{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:R3NI6IOHAZPC2DJFXXQAQSZESZ","short_pith_number":"pith:R3NI6IOH","canonical_record":{"source":{"id":"1607.06897","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2016-07-23T06:48:18Z","cross_cats_sorted":[],"title_canon_sha256":"909a4e6c24e6cf7e0ea8dd60f4d618bfa47cd3d38cbc94a85887107f7fe2658b","abstract_canon_sha256":"7b6663d109ab1f4e3db58b487bd4d28c29fe785908e744d2c1953c7609929a42"},"schema_version":"1.0"},"canonical_sha256":"8eda8f21c7065e2d0d25bde0084b24967d2c94f715f15af01098b0b29d3edebd","source":{"kind":"arxiv","id":"1607.06897","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1607.06897","created_at":"2026-05-18T01:10:35Z"},{"alias_kind":"arxiv_version","alias_value":"1607.06897v1","created_at":"2026-05-18T01:10:35Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1607.06897","created_at":"2026-05-18T01:10:35Z"},{"alias_kind":"pith_short_12","alias_value":"R3NI6IOHAZPC","created_at":"2026-05-18T12:30:41Z"},{"alias_kind":"pith_short_16","alias_value":"R3NI6IOHAZPC2DJF","created_at":"2026-05-18T12:30:41Z"},{"alias_kind":"pith_short_8","alias_value":"R3NI6IOH","created_at":"2026-05-18T12:30:41Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:R3NI6IOHAZPC2DJFXXQAQSZESZ","target":"record","payload":{"canonical_record":{"source":{"id":"1607.06897","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2016-07-23T06:48:18Z","cross_cats_sorted":[],"title_canon_sha256":"909a4e6c24e6cf7e0ea8dd60f4d618bfa47cd3d38cbc94a85887107f7fe2658b","abstract_canon_sha256":"7b6663d109ab1f4e3db58b487bd4d28c29fe785908e744d2c1953c7609929a42"},"schema_version":"1.0"},"canonical_sha256":"8eda8f21c7065e2d0d25bde0084b24967d2c94f715f15af01098b0b29d3edebd","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:10:35.520390Z","signature_b64":"A4BexEaWU9aivbTTn1dprBoM1yDMsVHSCat3ys2m+0/0FfQp1Pnb7kJSpa1BAkVCIINpUDHxBIpenLnaWSlEBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8eda8f21c7065e2d0d25bde0084b24967d2c94f715f15af01098b0b29d3edebd","last_reissued_at":"2026-05-18T01:10:35.519799Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:10:35.519799Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1607.06897","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":"2026-05-18T01:10:35Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"xkWr9NXMli4o+CZ3eUyn7EWnCMsWdWtyytT7EkXxrF0zue2sIKB+YZ1ELzfQ5sMTb6Iqv0oFlzYuJbjbXUYZAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T15:31:17.294092Z"},"content_sha256":"1b9c59fa0a19002fd8bc581c513484cf71e1d51825de963c6da8a1ea85a33ba7","schema_version":"1.0","event_id":"sha256:1b9c59fa0a19002fd8bc581c513484cf71e1d51825de963c6da8a1ea85a33ba7"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:R3NI6IOHAZPC2DJFXXQAQSZESZ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Efficient spectral sparse grid approximations for solving multi-dimensional forward backward SDEs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NA","authors_text":"Tao Zhou, Weidong Zhao, Yu Fu","submitted_at":"2016-07-23T06:48:18Z","abstract_excerpt":"This is the second part in a series of papers on multi-step schemes for solving coupled forward backward stochastic differential equations (FBSDEs). We extend the basic idea in our former paper [W. Zhao, Y. Fu and T. Zhou, SIAM J. Sci. Comput., 36 (2014), pp. A1731-A1751] to solve high-dimensional FBSDEs, by using the spectral sparse grid approximations. The main issue for solving high dimensional FBSDEs is to build an efficient spatial discretization, and deal with the related high dimensional conditional expectations and interpolations. In this work, we propose the sparse grid spatial discre"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.06897","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":"2026-05-18T01:10:35Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"CiZOakZ8tWaZ6iErsEagTVmD4T6TD57wPZHztvUwpAhd2SEj3CQWJoEPXrR/EfD/ESjh3ztkJ/9brO/y6bftCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T15:31:17.294834Z"},"content_sha256":"eddf9e1d0f1c31c2fd6b9cfb577d082b3f7c3865252b13475986abb886c637a1","schema_version":"1.0","event_id":"sha256:eddf9e1d0f1c31c2fd6b9cfb577d082b3f7c3865252b13475986abb886c637a1"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/R3NI6IOHAZPC2DJFXXQAQSZESZ/bundle.json","state_url":"https://pith