{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2024:TZ2PY2KXAK47U4VVQSIJI3BZT5","short_pith_number":"pith:TZ2PY2KX","canonical_record":{"source":{"id":"2403.14229","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2024-03-21T08:40:16Z","cross_cats_sorted":["cs.NA"],"title_canon_sha256":"b0800f9330104420f3573be990e7d911b8eb36bdadf0b487d78d2f086cbce602","abstract_canon_sha256":"0a273360565aad41cf4f2f18883a6831d226ebc801b17a097604273c50cae530"},"schema_version":"1.0"},"canonical_sha256":"9e74fc695702b9fa72b58490946c399f5a4bef0e56b03ca703223b2e4ca126c0","source":{"kind":"arxiv","id":"2403.14229","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2403.14229","created_at":"2026-07-05T11:09:58Z"},{"alias_kind":"arxiv_version","alias_value":"2403.14229v2","created_at":"2026-07-05T11:09:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2403.14229","created_at":"2026-07-05T11:09:58Z"},{"alias_kind":"pith_short_12","alias_value":"TZ2PY2KXAK47","created_at":"2026-07-05T11:09:58Z"},{"alias_kind":"pith_short_16","alias_value":"TZ2PY2KXAK47U4VV","created_at":"2026-07-05T11:09:58Z"},{"alias_kind":"pith_short_8","alias_value":"TZ2PY2KX","created_at":"2026-07-05T11:09:58Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2024:TZ2PY2KXAK47U4VVQSIJI3BZT5","target":"record","payload":{"canonical_record":{"source":{"id":"2403.14229","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2024-03-21T08:40:16Z","cross_cats_sorted":["cs.NA"],"title_canon_sha256":"b0800f9330104420f3573be990e7d911b8eb36bdadf0b487d78d2f086cbce602","abstract_canon_sha256":"0a273360565aad41cf4f2f18883a6831d226ebc801b17a097604273c50cae530"},"schema_version":"1.0"},"canonical_sha256":"9e74fc695702b9fa72b58490946c399f5a4bef0e56b03ca703223b2e4ca126c0","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:09:58.239494Z","signature_b64":"otOJgssPLSba1nplRpUf6RrtSMWRXx5jYAxM6pJe+0X0ZcI0ApohQXxVs81OYRNLxabQo+JpUhbUxB/PYpfkAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9e74fc695702b9fa72b58490946c399f5a4bef0e56b03ca703223b2e4ca126c0","last_reissued_at":"2026-07-05T11:09:58.239018Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:09:58.239018Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2403.14229","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":"2026-07-05T11:09:58Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"3ABDTe3Ij7OLIqtZA7Nta+trgApQ8D/bYDX+U2CDQGCMBV7UtvU5rfhXV9O0WGlpPwM1mxMH10UPQhslNrpEBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-10T12:45:36.270977Z"},"content_sha256":"07dade265d1ce30be65835a49916a9db5853553465f9f00384a3ef574f7a4cc1","schema_version":"1.0","event_id":"sha256:07dade265d1ce30be65835a49916a9db5853553465f9f00384a3ef574f7a4cc1"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2024:TZ2PY2KXAK47U4VVQSIJI3BZT5","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Low-rank tensor product Richardson iteration for radiative transfer in plane-parallel geometry","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.NA"],"primary_cat":"math.NA","authors_text":"Markus Bachmayr, Matthias Schlottbom, Riccardo Bardin","submitted_at":"2024-03-21T08:40:16Z","abstract_excerpt":"The radiative transfer equation (RTE) has been established as a fundamental tool for the description of energy transport, absorption and scattering in many relevant societal applications, and requires numerical approximations. However, classical numerical algorithms scale unfavorably with respect to the dimensionality of such radiative transfer problems, where solutions depend on physical as well as angular variables. In this paper we address this dimensionality issue by developing a low-rank tensor product framework for the RTE in plane-parallel geometry. We exploit the tensor product nature "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2403.14229","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/2403.14229/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":"2026-07-05T11:09:58Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"WMxUarQ3RL7y3v3SJxE865v0uaTwK7C6clKks38OLw3eeamn8MnBeUr9EuVZdwEYnpApRTUsH/gW52stnE2eBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-10T12:45:36.272458Z"},"content_sha256":"f30db4d1f29068678337afbdce8d31617f90493310f2f7be81ee9cfa3d281356","schema_version":"1.0","event_id":"sha256:f30db4d1f29068678337afbdce8d31617f90493310f2f7be81ee9cfa3d281356"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/TZ2PY2KXAK47U4VVQSIJI3BZT5/bundle.json","state_url":"https://pith.science/pith/TZ2