{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:76EPMKCDNHJWYAXOY7A7SPOCNS","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":"1ae5505a2d64272be0f684a00561077ad0d73ff2b6f60ae79eeff3cdcc5ae446","cross_cats_sorted":["math.AP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2014-07-23T13:35:14Z","title_canon_sha256":"27690658319f8f2f77fbff2ad453644ef45a17fec9662c02ad12365aca0341d7"},"schema_version":"1.0","source":{"id":"1407.6208","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1407.6208","created_at":"2026-05-18T02:46:58Z"},{"alias_kind":"arxiv_version","alias_value":"1407.6208v1","created_at":"2026-05-18T02:46:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1407.6208","created_at":"2026-05-18T02:46:58Z"},{"alias_kind":"pith_short_12","alias_value":"76EPMKCDNHJW","created_at":"2026-05-18T12:28:16Z"},{"alias_kind":"pith_short_16","alias_value":"76EPMKCDNHJWYAXO","created_at":"2026-05-18T12:28:16Z"},{"alias_kind":"pith_short_8","alias_value":"76EPMKCD","created_at":"2026-05-18T12:28:16Z"}],"graph_snapshots":[{"event_id":"sha256:1b54502e0ffed2fa89f8544f768aa0cbf45019a7262cc0416463f6035eebe26d","target":"graph","created_at":"2026-05-18T02:46:58Z","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":"A recurring theme in attempts to break the curse of dimensionality in the numerical approximations of solutions to high-dimensional partial differential equations (PDEs) is to employ some form of sparse tensor approximation. Unfortunately, there are only a few results that quantify the possible advantages of such an approach. This paper introduces a class $\\Sigma_n$ of functions, which can be written as a sum of rank-one tensors using a total of at most $n$ parameters and then uses this notion of sparsity to prove a regularity theorem for certain high-dimensional elliptic PDEs. It is shown, am","authors_text":"Endre S\\\"uli, Lars Grasedyck, Ronald DeVore, Wolfgang Dahmen","cross_cats":["math.AP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2014-07-23T13:35:14Z","title":"Tensor-Sparsity of Solutions to High-Dimensional Elliptic Partial Differential Equations"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1407.6208","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":"sha256:538bc8febe9e2060fcf30534ee670f68d943907cab6c9cf0ff374eafb55ca280","target":"record","created_at":"2026-05-18T02:46:58Z","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":"1ae5505a2d64272be0f684a00561077ad0d73ff2b6f60ae79eeff3cdcc5ae446","cross_cats_sorted":["math.AP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2014-07-23T13:35:14Z","title_canon_sha256":"27690658319f8f2f77fbff2ad453644ef45a17fec9662c02ad12365aca0341d7"},"schema_version":"1.0","source":{"id":"1407.6208","kind":"arxiv","version":1}},"canonical_sha256":"ff88f6284369d36c02eec7c1f93dc26c8890e80a42287ca7a1641bbf470c4508","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ff88f6284369d36c02eec7c1f93dc26c8890e80a42287ca7a1641bbf470c4508","first_computed_at":"2026-05-18T02:46:58.654684Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:46:58.654684Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"/BvZU9V1srXjLEflVB3fxnbd3ufMZiXUlMgad6DWTwoT+4YDQfQohKt2+rNGwVSz2uteW1kpRTriEiua294wAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T02:46:58.655390Z","signed_message":"canonical_sha256_bytes"},"source_id":"1407.6208","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:538bc8febe9e2060fcf30534ee670f68d943907cab6c9cf0ff374eafb55ca280","sha256:1b54502e0ffed2fa89f8544f768aa0cbf45019a7262cc0416463f6035eebe26d"],"state_sha256":"07cd6a6046f76af376436d2987076731dbb91dca1c37a55a169d3934073c4fdc"}