{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:5EFWZG2CFVBMYL3G44GFQAPNJU","short_pith_number":"pith:5EFWZG2C","canonical_record":{"source":{"id":"1108.4213","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2011-08-21T21:19:28Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"977eed3274178dd2553ed45b3203304f17223c360d176adbe741dde9da777b10","abstract_canon_sha256":"f2cc2a6a8c60380bfda300d2ef2f9385f0ca73f7c3dec9386903fb777f3afa7f"},"schema_version":"1.0"},"canonical_sha256":"e90b6c9b422d42cc2f66e70c5801ed4d3eacc26e95f68548e8c9212a30b8ee56","source":{"kind":"arxiv","id":"1108.4213","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1108.4213","created_at":"2026-05-18T03:46:03Z"},{"alias_kind":"arxiv_version","alias_value":"1108.4213v3","created_at":"2026-05-18T03:46:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1108.4213","created_at":"2026-05-18T03:46:03Z"},{"alias_kind":"pith_short_12","alias_value":"5EFWZG2CFVBM","created_at":"2026-05-18T12:26:20Z"},{"alias_kind":"pith_short_16","alias_value":"5EFWZG2CFVBMYL3G","created_at":"2026-05-18T12:26:20Z"},{"alias_kind":"pith_short_8","alias_value":"5EFWZG2C","created_at":"2026-05-18T12:26:20Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:5EFWZG2CFVBMYL3G44GFQAPNJU","target":"record","payload":{"canonical_record":{"source":{"id":"1108.4213","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2011-08-21T21:19:28Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"977eed3274178dd2553ed45b3203304f17223c360d176adbe741dde9da777b10","abstract_canon_sha256":"f2cc2a6a8c60380bfda300d2ef2f9385f0ca73f7c3dec9386903fb777f3afa7f"},"schema_version":"1.0"},"canonical_sha256":"e90b6c9b422d42cc2f66e70c5801ed4d3eacc26e95f68548e8c9212a30b8ee56","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:46:03.838196Z","signature_b64":"cMpmzi5SyJWuTceM76OK3YLCIO+JV6h7RQ/Z9y1kPhM3cfu31SydW42jDsjIqPLP23wOpCwWxCHun8Ft2mgsAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e90b6c9b422d42cc2f66e70c5801ed4d3eacc26e95f68548e8c9212a30b8ee56","last_reissued_at":"2026-05-18T03:46:03.837561Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:46:03.837561Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1108.4213","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:46:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"SVKaP5hOHXc7AeqD0DnqIndvS5ZDLuD+FWqHsvXn4d9DfGz5QhQilxTHSD9FKN9wM/NtqTepTl5DbAdrSXKFDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T06:24:42.235726Z"},"content_sha256":"95b40e68dbaaf7b1584e934f6ed45a009910b728e6799b9778ac4c2821f091c7","schema_version":"1.0","event_id":"sha256:95b40e68dbaaf7b1584e934f6ed45a009910b728e6799b9778ac4c2821f091c7"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:5EFWZG2CFVBMYL3G44GFQAPNJU","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Approximation of Stochastic Partial Differential Equations by a Kernel-based Collocation Method","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.NA","authors_text":"Gregory E. Fasshauer, Igor Cialenco, Qi Ye","submitted_at":"2011-08-21T21:19:28Z","abstract_excerpt":"In this paper we present the theoretical framework needed to justify the use of a kernel-based collocation method (meshfree approximation method) to estimate the solution of high-dimensional stochastic partial differential equations (SPDEs). Using an implicit time stepping scheme, we transform stochastic parabolic equations into stochastic elliptic equations. Our main attention is concentrated on the numerical solution of the elliptic equations at each time step. The estimator of the solution of the elliptic equations is given as a linear combination of reproducing kernels derived from the dif"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1108.4213","kind":"arxiv","version":3},"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":"2026-05-18T03:46:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/TAdwlSKGxpdr+6Ez6/wtvth+SOOVaU6ggUuDOfuqc/Cj4ba3vnBZzuuZvSNZf7tht3beb1xrEpv5EJqfbf+AA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T06:24:42.236090Z"},"content_sha256":"2e8a847e12ba7f8b06971723904acb1e3c07631616b4ce6657b97b3dfa19e421","schema_version":"1.0","event_id":"sha256:2e8a847e12ba7f8b06971723904acb1e3c07631616b4ce6657b97b3dfa19e421"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/5EFWZG2CFVBMYL3G44GFQAPNJU/bundle.json","state_url":"https://pith.science/pith/5EFWZG2CFVBMYL3G44GFQAPNJU/