{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:OD6X4YPJ2CSPK4E4IQY2OW7X5F","short_pith_number":"pith:OD6X4YPJ","canonical_record":{"source":{"id":"1601.02500","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2016-01-11T16:08:40Z","cross_cats_sorted":["cs.NA","math.NA"],"title_canon_sha256":"78478f872c4f2ca317c3f25af751c3ee24a8d479b00f3e85e927acb5fba9e12a","abstract_canon_sha256":"184e016824067ee6dcbced0f526d4cb3e01be01cd2ec8e5e6fe1caace895adaf"},"schema_version":"1.0"},"canonical_sha256":"70fd7e61e9d0a4f5709c4431a75bf7e9709b90fc95771a549618858d38946540","source":{"kind":"arxiv","id":"1601.02500","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1601.02500","created_at":"2026-06-04T17:09:52Z"},{"alias_kind":"arxiv_version","alias_value":"1601.02500v2","created_at":"2026-06-04T17:09:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1601.02500","created_at":"2026-06-04T17:09:52Z"},{"alias_kind":"pith_short_12","alias_value":"OD6X4YPJ2CSP","created_at":"2026-06-04T17:09:52Z"},{"alias_kind":"pith_short_16","alias_value":"OD6X4YPJ2CSPK4E4","created_at":"2026-06-04T17:09:52Z"},{"alias_kind":"pith_short_8","alias_value":"OD6X4YPJ","created_at":"2026-06-04T17:09:52Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:OD6X4YPJ2CSPK4E4IQY2OW7X5F","target":"record","payload":{"canonical_record":{"source":{"id":"1601.02500","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2016-01-11T16:08:40Z","cross_cats_sorted":["cs.NA","math.NA"],"title_canon_sha256":"78478f872c4f2ca317c3f25af751c3ee24a8d479b00f3e85e927acb5fba9e12a","abstract_canon_sha256":"184e016824067ee6dcbced0f526d4cb3e01be01cd2ec8e5e6fe1caace895adaf"},"schema_version":"1.0"},"canonical_sha256":"70fd7e61e9d0a4f5709c4431a75bf7e9709b90fc95771a549618858d38946540","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-04T17:09:52.735551Z","signature_b64":"z/LRXOEjlL7vszgO6xQt+kfMsLSi/944rWxjkX2g9qoTxZygEMrCqpp1xrKRTXbSnBBfWG1NqDQ4bgwE10iqAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"70fd7e61e9d0a4f5709c4431a75bf7e9709b90fc95771a549618858d38946540","last_reissued_at":"2026-06-04T17:09:52.735087Z","signature_status":"signed_v1","first_computed_at":"2026-06-04T17:09:52.735087Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1601.02500","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":"2026-06-04T17:09:52Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"6X78ttJsgOX7hQ+664j0snsJfRhuU0i081H8CkYKBGGDk01rfbn2cbQhVL2P30MHpP0/Mzl8Y/7JdfFt7t4rDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T02:39:03.103253Z"},"content_sha256":"6e9c202452077c2bd02b0389177c8efebc656fd73f783f62fe242e135cb6bc23","schema_version":"1.0","event_id":"sha256:6e9c202452077c2bd02b0389177c8efebc656fd73f783f62fe242e135cb6bc23"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:OD6X4YPJ2CSPK4E4IQY2OW7X5F","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Numerical analysis of lognormal diffusions on the sphere","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.NA","math.NA"],"primary_cat":"math.PR","authors_text":"Annika Lang, Christoph Schwab, Lukas Herrmann","submitted_at":"2016-01-11T16:08:40Z","abstract_excerpt":"Numerical solutions of stationary diffusion equations on the unit sphere with isotropic lognormal diffusion coefficients are considered. H\\\"older regularity in $L^p$ sense for isotropic Gaussian random fields is obtained and related to the regularity of the driving lognormal coefficients. This yields regularity in $L^p$ sense of the solution to the diffusion problem in Sobolev spaces. Convergence rate estimates of multilevel Monte Carlo Finite and Spectral Element discretizations of these problems are then deduced. Specifically, a convergence analysis is provided with convergence rate estimate"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.02500","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1601.02500/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-06-04T17:09:52Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FPZZ/A4N8AjhmqMs3tTZvxKUMiM7d7WRhwJQIkOYZHIvbL36eU8Y4UGVZYmif4rrMhurtzzbOfIuApxPHLZHBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T02:39:03.104070Z"},"content_sha256":"81cb0f6f04f01320675696e70c7d431ec800ba158543c52d31a7ce693ab1da45","schema_version":"1.0","event_id":"sha256:81cb0f6f04f01320675696e70c7d431ec800ba158543c52d31a7ce693ab1da45"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/OD6X4YPJ2CSPK4E4IQY2OW7X5F/bundle.json","state_url":"https://pith.science/pith/OD6X4YPJ2CSPK4E4IQY2OW7X5F/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/OD6X4YPJ2CSPK4E4IQY2OW7X5F/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":"2026-06-11T02:39:03Z","links":{"resolver":"https://pith.science/pith/OD6X4YPJ2CSPK4E4IQY2OW7X5F","bundle":"https://pith.science/pith/OD6X4YPJ2CSPK4E4IQY2OW7X5F/bundle.json","state":"https://pith.science/pith/OD6X4YPJ2CSPK4E4IQY2OW7X5F/state.json","well_known_bundle":"https://pith.science/.well-known/pith/OD6X4YPJ2CSPK4E4IQY2OW7X5F/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:OD6X4YPJ2CSPK4E4IQY2OW7X5F","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":"184e016824067ee6dcbced0f526d4cb3e01be01cd2ec8e5e6fe1caace895adaf","cross_cats_sorted":["cs.NA","math.NA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2016-01-11T16:08:40Z","title_canon_sha256":"78478f872c4f2ca317c3f25af751c3ee24a8d479b00f3e85e927acb5fba9e12a"},"schema_version":"1.0","source":{"id":"1601.02500","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1601.02500","created_at":"2026-06-04T17:09:52Z"},{"alias_kind":"arxiv_version","alias_value":"1601.02500v2","created_at":"2026-06-04T17:09:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1601.02500","created_at":"2026-06-04T17:09:52Z"},{"alias_kind":"pith_short_12","alias_value":"OD6X4YPJ2CSP","created_at":"2026-06-04T17:09:52Z"},{"alias_kind":"pith_short_16","alias_value":"OD6X4YPJ2CSPK4E4","created_at":"2026-06-04T17:09:52Z"},{"alias_kind":"pith_short_8","alias_value":"OD6X4YPJ","created_at":"2026-06-04T17:09:52Z"}],"graph_snapshots":[{"event_id":"sha256:81cb0f6f04f01320675696e70c7d431ec800ba158543c52d31a7ce693ab1da45","target":"graph","created_at":"2026-06-04T17:09:52Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/1601.02500/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Numerical solutions of stationary diffusion equations on the unit sphere with isotropic lognormal diffusion coefficients are considered. H\\\"older regularity in $L^p$ sense for isotropic Gaussian random fields is obtained and related to the regularity of the driving lognormal coefficients. This yields regularity in $L^p$ sense of the solution to the diffusion problem in Sobolev spaces. Convergence rate estimates of multilevel Monte Carlo Finite and Spectral Element discretizations of these problems are then deduced. Specifically, a convergence analysis is provided with convergence rate estimate","authors_text":"Annika Lang, Christoph Schwab, Lukas Herrmann","cross_cats":["cs.NA","math.NA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2016-01-11T16:08:40Z","title":"Numerical analysis of lognormal diffusions on the sphere"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.02500","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":"sha256:6e9c202452077c2bd02b0389177c8efebc656fd73f783f62fe242e135cb6bc23","target":"record","created_at":"2026-06-04T17:09:52Z","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":"184e016824067ee6dcbced0f526d4cb3e01be01cd2ec8e5e6fe1caace895adaf","cross_cats_sorted":["cs.NA","math.NA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2016-01-11T16:08:40Z","title_canon_sha256":"78478f872c4f2ca317c3f25af751c3ee24a8d479b00f3e85e927acb5fba9e12a"},"schema_version":"1.0","source":{"id":"1601.02500","kind":"arxiv","version":2}},"canonical_sha256":"70fd7e61e9d0a4f5709c4431a75bf7e9709b90fc95771a549618858d38946540","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"70fd7e61e9d0a4f5709c4431a75bf7e9709b90fc95771a549618858d38946540","first_computed_at":"2026-06-04T17:09:52.735087Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-04T17:09:52.735087Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"z/LRXOEjlL7vszgO6xQt+kfMsLSi/944rWxjkX2g9qoTxZygEMrCqpp1xrKRTXbSnBBfWG1NqDQ4bgwE10iqAA==","signature_status":"signed_v1","signed_at":"2026-06-04T17:09:52.735551Z","signed_message":"canonical_sha256_bytes"},"source_id":"1601.02500","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6e9c202452077c2bd02b0389177c8efebc656fd73f783f62fe242e135cb6bc23","sha256:81cb0f6f04f01320675696e70c7d431ec800ba158543c52d31a7ce693ab1da45"],"state_sha256":"2a0a3125119e88858ec8a2e67e4179fe70a016c7727579d92c4065c5da360243"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"nateexMw3MH6oPjOg1I8fkl/y6kAK2Iu3UZUVkgPfl+vcReeSBFkpZ/fZCVY2Sq+YpgNh6XkPUjnLDDXxdbEAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-11T02:39:03.107636Z","bundle_sha256":"45f40940d9ddf42b5dddb493cd6d0edce6347610409e80a315a4d7cf07943ac7"}}