{"paper":{"title":"A Two Dimensional Backward Heat Problem With Statistical Discrete Data","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Dang Duc Trong, Nguyen Dang Minh, Nguyen Huy Tuan, To Duc Khanh","submitted_at":"2016-06-17T09:29:53Z","abstract_excerpt":"In this paper, we focus on the backward heat problem of finding the function $\\theta(x,y)=u(x,y,0)$ such that \\[ {l l l} u_t - a(t)(u_{xx} + u_{yy}) & = f(x,y,t), & \\qquad (x,y,t) \\in \\Omega\\times (0,T), u(x,y,T) & = h(x,y), & \\qquad (x,y) \\in\\bar{\\Omega}. \\] where $\\Omega = (0,\\pi) \\times (0,\\pi)$ and the heat transfer coefficient $a(t)$ is known. In our problem, the source $f = f(x,y,t)$ and the final data $h(x,y)$ are unknown. We only know random noise data $g_{ij}(t)$ and $d_{ij}$ satisfying the regression models g_{ij}(t) &=& f(x_i,y_j,t) + \\vartheta\\xi_{ij}(t), d_{ij} &=& h(x_i,y_j) + \\s"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1606.05463","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"}