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/5EFWZG2CFVBMYL3G44GFQAPNJU/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":"2026-05-27T06:24:42Z","links":{"resolver":"https://pith.science/pith/5EFWZG2CFVBMYL3G44GFQAPNJU","bundle":"https://pith.science/pith/5EFWZG2CFVBMYL3G44GFQAPNJU/bundle.json","state":"https://pith.science/pith/5EFWZG2CFVBMYL3G44GFQAPNJU/state.json","well_known_bundle":"https://pith.science/.well-known/pith/5EFWZG2CFVBMYL3G44GFQAPNJU/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:5EFWZG2CFVBMYL3G44GFQAPNJU","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":"f2cc2a6a8c60380bfda300d2ef2f9385f0ca73f7c3dec9386903fb777f3afa7f","cross_cats_sorted":["math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2011-08-21T21:19:28Z","title_canon_sha256":"977eed3274178dd2553ed45b3203304f17223c360d176adbe741dde9da777b10"},"schema_version":"1.0","source":{"id":"1108.4213","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1108.4213","created_at":"2026-05-18T03:46:03Z"},{"alias_kind":"arxiv_version","alias_value":"1108.4213v3","created_at":"2026-05-18T03:46:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1108.4213","created_at":"2026-05-18T03:46:03Z"},{"alias_kind":"pith_short_12","alias_value":"5EFWZG2CFVBM","created_at":"2026-05-18T12:26:20Z"},{"alias_kind":"pith_short_16","alias_value":"5EFWZG2CFVBMYL3G","created_at":"2026-05-18T12:26:20Z"},{"alias_kind":"pith_short_8","alias_value":"5EFWZG2C","created_at":"2026-05-18T12:26:20Z"}],"graph_snapshots":[{"event_id":"sha256:2e8a847e12ba7f8b06971723904acb1e3c07631616b4ce6657b97b3dfa19e421","target":"graph","created_at":"2026-05-18T03:46:03Z","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":"In this paper we present the theoretical framework needed to justify the use of a kernel-based collocation method (meshfree approximation method) to estimate the solution of high-dimensional stochastic partial differential equations (SPDEs). Using an implicit time stepping scheme, we transform stochastic parabolic equations into stochastic elliptic equations. Our main attention is concentrated on the numerical solution of the elliptic equations at each time step. The estimator of the solution of the elliptic equations is given as a linear combination of reproducing kernels derived from the dif","authors_text":"Gregory E. Fasshauer, Igor Cialenco, Qi Ye","cross_cats":["math.PR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2011-08-21T21:19:28Z","title":"Approximation of Stochastic Partial Differential Equations by a Kernel-based Collocation Method"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1108.4213","kind":"arxiv","version":3},"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":"sha256:95b40e68dbaaf7b1584e934f6ed45a009910b728e6799b9778ac4c2821f091c7","target":"record","created_at":"2026-05-18T03:46:03Z","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":"f2cc2a6a8c60380bfda300d2ef2f9385f0ca73f7c3dec9386903fb777f3afa7f","cross_cats_sorted":["math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2011-08-21T21:19:28Z","title_canon_sha256":"977eed3274178dd2553ed45b3203304f17223c360d176adbe741dde9da777b10"},"schema_version":"1.0","source":{"id":"1108.4213","kind":"arxiv","version":3}},"canonical_sha256":"e90b6c9b422d42cc2f66e70c5801ed4d3eacc26e95f68548e8c9212a30b8ee56","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e90b6c9b422d42cc2f66e70c5801ed4d3eacc26e95f68548e8c9212a30b8ee56","first_computed_at":"2026-05-18T03:46:03.837561Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:46:03.837561Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"cMpmzi5SyJWuTceM76OK3YLCIO+JV6h7RQ/Z9y1kPhM3cfu31SydW42jDsjIqPLP23wOpCwWxCHun8Ft2mgsAg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:46:03.838196Z","signed_message":"canonical_sha256_bytes"},"source_id":"1108.4213","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:95b40e68dbaaf7b1584e934f6ed45a009910b728e6799b9778ac4c2821f091c7","sha256:2e8a847e12ba7f8b06971723904acb1e3c07631616b4ce6657b97b3dfa19e421"],"state_sha256":"c58108163be4e2a1de359931cc1e50864b13f67ee3626d8f7383176cc93e4cf2"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"TjsldxseC/nwrO6Obqnuv3f570MKN4PnExA7JaN7myisYLZO2fXMuzYc0BpStR5ksupvpOoCVU825GwVR+zsAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T06:24:42.238503Z","bundle_sha256":"5ca1ef82e87c0505433038497ad358717a225af303f2d94464f856a39fb15d9f"}}