{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:CYGMVUY5J6X75IKK7ZEPUZXSBB","short_pith_number":"pith:CYGMVUY5","canonical_record":{"source":{"id":"1805.01636","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2018-05-04T07:47:41Z","cross_cats_sorted":["math.AP","math.OC"],"title_canon_sha256":"a4390c37a46d60ab59691a8c9b7ded6be262c52f3575ea56b4cc6297d276948f","abstract_canon_sha256":"d64a5d6a789cec12b4b38bd2fd80ecee65a287914da6d7c2f26beae79eef75c6"},"schema_version":"1.0"},"canonical_sha256":"160ccad31d4faffea14afe48fa66f2085c4d1512e1695efc3fb7fada7bee08a0","source":{"kind":"arxiv","id":"1805.01636","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1805.01636","created_at":"2026-05-18T00:16:47Z"},{"alias_kind":"arxiv_version","alias_value":"1805.01636v1","created_at":"2026-05-18T00:16:47Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1805.01636","created_at":"2026-05-18T00:16:47Z"},{"alias_kind":"pith_short_12","alias_value":"CYGMVUY5J6X7","created_at":"2026-05-18T12:32:19Z"},{"alias_kind":"pith_short_16","alias_value":"CYGMVUY5J6X75IKK","created_at":"2026-05-18T12:32:19Z"},{"alias_kind":"pith_short_8","alias_value":"CYGMVUY5","created_at":"2026-05-18T12:32:19Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:CYGMVUY5J6X75IKK7ZEPUZXSBB","target":"record","payload":{"canonical_record":{"source":{"id":"1805.01636","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2018-05-04T07:47:41Z","cross_cats_sorted":["math.AP","math.OC"],"title_canon_sha256":"a4390c37a46d60ab59691a8c9b7ded6be262c52f3575ea56b4cc6297d276948f","abstract_canon_sha256":"d64a5d6a789cec12b4b38bd2fd80ecee65a287914da6d7c2f26beae79eef75c6"},"schema_version":"1.0"},"canonical_sha256":"160ccad31d4faffea14afe48fa66f2085c4d1512e1695efc3fb7fada7bee08a0","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:16:47.226947Z","signature_b64":"2uLXgJVtouEIiRSyRixUvZlN7jGGRfhybirPZSbz+9bKk/S6B74pTXbcHANjNBZzLJL74VxMjg0eZbq9+yX2CQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"160ccad31d4faffea14afe48fa66f2085c4d1512e1695efc3fb7fada7bee08a0","last_reissued_at":"2026-05-18T00:16:47.226212Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:16:47.226212Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1805.01636","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":"2026-05-18T00:16:47Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"v/JqJTDv39WCwiM0VxZvbXx7vYEwHcknYeKSsGWEvk2fqIyvIpElZDMqTk+IchXG8g8eaxMF8sp7fJBMKye5Cg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T14:53:08.387698Z"},"content_sha256":"6972efc004da9bccc32efdb703c813a9207ae0471dd1f2777463f9bde99213e5","schema_version":"1.0","event_id":"sha256:6972efc004da9bccc32efdb703c813a9207ae0471dd1f2777463f9bde99213e5"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:CYGMVUY5J6X75IKK7ZEPUZXSBB","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Algorithm for Hamilton-Jacobi equations in density space via a generalized Hopf formula","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP","math.OC"],"primary_cat":"math.NA","authors_text":"Stanley Osher, Wotao Yin, Wuchen Li, Yat Tin Chow","submitted_at":"2018-05-04T07:47:41Z","abstract_excerpt":"We design fast numerical methods for Hamilton-Jacobi equations in density space (HJD), which arises in optimal transport and mean field games. We overcome the curse-of-infinite-dimensionality nature of HJD by proposing a generalized Hopf formula in density space. The formula transfers optimal control problems in density space, which are constrained minimizations supported on both spatial and time variables, to optimization problems over only spatial variables. This transformation allows us to compute HJD efficiently via multi-level approaches and coordinate descent methods."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1805.01636","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":""},"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":"2026-05-18T00:16:47Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"x+aqxLa+A7cExMr1HEbHG8hHUtpGH3qUgRTtVmQx6nN3R7LuRcRfDQKLJymYefRFcIv6TvY+gH8XwMkSSkPyAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T14:53:08.388047Z"},"content_sha256":"a488e27261a7eb04cb96260ad44c84424ae88363efc2766c829c2c880cbbc92e","schema_version":"1.0","event_id":"sha256:a488e27261a7eb04cb96260ad44c84424ae88363efc2766c829c2c880cbbc92e"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/CYGMVUY5J6X75IKK7ZEPUZXSBB/bundle.json","state_url":"https