{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:V26Q4KKRBKUMIHGOXPXUV7SJR6","short_pith_number":"pith:V26Q4KKR","schema_version":"1.0","canonical_sha256":"aebd0e29510aa8c41ccebbef4afe498fbdacbf6d5337408004e69069a9485b58","source":{"kind":"arxiv","id":"1507.04782","version":1},"attestation_state":"computed","paper":{"title":"Fast Convergence of an Inertial Gradient-like System with Vanishing Viscosity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.OC","authors_text":"Hedy Attouch, Juan Peypouquet, Patrick Redont","submitted_at":"2015-07-16T22:01:11Z","abstract_excerpt":"In a real Hilbert space $\\mathcal H$, we study the fast convergence properties as $t \\to + \\infty$ of the trajectories of the second-order evolution equation $$ \\ddot{x}(t) + \\frac{\\alpha}{t} \\dot{x}(t) + \\nabla \\Phi (x(t)) = 0, $$ where $\\nabla \\Phi$ is the gradient of a convex continuously differentiable function $\\Phi : \\mathcal H \\rightarrow \\mathbb R$, and $\\alpha$ is a positive parameter. In this inertial system, the viscous damping coefficient $\\frac{\\alpha}{t}$ vanishes asymptotically in a moderate way. For $\\alpha > 3$, we show that any trajectory converges weakly to a minimizer of $\\"},"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":"1507.04782","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2015-07-16T22:01:11Z","cross_cats_sorted":[],"title_canon_sha256":"70ff11335ea7522dbbe6e9dbbcad1dfa04fdae58d640b8a538966964e372194d","abstract_canon_sha256":"f5a74e03f0ae78489b13d690ebf14d473539eaf8ae67144a3ba3eab6a44b3bdf"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:36:42.842963Z","signature_b64":"uYPAqEETP1ykb7CsVtd2zzmPlmnFI0WHYaKiFciS4aB8mO3UOsf2Q0ZiTeQrD9HroBfLUwxQOgG7BoqI7HwxBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"aebd0e29510aa8c41ccebbef4afe498fbdacbf6d5337408004e69069a9485b58","last_reissued_at":"2026-05-18T01:36:42.842263Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:36:42.842263Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Fast Convergence of an Inertial Gradient-like System with Vanishing Viscosity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.OC","authors_text":"Hedy Attouch, Juan Peypouquet, Patrick Redont","submitted_at":"2015-07-16T22:01:11Z","abstract_excerpt":"In a real Hilbert space $\\mathcal H$, we study the fast convergence properties as $t \\to + \\infty$ of the trajectories of the second-order evolution equation $$ \\ddot{x}(t) + \\frac{\\alpha}{t} \\dot{x}(t) + \\nabla \\Phi (x(t)) = 0, $$ where $\\nabla \\Phi$ is the gradient of a convex continuously differentiable function $\\Phi : \\mathcal H \\rightarrow \\mathbb R$, and $\\alpha$ is a positive parameter. In this inertial system, the viscous damping coefficient $\\frac{\\alpha}{t}$ vanishes asymptotically in a moderate way. For $\\alpha > 3$, we show that any trajectory converges weakly to a minimizer of $\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.04782","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":"1507.04782","created_at":"2026-05-18T01:36:42.842396+00:00"},{"alias_kind":"arxiv_version","alias_value":"1507.04782v1","created_at":"2026-05-18T01:36:42.842396+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1507.04782","created_at":"2026-05-18T01:36:42.842396+00:00"},{"alias_kind":"pith_short_12","alias_value":"V26Q4KKRBKUM","created_at":"2026-05-18T12:29:44.643036+00:00"},{"alias_kind":"pith_short_16","alias_value":"V26Q4KKRBKUMIHGO","created_at":"2026-05-18T12:29:44.643036+00:00"},{"alias_kind":"pith_short_8","alias_value":"V26Q4KKR","created_at":"2026-05-18T12:29:44.643036+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":0,"sample":[{"citing_arxiv_id":"2604.06651","citing_title":"Nesterov Flow May Travel Infinitely Long to Converge to a Minimizer","ref_index":3,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/V26Q4KKRBKUMIHGOXPXUV7SJR6","json":"https://pith.science/pith/V26Q4KKRBKUMIHGOXPXUV7SJR6.json","graph_json":"https://pith.science/api/pith-number/V26Q4KKRBKUMIHGOXPXUV7SJR6/graph.json","events_json":"https://pith.science/api/pith-number/V26Q4KKRBKUMIHGOXPXUV7SJR6/events.json","paper":"https://pith.science/paper/V26Q4KKR"},"agent_actions":{"view_html":"https://pith.science/pith/V26Q4KKRBKUMIHGOXPXUV7SJR6","download_json":"https://pith.science/pith/V26Q4KKRBKUMIHGOXPXUV7SJR6.json","view_paper":"https://pith.science/paper/V26Q4KKR","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1507.04782&json=true","fetch_graph":"https://pith.science/api/pith-number/V26Q4KKRBKUMIHGOXPXUV7SJR6/graph.json","fetch_events":"https://pith.science/api/pith-number/V26Q4KKRBKUMIHGOXPXUV7SJR6/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/V26Q4KKRBKUMIHGOXPXUV7SJR6/action/timestamp_anchor","attest_storage":"https://pith.science/pith/V26Q4KKRBKUMIHGOXPXUV7SJR6/action/storage_attestation","attest_author":"https://pith.science/pith/V26Q4KKRBKUMIHGOXPXUV7SJR6/action/author_attestation","sign_citation":"https://pith.science/pith/V26Q4KKRBKUMIHGOXPXUV7SJR6/action/citation_signature","submit_replication":"https://pith.science/pith/V26Q4KKRBKUMIHGOXPXUV7SJR6/action/replication_record"}},"created_at":"2026-05-18T01:36:42.842396+00:00","updated_at":"2026-05-18T01:36:42.842396+00:00"}