{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:6NME6JVYTAVGRCNNFFSFRVWYM4","short_pith_number":"pith:6NME6JVY","canonical_record":{"source":{"id":"1908.04498","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2019-08-13T05:49:53Z","cross_cats_sorted":["cs.NA"],"title_canon_sha256":"d19118526ad8d8e8c1063b75c1873bac4c6cb4426247ace59ee69105d89f5765","abstract_canon_sha256":"d20f55c54d58150aed16aedcaf08946417965b2a28b4552383a57661614759fb"},"schema_version":"1.0"},"canonical_sha256":"f3584f26b8982a6889ad296458d6d8670692454e74ac2d9add03c68002defe11","source":{"kind":"arxiv","id":"1908.04498","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1908.04498","created_at":"2026-07-04T23:55:14Z"},{"alias_kind":"arxiv_version","alias_value":"1908.04498v1","created_at":"2026-07-04T23:55:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1908.04498","created_at":"2026-07-04T23:55:14Z"},{"alias_kind":"pith_short_12","alias_value":"6NME6JVYTAVG","created_at":"2026-07-04T23:55:14Z"},{"alias_kind":"pith_short_16","alias_value":"6NME6JVYTAVGRCNN","created_at":"2026-07-04T23:55:14Z"},{"alias_kind":"pith_short_8","alias_value":"6NME6JVY","created_at":"2026-07-04T23:55:14Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:6NME6JVYTAVGRCNNFFSFRVWYM4","target":"record","payload":{"canonical_record":{"source":{"id":"1908.04498","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2019-08-13T05:49:53Z","cross_cats_sorted":["cs.NA"],"title_canon_sha256":"d19118526ad8d8e8c1063b75c1873bac4c6cb4426247ace59ee69105d89f5765","abstract_canon_sha256":"d20f55c54d58150aed16aedcaf08946417965b2a28b4552383a57661614759fb"},"schema_version":"1.0"},"canonical_sha256":"f3584f26b8982a6889ad296458d6d8670692454e74ac2d9add03c68002defe11","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T23:55:14.177432Z","signature_b64":"kzkhBKPtIm3nHTQfRjL/V698ORYuJuqv6fFy7hNAeyPIo3iik3TaWSiPEnGaD3MuE2AmJVYuVOichZ32zASFCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f3584f26b8982a6889ad296458d6d8670692454e74ac2d9add03c68002defe11","last_reissued_at":"2026-07-04T23:55:14.176976Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T23:55:14.176976Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1908.04498","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-04T23:55:14Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"M5E25w6pG/SLQZ/7xr+LNGKSZ0bzgvnyNNMfZduakk3XnhL25f1J+DoH15IH4dNPU+JO9+D6zeA0C4c9masaAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-16T05:01:30.157397Z"},"content_sha256":"0716e9ff3c2885d2ae10118874c3ede1bbe769319b93d98e4ca4883a549d2639","schema_version":"1.0","event_id":"sha256:0716e9ff3c2885d2ae10118874c3ede1bbe769319b93d98e4ca4883a549d2639"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:6NME6JVYTAVGRCNNFFSFRVWYM4","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"An Auxiliary Space Preconditioner for Fractional Laplacian of Negative Order","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.NA"],"primary_cat":"math.NA","authors_text":"Trygve B{\\ae}rland","submitted_at":"2019-08-13T05:49:53Z","abstract_excerpt":"Coupled multiphysics problems often give rise to interface conditions naturally formulated in fractional Sobolev spaces. Here, both positive and negative fractionality are common. When designing efficient solvers for discretizations of such problems it would then be useful to have a preconditioner for the fractional Laplacian, $(-\\Delta)^s$, with $s \\in [-1,1]$. Previously, additive multigrid preconditioners for the case when $s \\geq 0$ have been proposed. In this work we complement this construction with auxiliary space preconditioners suitable when $s \\leq 0$. These preconditioners are shown"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.04498","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1908.04498/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":"2026-07-04T23:55:14Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"nEzm/ApWXCJtXpYL3G7hxDhwxef3wcrO1+GeWwkN/+93dd9es37lD1qFTAW/rBMG49AY0melcO+dUkKWoYIwAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-16T05:01:30.157936Z"},"content_sha256":"e8055412f2bcb7e8556e88f6206441116b3b1cfee00367b00a206a10cf7dbf73","schema_version":"1.0","event_id":"sha256:e8055412f2bcb7e8556e88f6206441116b3b1cfee00367b00a206a10cf7dbf73"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/6NME6JVYTAVGRCNNFFSFRVWYM4/bundle.json","state_url":"https://pith.science/pith/6NME6JVYTAVGRCNNFFSFRVWYM4/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/6NME6JVYTAVGRCNNFFSFRVWYM4/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":"2026-08-16T05:01:30Z","links":{"resolver":"https://pith.science/pith/6NME6JVYTAVGRCNNFFSFRVWYM4","bundle":"https://pith.science/pith/6NME6JVYTAVGRCNNFFSFRVWYM4/bundle.json","state":"https://pith.science/pith/6NME6JVYTAVGRCNNFFSFRVWYM4/state.json","well_known_bundle":"https://pith.science/.well-known/pith/6NME6JVYTAVGRCNNFFSFRVWYM4/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:6NME6JVYTAVGRCNNFFSFRVWYM4","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":"d20f55c54d58150aed16aedcaf08946417965b2a28b4552383a57661614759fb","cross_cats_sorted":["cs.NA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2019-08-13T05:49:53Z","title_canon_sha256":"d19118526ad8d8e8c1063b75c1873bac4c6cb4426247ace59ee69105d89f5765"},"schema_version":"1.0","source":{"id":"1908.04498","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1908.04498","created_at":"2026-07-04T23:55:14Z"},{"alias_kind":"arxiv_version","alias_value":"1908.04498v1","created_at":"2026-07-04T23:55:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1908.04498","created_at":"2026-07-04T23:55:14Z"},{"alias_kind":"pith_short_12","alias_value":"6NME6JVYTAVG","created_at":"2026-07-04T23:55:14Z"},{"alias_kind":"pith_short_16","alias_value":"6NME6JVYTAVGRCNN","created_at":"2026-07-04T23:55:14Z"},{"alias_kind":"pith_short_8","alias_value":"6NME6JVY","created_at":"2026-07-04T23:55:14Z"}],"graph_snapshots":[{"event_id":"sha256:e8055412f2bcb7e8556e88f6206441116b3b1cfee00367b00a206a10cf7dbf73","target":"graph","created_at":"2026-07-04T23:55:14Z","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/1908.04498/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Coupled multiphysics problems often give rise to interface conditions naturally formulated in fractional Sobolev spaces. Here, both positive and negative fractionality are common. When designing efficient solvers for discretizations of such problems it would then be useful to have a preconditioner for the fractional Laplacian, $(-\\Delta)^s$, with $s \\in [-1,1]$. Previously, additive multigrid preconditioners for the case when $s \\geq 0$ have been proposed. In this work we complement this construction with auxiliary space preconditioners suitable when $s \\leq 0$. These preconditioners are shown","authors_text":"Trygve B{\\ae}rland","cross_cats":["cs.NA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2019-08-13T05:49:53Z","title":"An Auxiliary Space Preconditioner for Fractional Laplacian of Negative Order"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.04498","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":"sha256:0716e9ff3c2885d2ae10118874c3ede1bbe769319b93d98e4ca4883a549d2639","target":"record","created_at":"2026-07-04T23:55:14Z","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":"d20f55c54d58150aed16aedcaf08946417965b2a28b4552383a57661614759fb","cross_cats_sorted":["cs.NA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2019-08-13T05:49:53Z","title_canon_sha256":"d19118526ad8d8e8c1063b75c1873bac4c6cb4426247ace59ee69105d89f5765"},"schema_version":"1.0","source":{"id":"1908.04498","kind":"arxiv","version":1}},"canonical_sha256":"f3584f26b8982a6889ad296458d6d8670692454e74ac2d9add03c68002defe11","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f3584f26b8982a6889ad296458d6d8670692454e74ac2d9add03c68002defe11","first_computed_at":"2026-07-04T23:55:14.176976Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-04T23:55:14.176976Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"kzkhBKPtIm3nHTQfRjL/V698ORYuJuqv6fFy7hNAeyPIo3iik3TaWSiPEnGaD3MuE2AmJVYuVOichZ32zASFCQ==","signature_status":"signed_v1","signed_at":"2026-07-04T23:55:14.177432Z","signed_message":"canonical_sha256_bytes"},"source_id":"1908.04498","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0716e9ff3c2885d2ae10118874c3ede1bbe769319b93d98e4ca4883a549d2639","sha256:e8055412f2bcb7e8556e88f6206441116b3b1cfee00367b00a206a10cf7dbf73"],"state_sha256":"1614d6d79ba448975880b79742571d2e79411175cffcb5ca550b370ba4b5ebec"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"+G7LYXn1JcFxjPOBKTdk94W1fE8GjBejBU0fOstpyz8Bmepuem7u2n71l4cYB7/XPPdVTCZbFYg47SJx1sTbCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-16T05:01:30.165385Z","bundle_sha256":"268043718cd6a179ee1815847e4ebc56191bfa8d64475de28cef267585bef40e"}}