PY2KXAK47U4VVQSIJI3BZT5/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/TZ2PY2KXAK47U4VVQSIJI3BZT5/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":"2026-08-10T12:45:36Z","links":{"resolver":"https://pith.science/pith/TZ2PY2KXAK47U4VVQSIJI3BZT5","bundle":"https://pith.science/pith/TZ2PY2KXAK47U4VVQSIJI3BZT5/bundle.json","state":"https://pith.science/pith/TZ2PY2KXAK47U4VVQSIJI3BZT5/state.json","well_known_bundle":"https://pith.science/.well-known/pith/TZ2PY2KXAK47U4VVQSIJI3BZT5/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:TZ2PY2KXAK47U4VVQSIJI3BZT5","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":"0a273360565aad41cf4f2f18883a6831d226ebc801b17a097604273c50cae530","cross_cats_sorted":["cs.NA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2024-03-21T08:40:16Z","title_canon_sha256":"b0800f9330104420f3573be990e7d911b8eb36bdadf0b487d78d2f086cbce602"},"schema_version":"1.0","source":{"id":"2403.14229","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2403.14229","created_at":"2026-07-05T11:09:58Z"},{"alias_kind":"arxiv_version","alias_value":"2403.14229v2","created_at":"2026-07-05T11:09:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2403.14229","created_at":"2026-07-05T11:09:58Z"},{"alias_kind":"pith_short_12","alias_value":"TZ2PY2KXAK47","created_at":"2026-07-05T11:09:58Z"},{"alias_kind":"pith_short_16","alias_value":"TZ2PY2KXAK47U4VV","created_at":"2026-07-05T11:09:58Z"},{"alias_kind":"pith_short_8","alias_value":"TZ2PY2KX","created_at":"2026-07-05T11:09:58Z"}],"graph_snapshots":[{"event_id":"sha256:f30db4d1f29068678337afbdce8d31617f90493310f2f7be81ee9cfa3d281356","target":"graph","created_at":"2026-07-05T11:09:58Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2403.14229/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"The radiative transfer equation (RTE) has been established as a fundamental tool for the description of energy transport, absorption and scattering in many relevant societal applications, and requires numerical approximations. However, classical numerical algorithms scale unfavorably with respect to the dimensionality of such radiative transfer problems, where solutions depend on physical as well as angular variables. In this paper we address this dimensionality issue by developing a low-rank tensor product framework for the RTE in plane-parallel geometry. We exploit the tensor product nature ","authors_text":"Markus Bachmayr, Matthias Schlottbom, Riccardo Bardin","cross_cats":["cs.NA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2024-03-21T08:40:16Z","title":"Low-rank tensor product Richardson iteration for radiative transfer in plane-parallel geometry"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2403.14229","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":"sha256:07dade265d1ce30be65835a49916a9db5853553465f9f00384a3ef574f7a4cc1","target":"record","created_at":"2026-07-05T11:09:58Z","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":"0a273360565aad41cf4f2f18883a6831d226ebc801b17a097604273c50cae530","cross_cats_sorted":["cs.NA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2024-03-21T08:40:16Z","title_canon_sha256":"b0800f9330104420f3573be990e7d911b8eb36bdadf0b487d78d2f086cbce602"},"schema_version":"1.0","source":{"id":"2403.14229","kind":"arxiv","version":2}},"canonical_sha256":"9e74fc695702b9fa72b58490946c399f5a4bef0e56b03ca703223b2e4ca126c0","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"9e74fc695702b9fa72b58490946c399f5a4bef0e56b03ca703223b2e4ca126c0","first_computed_at":"2026-07-05T11:09:58.239018Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T11:09:58.239018Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"otOJgssPLSba1nplRpUf6RrtSMWRXx5jYAxM6pJe+0X0ZcI0ApohQXxVs81OYRNLxabQo+JpUhbUxB/PYpfkAg==","signature_status":"signed_v1","signed_at":"2026-07-05T11:09:58.239494Z","signed_message":"canonical_sha256_bytes"},"source_id":"2403.14229","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:07dade265d1ce30be65835a49916a9db5853553465f9f00384a3ef574f7a4cc1","sha256:f30db4d1f29068678337afbdce8d31617f90493310f2f7be81ee9cfa3d281356"],"state_sha256":"244c5f7b44d57d3f2046590ad1f6204a02ea594a18be7c13f15387eb2aac210e"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"w6xkgny8lnXMO/6dk0yKS3ciplJMAqyZzyh/RVjT7Rv1FwTRGygG6Hg3Z1BfjFM4RZcFQFwS5MV3ceeJmz3xBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-10T12:45:36.283047Z","bundle_sha256":"f4a9b0f0c21454df87ed67a527fcda761b60d77e86c7c0122a12783997d6c127"}}