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Improved efficiency of a multi-index FEM for computational uncertainty quantification

Dick, Josef; Feischl, Michael; Schwab, Christoph

We propose a multi-index algorithm for the Monte Carlo discretization of a linear, elliptic PDE with affine-parametric input. We prove an error vs. work analysis which allows a multi-level finite-element approximation in the physical domain, and apply the multi-index analysis with isotropic, unstructured mesh refinement in the physical domain for the solution of the forward problem, for the approximation of the random field, and for the Monte-Carlo quadrature error. Our approach allows general spatial domains and unstructured mesh hierarchies. The improvement in complexity is obtained from combining spacial discretization, dimension truncation and MC sampling in a multi-index fashion. Our analysis improves cost estimates compared to multi-level algorithms for similar problems and mathematically underpins the outstanding practical performance of multi-index algorithms for partial differential equations with random coefficients.

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Volltext §
DOI: 10.5445/IR/1000086907
Veröffentlicht am 24.10.2018
Cover der Publikation
Zugehörige Institution(en) am KIT Institut für Angewandte und Numerische Mathematik (IANM)
Sonderforschungsbereich 1173 (SFB 1173)
Publikationstyp Forschungsbericht/Preprint
Publikationsjahr 2018
Sprache Englisch
Identifikator ISSN: 2365-662X
KITopen-ID: 1000086907
Verlag KIT, Karlsruhe
Umfang 20 S.
Serie CRC 1173 ; 2018/22
Schlagwörter multi-index, Monte Carlo, high-dimensional, uncertainty quantification
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