{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:EY3Y7XNYE4UL5B2CVNXGKD4JFI","short_pith_number":"pith:EY3Y7XNY","schema_version":"1.0","canonical_sha256":"26378fddb82728be8742ab6e650f892a12f1dce338a38936659a212bf935a342","source":{"kind":"arxiv","id":"2607.22206","version":1},"attestation_state":"computed","paper":{"title":"Shape optimisation for adaptive $r$-refinement: the one-dimensional case with residual based error estimators","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.NA"],"primary_cat":"math.NA","authors_text":"Philip J. Herbert","submitted_at":"2026-07-24T11:22:36Z","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"},"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":"2607.22206","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NA","submitted_at":"2026-07-24T11:22:36Z","cross_cats_sorted":["cs.NA"],"title_canon_sha256":"18d35a51a56cf63c0a2e4e6d6e615bfb66398d27718e7dc4d4b723b94e36b28b","abstract_canon_sha256":"aef1408ba064640988c575acb02c602750a37889fc8d66cdcde626be78dbb5b9"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-27T01:20:53.811061Z","signature_b64":"/5AO4BJyVkHOoGOcUaGUTxJqsRAgFenFyX0mLSJ422UnxJcC/05LaY7HAnkC2qmxUWcpOpV0lmm7am19X2beCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"26378fddb82728be8742ab6e650f892a12f1dce338a38936659a212bf935a342","last_reissued_at":"2026-07-27T01:20:53.810135Z","signature_status":"signed_v1","first_computed_at":"2026-07-27T01:20:53.810135Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Shape optimisation for adaptive $r$-refinement: the one-dimensional case with residual based error estimators","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.NA"],"primary_cat":"math.NA","authors_text":"Philip J. Herbert","submitted_at":"2026-07-24T11:22:36Z","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"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.22206","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/2607.22206/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":"2607.22206","created_at":"2026-07-27T01:20:53.810583+00:00"},{"alias_kind":"arxiv_version","alias_value":"2607.22206v1","created_at":"2026-07-27T01:20:53.810583+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.22206","created_at":"2026-07-27T01:20:53.810583+00:00"},{"alias_kind":"pith_short_12","alias_value":"EY3Y7XNYE4UL","created_at":"2026-07-27T01:20:53.810583+00:00"},{"alias_kind":"pith_short_16","alias_value":"EY3Y7XNYE4UL5B2C","created_at":"2026-07-27T01:20:53.810583+00:00"},{"alias_kind":"pith_short_8","alias_value":"EY3Y7XNY","created_at":"2026-07-27T01:20:53.810583+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/EY3Y7XNYE4UL5B2CVNXGKD4JFI","json":"https://pith.science/pith/EY3Y7XNYE4UL5B2CVNXGKD4JFI.json","graph_json":"https://pith.science/api/pith-number/EY3Y7XNYE4UL5B2CVNXGKD4JFI/graph.json","events_json":"https://pith.science/api/pith-number/EY3Y7XNYE4UL5B2CVNXGKD4JFI/events.json","paper":"https://pith.science/paper/EY3Y7XNY"},"agent_actions":{"view_html":"https://pith.science/pith/EY3Y7XNYE4UL5B2CVNXGKD4JFI","download_json":"https://pith.science/pith/EY3Y7XNYE4UL5B2CVNXGKD4JFI.json","view_paper":"https://pith.science/paper/EY3Y7XNY","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2607.22206&json=true","fetch_graph":"https://pith.science/api/pith-number/EY3Y7XNYE4UL5B2CVNXGKD4JFI/graph.json","fetch_events":"https://pith.science/api/pith-number/EY3Y7XNYE4UL5B2CVNXGKD4JFI/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/EY3Y7XNYE4UL5B2CVNXGKD4JFI/action/timestamp_anchor","attest_storage":"https://pith.science/pith/EY3Y7XNYE4UL5B2CVNXGKD4JFI/action/storage_attestation","attest_author":"https://pith.science/pith/EY3Y7XNYE4UL5B2CVNXGKD4JFI/action/author_attestation","sign_citation":"https://pith.science/pith/EY3Y7XNYE4UL5B2CVNXGKD4JFI/action/citation_signature","submit_replication":"https://pith.science/pith/EY3Y7XNYE4UL5B2CVNXGKD4JFI/action/replication_record"}},"created_at":"2026-07-27T01:20:53.810583+00:00","updated_at":"2026-07-27T01:20:53.810583+00:00"}