{"paper":{"title":"Numerical Approximation of Fractional Powers of Elliptic Operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.NA","authors_text":"Andrea Bonito, Joseph E. Pasciak","submitted_at":"2013-07-03T00:49:09Z","abstract_excerpt":"We present and study a novel numerical algorithm to approximate the action of $T^\\beta:=L^{-\\beta}$ where $L$ is a symmetric and positive definite unbounded operator on a Hilbert space $H_0$. The numerical method is based on a representation formula for $T^{-\\beta}$ in terms of Bochner integrals involving $(I+t^2L)^{-1}$ for $t\\in(0,\\infty)$.\n  To develop an approximation to $T^\\beta$, we introduce a finite element approximation $L_h$ to $L$ and base our approximation to $T^\\beta$ on $T_h^\\beta:= L_h^{-\\beta}$. The direct evaluation of $T_h^{\\beta}$ is extremely expensive as it involves expans"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1307.0888","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":""},"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"}