33SRE/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/IXW663BKT6HGNWQKRLV2V33SRE/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":"2026-06-23T07:28:38Z","links":{"resolver":"https://pith.science/pith/IXW663BKT6HGNWQKRLV2V33SRE","bundle":"https://pith.science/pith/IXW663BKT6HGNWQKRLV2V33SRE/bundle.json","state":"https://pith.science/pith/IXW663BKT6HGNWQKRLV2V33SRE/state.json","well_known_bundle":"https://pith.science/.well-known/pith/IXW663BKT6HGNWQKRLV2V33SRE/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:IXW663BKT6HGNWQKRLV2V33SRE","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":"8d9602e8f7db98161139429d02fc347cac02ef9f6874ebef014164acdf15ee99","cross_cats_sorted":["cs.NA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2010-01-08T21:13:53Z","title_canon_sha256":"77b5dfd18ddd05e86509618d9379fab628bba370146e954d5a98eb49bfc5efb0"},"schema_version":"1.0","source":{"id":"1001.1362","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1001.1362","created_at":"2026-06-03T23:06:32Z"},{"alias_kind":"arxiv_version","alias_value":"1001.1362v1","created_at":"2026-06-03T23:06:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1001.1362","created_at":"2026-06-03T23:06:32Z"},{"alias_kind":"pith_short_12","alias_value":"IXW663BKT6HG","created_at":"2026-06-03T23:06:32Z"},{"alias_kind":"pith_short_16","alias_value":"IXW663BKT6HGNWQK","created_at":"2026-06-03T23:06:32Z"},{"alias_kind":"pith_short_8","alias_value":"IXW663BK","created_at":"2026-06-03T23:06:32Z"}],"graph_snapshots":[{"event_id":"sha256:0cc06c77f704df5e1a1d2f69bfa5a2f684b72760d45128290e336950ea76280b","target":"graph","created_at":"2026-06-03T23:06:32Z","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/1001.1362/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"A preconditioning theory is presented which establishes sufficient conditions for multiplicative and additive Schwarz algorithms to yield self-adjoint positive definite preconditioners. It allows for the analysis and use of non-variational and non-convergent linear methods as preconditioners for conjugate gradient methods, and it is applied to domain decomposition and multigrid. It is illustrated why symmetrizing may be a bad idea for linear methods. It is conjectured that enforcing minimal symmetry achieves the best results when combined with conjugate gradient acceleration. Also, it is shown","authors_text":"Michael Holst, Stefan Vandewalle","cross_cats":["cs.NA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2010-01-08T21:13:53Z","title":"Schwarz Methods: To Symmetrize or Not to Symmetrize"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1001.1362","kind":"arxiv","version":1},"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":"sha256:93a0a4ef1c496028d27fa1503ce3c2c199e3407c90c28cfc687bad058bfddda1","target":"record","created_at":"2026-06-03T23:06:32Z","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":"8d9602e8f7db98161139429d02fc347cac02ef9f6874ebef014164acdf15ee99","cross_cats_sorted":["cs.NA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2010-01-08T21:13:53Z","title_canon_sha256":"77b5dfd18ddd05e86509618d9379fab628bba370146e954d5a98eb49bfc5efb0"},"schema_version":"1.0","source":{"id":"1001.1362","kind":"arxiv","version":1}},"canonical_sha256":"45edef6c2a9f8e66da0a8aebaaef72892f38822dbfb6ac7c10561ba40a5b7c37","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"45edef6c2a9f8e66da0a8aebaaef72892f38822dbfb6ac7c10561ba40a5b7c37","first_computed_at":"2026-06-03T23:06:32.340759Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-03T23:06:32.340759Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"2tkYwqSEOK05y2So5mAVh6p1I4hzvT3poEz0QlfwVPgivx/z3opz0EAVwEKyd7IbcbwSchAJQ3Qo2ZFtZhfJDQ==","signature_status":"signed_v1","signed_at":"2026-06-03T23:06:32.341161Z","signed_message":"canonical_sha256_bytes"},"source_id":"1001.1362","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:93a0a4ef1c496028d27fa1503ce3c2c199e3407c90c28cfc687bad058bfddda1","sha256:0cc06c77f704df5e1a1d2f69bfa5a2f684b72760d45128290e336950ea76280b"],"state_sha256":"35a5cd770ef3c50c92f8dd152e940e3efbc45ee36ce6376355557e6c75b2bb89"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"12YgfIICybTOaFjrphzlicNtgaWhPhZCpT47yvFV5+z8pyCBW77kPWmXGHw/c7M24EE9lixu0chreLOwx0z2Ag==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-23T07:28:38.596917Z","bundle_sha256":"bdc4bcfa2bc4e62e1f824580aa8d353477c57c705feb57e6d8c5d9dc01eff2d0"}}