{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:NDMVIBJQZXCRGLKKFC5YLZ2XNU","short_pith_number":"pith:NDMVIBJQ","canonical_record":{"source":{"id":"1109.2936","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2011-09-13T21:32:45Z","cross_cats_sorted":[],"title_canon_sha256":"2010e8b1ca1aad83a0fa6b8e50a9473e958aee9efd4deee9f25ca7dd71129f1e","abstract_canon_sha256":"17399f1573f02c9f374a77a60383221ceba28757277137f446d819be84639a53"},"schema_version":"1.0"},"canonical_sha256":"68d9540530cdc5132d4a28bb85e7576d1aaf7bd9ac4598d766ee0de78bd73925","source":{"kind":"arxiv","id":"1109.2936","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1109.2936","created_at":"2026-05-18T03:28:54Z"},{"alias_kind":"arxiv_version","alias_value":"1109.2936v2","created_at":"2026-05-18T03:28:54Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1109.2936","created_at":"2026-05-18T03:28:54Z"},{"alias_kind":"pith_short_12","alias_value":"NDMVIBJQZXCR","created_at":"2026-05-18T12:26:37Z"},{"alias_kind":"pith_short_16","alias_value":"NDMVIBJQZXCRGLKK","created_at":"2026-05-18T12:26:37Z"},{"alias_kind":"pith_short_8","alias_value":"NDMVIBJQ","created_at":"2026-05-18T12:26:37Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:NDMVIBJQZXCRGLKKFC5YLZ2XNU","target":"record","payload":{"canonical_record":{"source":{"id":"1109.2936","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2011-09-13T21:32:45Z","cross_cats_sorted":[],"title_canon_sha256":"2010e8b1ca1aad83a0fa6b8e50a9473e958aee9efd4deee9f25ca7dd71129f1e","abstract_canon_sha256":"17399f1573f02c9f374a77a60383221ceba28757277137f446d819be84639a53"},"schema_version":"1.0"},"canonical_sha256":"68d9540530cdc5132d4a28bb85e7576d1aaf7bd9ac4598d766ee0de78bd73925","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:28:54.208421Z","signature_b64":"V6Hy3Mf9BdqNTdPJatkmyfEohNL1fm9PaLn3MMKLh22khoI7wZG7d8CMpKA1VUYlywxX8QztiOJd1Fw74rgTAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"68d9540530cdc5132d4a28bb85e7576d1aaf7bd9ac4598d766ee0de78bd73925","last_reissued_at":"2026-05-18T03:28:54.207658Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:28:54.207658Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1109.2936","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:28:54Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"aOQ78HekInth+LgcK8yu+IABc6AkTO2TanKvpSvPcx8uW0+wB1W4o781nuoEoSbDAFoB7J7AX2g8qhT1zlhKBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T12:50:57.988193Z"},"content_sha256":"b3a4818f39ebc9ddb7bb48fef2a52680ef3591f5e7902a7ed690eee1423ff120","schema_version":"1.0","event_id":"sha256:b3a4818f39ebc9ddb7bb48fef2a52680ef3591f5e7902a7ed690eee1423ff120"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:NDMVIBJQZXCRGLKKFC5YLZ2XNU","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Sparse Pseudospectral Approximation Method","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NA","authors_text":"Eric T. Phipps, Michael S. Eldred, Paul G. Constantine","submitted_at":"2011-09-13T21:32:45Z","abstract_excerpt":"Multivariate global polynomial approximations - such as polynomial chaos or stochastic collocation methods - are now in widespread use for sensitivity analysis and uncertainty quantification. The pseudospectral variety of these methods uses a numerical integration rule to approximate the Fourier-type coefficients of a truncated expansion in orthogonal polynomials. For problems in more than two or three dimensions, a sparse grid numerical integration rule offers accuracy with a smaller node set compared to tensor product approximation. However, when using a sparse rule to approximately integrat"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1109.2936","kind":"arxiv","version":2},"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":"2026-05-18T03:28:54Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"NvQ3Es8dj1z+w+Hh/LhdZv2/7JmFJj7WMfrtA1e/hiioYPA6BDvmWvymiLfh01MPcFr+lmzAzg+eVmJ+yq4mDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T12:50:57.988564Z"},"content_sha256":"4e0cbcd49111fb65e2b5e31a50f97d7b09c1092cb6c9d949a2c8146611df56e0","schema_version":"1.0","event_id":"sha256:4e0cbcd49111fb65e2b5e31a50f97d7b09c1092cb6c9d949a2c8146611df56e0"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/NDMVIBJQZXCRGLKKFC5YLZ2XNU/bundle.json","state_url":"https://pith.science/pith