{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:EY3Y7XNYE4UL5B2CVNXGKD4JFI","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":"aef1408ba064640988c575acb02c602750a37889fc8d66cdcde626be78dbb5b9","cross_cats_sorted":["cs.NA"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NA","submitted_at":"2026-07-24T11:22:36Z","title_canon_sha256":"18d35a51a56cf63c0a2e4e6d6e615bfb66398d27718e7dc4d4b723b94e36b28b"},"schema_version":"1.0","source":{"id":"2607.22206","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.22206","created_at":"2026-07-27T01:20:53Z"},{"alias_kind":"arxiv_version","alias_value":"2607.22206v1","created_at":"2026-07-27T01:20:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.22206","created_at":"2026-07-27T01:20:53Z"},{"alias_kind":"pith_short_12","alias_value":"EY3Y7XNYE4UL","created_at":"2026-07-27T01:20:53Z"},{"alias_kind":"pith_short_16","alias_value":"EY3Y7XNYE4UL5B2C","created_at":"2026-07-27T01:20:53Z"},{"alias_kind":"pith_short_8","alias_value":"EY3Y7XNY","created_at":"2026-07-27T01:20:53Z"}],"graph_snapshots":[{"event_id":"sha256:748ef8af4fdb2860185f7a1dcd343c8c448fb5849694723ee9496b76ea1b2c4e","target":"graph","created_at":"2026-07-27T01:20:53Z","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/2607.22206/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We consider $r$-refinement for the finite element discretisation of a Poisson problem. The goal of $r$-refinement is to reposition the nodes of a computational mesh in order to better approximate the finite element error. Since the mesh is being moved, it naturally becomes linked to shape optimisation methods. We propose a standard optimisation algorithm for this $r$-refinement procedure and show that if one seeks to minimise the actual error - which one cannot generally calculate - one has an algorithm which will terminate. The most novel aspect of this work is to apply shape optimisation tec","authors_text":"Philip J. Herbert","cross_cats":["cs.NA"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NA","submitted_at":"2026-07-24T11:22:36Z","title":"Shape optimisation for adaptive $r$-refinement: the one-dimensional case with residual based error estimators"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.22206","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":"sha256:7bbeb854dad92d3643813cdd98e08ad8bbdd67d884b43bedbea78e6bdd64d496","target":"record","created_at":"2026-07-27T01:20:53Z","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":"aef1408ba064640988c575acb02c602750a37889fc8d66cdcde626be78dbb5b9","cross_cats_sorted":["cs.NA"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NA","submitted_at":"2026-07-24T11:22:36Z","title_canon_sha256":"18d35a51a56cf63c0a2e4e6d6e615bfb66398d27718e7dc4d4b723b94e36b28b"},"schema_version":"1.0","source":{"id":"2607.22206","kind":"arxiv","version":1}},"canonical_sha256":"26378fddb82728be8742ab6e650f892a12f1dce338a38936659a212bf935a342","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"26378fddb82728be8742ab6e650f892a12f1dce338a38936659a212bf935a342","first_computed_at":"2026-07-27T01:20:53.810135Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-27T01:20:53.810135Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"/5AO4BJyVkHOoGOcUaGUTxJqsRAgFenFyX0mLSJ422UnxJcC/05LaY7HAnkC2qmxUWcpOpV0lmm7am19X2beCw==","signature_status":"signed_v1","signed_at":"2026-07-27T01:20:53.811061Z","signed_message":"canonical_sha256_bytes"},"source_id":"2607.22206","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:7bbeb854dad92d3643813cdd98e08ad8bbdd67d884b43bedbea78e6bdd64d496","sha256:748ef8af4fdb2860185f7a1dcd343c8c448fb5849694723ee9496b76ea1b2c4e"],"state_sha256":"d636060280747fe6e4e9d984b1c09ff379b9b4b1072bc08e0c28a642fc94fcd3"}