{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2018:NWNFSBRCR6AKVKPAE4ZRRHCCH7","short_pith_number":"pith:NWNFSBRC","schema_version":"1.0","canonical_sha256":"6d9a5906228f80aaa9e02733189c423fe0efdfd1cb6ab6b77441dbfc08470034","source":{"kind":"arxiv","id":"1809.09890","version":1},"attestation_state":"computed","paper":{"title":"Optimal confidence for Monte Carlo integration of smooth functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NA","authors_text":"Daniel Rudolf, Robert J. Kunsch","submitted_at":"2018-09-26T10:25:41Z","abstract_excerpt":"We study the complexity of approximating integrals of smooth functions at absolute precision $\\varepsilon > 0$ with confidence level $1 - \\delta \\in (0,1)$. The optimal error rate for multivariate functions from classical isotropic Sobolev spaces $W_p^r(G)$ with sufficient smoothness on bounded Lipschitz domains $G \\subset \\mathbb{R}^d$ is determined. It turns out that the integrability index $p$ has an effect on the influence of the uncertainty $\\delta$ in the complexity. In the limiting case $p = 1$ we see that deterministic methods cannot be improved by randomization. In general, higher smo"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1809.09890","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2018-09-26T10:25:41Z","cross_cats_sorted":[],"title_canon_sha256":"c0f7af14d618030f671d6a12dc8195591379b2271d55d2e49c2047ae3e9ff292","abstract_canon_sha256":"097f655b18a469dd28e807301fa9268904b1f96c159479448703c3d140ff248e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:04:43.361505Z","signature_b64":"Wnxzb3it0PDyFteYnsxQQsZBF0c31sutgZ1ebsHWYt8wjFLQapePh+iL9hf8tDsbuq3zU+ErLPJ6/alNUzT/DA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6d9a5906228f80aaa9e02733189c423fe0efdfd1cb6ab6b77441dbfc08470034","last_reissued_at":"2026-05-18T00:04:43.360864Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:04:43.360864Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Optimal confidence for Monte Carlo integration of smooth functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NA","authors_text":"Daniel Rudolf, Robert J. Kunsch","submitted_at":"2018-09-26T10:25:41Z","abstract_excerpt":"We study the complexity of approximating integrals of smooth functions at absolute precision $\\varepsilon > 0$ with confidence level $1 - \\delta \\in (0,1)$. The optimal error rate for multivariate functions from classical isotropic Sobolev spaces $W_p^r(G)$ with sufficient smoothness on bounded Lipschitz domains $G \\subset \\mathbb{R}^d$ is determined. It turns out that the integrability index $p$ has an effect on the influence of the uncertainty $\\delta$ in the complexity. In the limiting case $p = 1$ we see that deterministic methods cannot be improved by randomization. In general, higher smo"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1809.09890","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1809.09890","created_at":"2026-05-18T00:04:43.360960+00:00"},{"alias_kind":"arxiv_version","alias_value":"1809.09890v1","created_at":"2026-05-18T00:04:43.360960+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1809.09890","created_at":"2026-05-18T00:04:43.360960+00:00"},{"alias_kind":"pith_short_12","alias_value":"NWNFSBRCR6AK","created_at":"2026-05-18T12:32:40.477152+00:00"},{"alias_kind":"pith_short_16","alias_value":"NWNFSBRCR6AKVKPA","created_at":"2026-05-18T12:32:40.477152+00:00"},{"alias_kind":"pith_short_8","alias_value":"NWNFSBRC","created_at":"2026-05-18T12:32:40.477152+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/NWNFSBRCR6AKVKPAE4ZRRHCCH7","json":"https://pith.science/pith/NWNFSBRCR6AKVKPAE4ZRRHCCH7.json","graph_json":"https://pith.science/api/pith-number/NWNFSBRCR6AKVKPAE4ZRRHCCH7/graph.json","events_json":"https://pith.science/api/pith-number/NWNFSBRCR6AKVKPAE4ZRRHCCH7/events.json","paper":"https://pith.science/paper/NWNFSBRC"},"agent_actions":{"view_html":"https://pith.science/pith/NWNFSBRCR6AKVKPAE4ZRRHCCH7","download_json":"https://pith.science/pith/NWNFSBRCR6AKVKPAE4ZRRHCCH7.json","view_paper":"https://pith.science/paper/NWNFSBRC","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1809.09890&json=true","fetch_graph":"https://pith.science/api/pith-number/NWNFSBRCR6AKVKPAE4ZRRHCCH7/graph.json","fetch_events":"https://pith.science/api/pith-number/NWNFSBRCR6AKVKPAE4ZRRHCCH7/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/NWNFSBRCR6AKVKPAE4ZRRHCCH7/action/timestamp_anchor","attest_storage":"https://pith.science/pith/NWNFSBRCR6AKVKPAE4ZRRHCCH7/action/storage_attestation","attest_author":"https://pith.science/pith/NWNFSBRCR6AKVKPAE4ZRRHCCH7/action/author_attestation","sign_citation":"https://pith.science/pith/NWNFSBRCR6AKVKPAE4ZRRHCCH7/action/citation_signature","submit_replication":"https://pith.science/pith/NWNFSBRCR6AKVKPAE4ZRRHCCH7/action/replication_record"}},"created_at":"2026-05-18T00:04:43.360960+00:00","updated_at":"2026-05-18T00:04:43.360960+00:00"}