/NDMVIBJQZXCRGLKKFC5YLZ2XNU/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/NDMVIBJQZXCRGLKKFC5YLZ2XNU/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":"2026-05-28T12:50:57Z","links":{"resolver":"https://pith.science/pith/NDMVIBJQZXCRGLKKFC5YLZ2XNU","bundle":"https://pith.science/pith/NDMVIBJQZXCRGLKKFC5YLZ2XNU/bundle.json","state":"https://pith.science/pith/NDMVIBJQZXCRGLKKFC5YLZ2XNU/state.json","well_known_bundle":"https://pith.science/.well-known/pith/NDMVIBJQZXCRGLKKFC5YLZ2XNU/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:NDMVIBJQZXCRGLKKFC5YLZ2XNU","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":"17399f1573f02c9f374a77a60383221ceba28757277137f446d819be84639a53","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2011-09-13T21:32:45Z","title_canon_sha256":"2010e8b1ca1aad83a0fa6b8e50a9473e958aee9efd4deee9f25ca7dd71129f1e"},"schema_version":"1.0","source":{"id":"1109.2936","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1109.2936","created_at":"2026-05-18T03:28:54Z"},{"alias_kind":"arxiv_version","alias_value":"1109.2936v2","created_at":"2026-05-18T03:28:54Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1109.2936","created_at":"2026-05-18T03:28:54Z"},{"alias_kind":"pith_short_12","alias_value":"NDMVIBJQZXCR","created_at":"2026-05-18T12:26:37Z"},{"alias_kind":"pith_short_16","alias_value":"NDMVIBJQZXCRGLKK","created_at":"2026-05-18T12:26:37Z"},{"alias_kind":"pith_short_8","alias_value":"NDMVIBJQ","created_at":"2026-05-18T12:26:37Z"}],"graph_snapshots":[{"event_id":"sha256:4e0cbcd49111fb65e2b5e31a50f97d7b09c1092cb6c9d949a2c8146611df56e0","target":"graph","created_at":"2026-05-18T03:28:54Z","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":"Multivariate global polynomial approximations - such as polynomial chaos or stochastic collocation methods - are now in widespread use for sensitivity analysis and uncertainty quantification. The pseudospectral variety of these methods uses a numerical integration rule to approximate the Fourier-type coefficients of a truncated expansion in orthogonal polynomials. For problems in more than two or three dimensions, a sparse grid numerical integration rule offers accuracy with a smaller node set compared to tensor product approximation. However, when using a sparse rule to approximately integrat","authors_text":"Eric T. Phipps, Michael S. Eldred, Paul G. Constantine","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2011-09-13T21:32:45Z","title":"Sparse Pseudospectral Approximation Method"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1109.2936","kind":"arxiv","version":2},"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:b3a4818f39ebc9ddb7bb48fef2a52680ef3591f5e7902a7ed690eee1423ff120","target":"record","created_at":"2026-05-18T03:28:54Z","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":"17399f1573f02c9f374a77a60383221ceba28757277137f446d819be84639a53","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2011-09-13T21:32:45Z","title_canon_sha256":"2010e8b1ca1aad83a0fa6b8e50a9473e958aee9efd4deee9f25ca7dd71129f1e"},"schema_version":"1.0","source":{"id":"1109.2936","kind":"arxiv","version":2}},"canonical_sha256":"68d9540530cdc5132d4a28bb85e7576d1aaf7bd9ac4598d766ee0de78bd73925","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"68d9540530cdc5132d4a28bb85e7576d1aaf7bd9ac4598d766ee0de78bd73925","first_computed_at":"2026-05-18T03:28:54.207658Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:28:54.207658Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"V6Hy3Mf9BdqNTdPJatkmyfEohNL1fm9PaLn3MMKLh22khoI7wZG7d8CMpKA1VUYlywxX8QztiOJd1Fw74rgTAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:28:54.208421Z","signed_message":"canonical_sha256_bytes"},"source_id":"1109.2936","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b3a4818f39ebc9ddb7bb48fef2a52680ef3591f5e7902a7ed690eee1423ff120","sha256:4e0cbcd49111fb65e2b5e31a50f97d7b09c1092cb6c9d949a2c8146611df56e0"],"state_sha256":"a56f4ed9ae9f86510f1e3014eccad68ae1c9b6f2ab76c4a7914953e12f9b3fc6"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"JRPCw1J9MohddzvYIs/jcmkwzevv241YZS0ajgao0pcGzFIXFtnj6tSAKD0Eqv7f0mKZIBaWlb/JsCs1FkwnCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-28T12:50:57.990982Z","bundle_sha256":"4d8c1cb7473f62dd8416f88d77f540535ddc9277f9bfb283c01304861f3fd8cb"}}