.science/pith/R3NI6IOHAZPC2DJFXXQAQSZESZ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/R3NI6IOHAZPC2DJFXXQAQSZESZ/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":"2026-06-07T15:31:17Z","links":{"resolver":"https://pith.science/pith/R3NI6IOHAZPC2DJFXXQAQSZESZ","bundle":"https://pith.science/pith/R3NI6IOHAZPC2DJFXXQAQSZESZ/bundle.json","state":"https://pith.science/pith/R3NI6IOHAZPC2DJFXXQAQSZESZ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/R3NI6IOHAZPC2DJFXXQAQSZESZ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:R3NI6IOHAZPC2DJFXXQAQSZESZ","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":"7b6663d109ab1f4e3db58b487bd4d28c29fe785908e744d2c1953c7609929a42","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2016-07-23T06:48:18Z","title_canon_sha256":"909a4e6c24e6cf7e0ea8dd60f4d618bfa47cd3d38cbc94a85887107f7fe2658b"},"schema_version":"1.0","source":{"id":"1607.06897","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1607.06897","created_at":"2026-05-18T01:10:35Z"},{"alias_kind":"arxiv_version","alias_value":"1607.06897v1","created_at":"2026-05-18T01:10:35Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1607.06897","created_at":"2026-05-18T01:10:35Z"},{"alias_kind":"pith_short_12","alias_value":"R3NI6IOHAZPC","created_at":"2026-05-18T12:30:41Z"},{"alias_kind":"pith_short_16","alias_value":"R3NI6IOHAZPC2DJF","created_at":"2026-05-18T12:30:41Z"},{"alias_kind":"pith_short_8","alias_value":"R3NI6IOH","created_at":"2026-05-18T12:30:41Z"}],"graph_snapshots":[{"event_id":"sha256:eddf9e1d0f1c31c2fd6b9cfb577d082b3f7c3865252b13475986abb886c637a1","target":"graph","created_at":"2026-05-18T01:10:35Z","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":"This is the second part in a series of papers on multi-step schemes for solving coupled forward backward stochastic differential equations (FBSDEs). We extend the basic idea in our former paper [W. Zhao, Y. Fu and T. Zhou, SIAM J. Sci. Comput., 36 (2014), pp. A1731-A1751] to solve high-dimensional FBSDEs, by using the spectral sparse grid approximations. The main issue for solving high dimensional FBSDEs is to build an efficient spatial discretization, and deal with the related high dimensional conditional expectations and interpolations. In this work, we propose the sparse grid spatial discre","authors_text":"Tao Zhou, Weidong Zhao, Yu Fu","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2016-07-23T06:48:18Z","title":"Efficient spectral sparse grid approximations for solving multi-dimensional forward backward SDEs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.06897","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":"sha256:1b9c59fa0a19002fd8bc581c513484cf71e1d51825de963c6da8a1ea85a33ba7","target":"record","created_at":"2026-05-18T01:10:35Z","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":"7b6663d109ab1f4e3db58b487bd4d28c29fe785908e744d2c1953c7609929a42","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2016-07-23T06:48:18Z","title_canon_sha256":"909a4e6c24e6cf7e0ea8dd60f4d618bfa47cd3d38cbc94a85887107f7fe2658b"},"schema_version":"1.0","source":{"id":"1607.06897","kind":"arxiv","version":1}},"canonical_sha256":"8eda8f21c7065e2d0d25bde0084b24967d2c94f715f15af01098b0b29d3edebd","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8eda8f21c7065e2d0d25bde0084b24967d2c94f715f15af01098b0b29d3edebd","first_computed_at":"2026-05-18T01:10:35.519799Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:10:35.519799Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"A4BexEaWU9aivbTTn1dprBoM1yDMsVHSCat3ys2m+0/0FfQp1Pnb7kJSpa1BAkVCIINpUDHxBIpenLnaWSlEBw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:10:35.520390Z","signed_message":"canonical_sha256_bytes"},"source_id":"1607.06897","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1b9c59fa0a19002fd8bc581c513484cf71e1d51825de963c6da8a1ea85a33ba7","sha256:eddf9e1d0f1c31c2fd6b9cfb577d082b3f7c3865252b13475986abb886c637a1"],"state_sha256":"9d43b1ad243037526de30fa46b768d360a688b450b2aa6db20ecdeb14dd532af"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ubmG8bH3O1PABLYMyvwkueq2S9vqzkEXJvvFDS1On9axGIrE8NYsEdY0HN/q42XoL2QtHO3L19mFyzPzdTw0DQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-07T15:31:17.299206Z","bundle_sha256":"b4880b887b12a96c8a40346897c9c3fde8a97d8f8b32edd6f72bf44653e2f95a"}}