{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:GBS5XSA5PL3HBYMED7XDRF2WX7","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":"e919c51bdd9617fc2a635dbf5334dcd56c179a965266e969e34f20214b77b5cd","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2018-11-29T19:09:30Z","title_canon_sha256":"3bd9b73ad316c9d3b5ac7433d99dceccd7bc0737c0f3a4c6cc5c46f137d8c8d2"},"schema_version":"1.0","source":{"id":"1811.12434","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1811.12434","created_at":"2026-05-17T23:59:31Z"},{"alias_kind":"arxiv_version","alias_value":"1811.12434v1","created_at":"2026-05-17T23:59:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1811.12434","created_at":"2026-05-17T23:59:31Z"},{"alias_kind":"pith_short_12","alias_value":"GBS5XSA5PL3H","created_at":"2026-05-18T12:32:25Z"},{"alias_kind":"pith_short_16","alias_value":"GBS5XSA5PL3HBYME","created_at":"2026-05-18T12:32:25Z"},{"alias_kind":"pith_short_8","alias_value":"GBS5XSA5","created_at":"2026-05-18T12:32:25Z"}],"graph_snapshots":[{"event_id":"sha256:542d9bc3a88f5e090929616f12f7aa2c2c5b4bcc6f3a79d844c714cfbe079dff","target":"graph","created_at":"2026-05-17T23:59:31Z","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"},"paper":{"abstract_excerpt":"We construct multigrid methods for an elliptic distributed optimal control problem that are robust with respect to a regularization parameter. We prove the uniform convergence of the $W$-cycle algorithm and demonstrate the performance of $V$-cycle and $W$-cycle algorithms in two and three dimensions through numerical experiments.","authors_text":"Li-yeng Sung, Sijing Liu, Susanne C. Brenner","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2018-11-29T19:09:30Z","title":"Multigrid methods for saddle point problems: Karush-Kuhn-Tucker systems"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1811.12434","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":"sha256:5fbe63106e293d8de61e89ce684ef209c4155f568ddb80d60889bab2dd90c7c8","target":"record","created_at":"2026-05-17T23:59:31Z","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":"e919c51bdd9617fc2a635dbf5334dcd56c179a965266e969e34f20214b77b5cd","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2018-11-29T19:09:30Z","title_canon_sha256":"3bd9b73ad316c9d3b5ac7433d99dceccd7bc0737c0f3a4c6cc5c46f137d8c8d2"},"schema_version":"1.0","source":{"id":"1811.12434","kind":"arxiv","version":1}},"canonical_sha256":"3065dbc81d7af670e1841fee389756bffce683ddf1393111405f564c705268cc","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3065dbc81d7af670e1841fee389756bffce683ddf1393111405f564c705268cc","first_computed_at":"2026-05-17T23:59:31.685867Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:59:31.685867Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"QBUcRpPcdVICH1xipFGE21lTn1ZdeTPAqIzS5ilvMhNUq5kqOv+DJq2F/C2BzBDX/44sku0mACY8hgTgCSyDCg==","signature_status":"signed_v1","signed_at":"2026-05-17T23:59:31.686486Z","signed_message":"canonical_sha256_bytes"},"source_id":"1811.12434","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5fbe63106e293d8de61e89ce684ef209c4155f568ddb80d60889bab2dd90c7c8","sha256:542d9bc3a88f5e090929616f12f7aa2c2c5b4bcc6f3a79d844c714cfbe079dff"],"state_sha256":"8564aaad01d3159c9bc69fce834cb718ad737f5b25701718f3f80a52166ba479"}