://pith.science/pith/CYGMVUY5J6X75IKK7ZEPUZXSBB/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/CYGMVUY5J6X75IKK7ZEPUZXSBB/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":"2026-06-25T14:53:08Z","links":{"resolver":"https://pith.science/pith/CYGMVUY5J6X75IKK7ZEPUZXSBB","bundle":"https://pith.science/pith/CYGMVUY5J6X75IKK7ZEPUZXSBB/bundle.json","state":"https://pith.science/pith/CYGMVUY5J6X75IKK7ZEPUZXSBB/state.json","well_known_bundle":"https://pith.science/.well-known/pith/CYGMVUY5J6X75IKK7ZEPUZXSBB/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:CYGMVUY5J6X75IKK7ZEPUZXSBB","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":"d64a5d6a789cec12b4b38bd2fd80ecee65a287914da6d7c2f26beae79eef75c6","cross_cats_sorted":["math.AP","math.OC"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2018-05-04T07:47:41Z","title_canon_sha256":"a4390c37a46d60ab59691a8c9b7ded6be262c52f3575ea56b4cc6297d276948f"},"schema_version":"1.0","source":{"id":"1805.01636","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1805.01636","created_at":"2026-05-18T00:16:47Z"},{"alias_kind":"arxiv_version","alias_value":"1805.01636v1","created_at":"2026-05-18T00:16:47Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1805.01636","created_at":"2026-05-18T00:16:47Z"},{"alias_kind":"pith_short_12","alias_value":"CYGMVUY5J6X7","created_at":"2026-05-18T12:32:19Z"},{"alias_kind":"pith_short_16","alias_value":"CYGMVUY5J6X75IKK","created_at":"2026-05-18T12:32:19Z"},{"alias_kind":"pith_short_8","alias_value":"CYGMVUY5","created_at":"2026-05-18T12:32:19Z"}],"graph_snapshots":[{"event_id":"sha256:a488e27261a7eb04cb96260ad44c84424ae88363efc2766c829c2c880cbbc92e","target":"graph","created_at":"2026-05-18T00:16:47Z","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 design fast numerical methods for Hamilton-Jacobi equations in density space (HJD), which arises in optimal transport and mean field games. We overcome the curse-of-infinite-dimensionality nature of HJD by proposing a generalized Hopf formula in density space. The formula transfers optimal control problems in density space, which are constrained minimizations supported on both spatial and time variables, to optimization problems over only spatial variables. This transformation allows us to compute HJD efficiently via multi-level approaches and coordinate descent methods.","authors_text":"Stanley Osher, Wotao Yin, Wuchen Li, Yat Tin Chow","cross_cats":["math.AP","math.OC"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2018-05-04T07:47:41Z","title":"Algorithm for Hamilton-Jacobi equations in density space via a generalized Hopf formula"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1805.01636","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":"sha256:6972efc004da9bccc32efdb703c813a9207ae0471dd1f2777463f9bde99213e5","target":"record","created_at":"2026-05-18T00:16:47Z","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":"d64a5d6a789cec12b4b38bd2fd80ecee65a287914da6d7c2f26beae79eef75c6","cross_cats_sorted":["math.AP","math.OC"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2018-05-04T07:47:41Z","title_canon_sha256":"a4390c37a46d60ab59691a8c9b7ded6be262c52f3575ea56b4cc6297d276948f"},"schema_version":"1.0","source":{"id":"1805.01636","kind":"arxiv","version":1}},"canonical_sha256":"160ccad31d4faffea14afe48fa66f2085c4d1512e1695efc3fb7fada7bee08a0","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"160ccad31d4faffea14afe48fa66f2085c4d1512e1695efc3fb7fada7bee08a0","first_computed_at":"2026-05-18T00:16:47.226212Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:16:47.226212Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"2uLXgJVtouEIiRSyRixUvZlN7jGGRfhybirPZSbz+9bKk/S6B74pTXbcHANjNBZzLJL74VxMjg0eZbq9+yX2CQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:16:47.226947Z","signed_message":"canonical_sha256_bytes"},"source_id":"1805.01636","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6972efc004da9bccc32efdb703c813a9207ae0471dd1f2777463f9bde99213e5","sha256:a488e27261a7eb04cb96260ad44c84424ae88363efc2766c829c2c880cbbc92e"],"state_sha256":"21530a3d584c98b52c4e92d02267319c5927f6e5bfb50e55a1cd8519e32adc31"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"eqAnWXlXi9Tdmd4UQe02BECvm04ipfT6EZBPDfyee4h/4jQCdQqXpAcRnMTKNcC8inJP4SI2amgGqkMolUg6BA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-25T14:53:08.390429Z","bundle_sha256":"0ac5a5c121cf9afdc975ee001e04564ac635353163f74563d20c75fab6414da2"}}