{"paper":{"title":"Linear collective collocation and Galerkin approximations for parametric and stochastic elliptic PDEs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NA","authors_text":"Dinh D\\~ung","submitted_at":"2015-11-11T03:46:54Z","abstract_excerpt":"Consider the parametric elliptic problem \\begin{equation} - \\operatorname{dv} \\big(a(y)(x)\\nabla u(y)(x)\\big) \\ = \\ f(x) \\quad x \\in D, \\ y \\in [-1,1]^\\infty, \\quad u|_{\\partial D} \\ = \\ 0, \\end{equation} where $D \\subset {\\mathbb R}^m$ is a bounded Lipschitz domain, $[-1,1]^\\infty$, $f \\in L_2(D)$, and the diffusions $a$ satisfy the uniform ellipticity assumption and are affinely dependent with respect to $y$. The parametric variable $y$ may be deterministic or random. In the present paper, a central question to be studied is as follows. Assume that we have an approximation property that ther"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.03377","kind":"arxiv","version":6},"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"}