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Computational Methods in Applied Mathematics

Editor-in-Chief: Carstensen, Carsten

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Volume 19, Issue 1

Issues

Non-intrusive Tensor Reconstruction for High-Dimensional Random PDEs

Martin Eigel / Johannes Neumann / Reinhold Schneider / Sebastian Wolf
Published Online: 2018-07-25 | DOI: https://doi.org/10.1515/cmam-2018-0028

Abstract

This paper examines a completely non-intrusive, sample-based method for the computation of functional low-rank solutions of high-dimensional parametric random PDEs, which have become an area of intensive research in Uncertainty Quantification (UQ). In order to obtain a generalized polynomial chaos representation of the approximate stochastic solution, a novel black-box rank-adapted tensor reconstruction procedure is proposed. The performance of the described approach is illustrated with several numerical examples and compared to (Quasi-)Monte Carlo sampling.

Keywords: Non-intrusive; Tensor Reconstruction; Partial Differential Equations with Random Coefficients; Tensor Representation; Tensor Train; Uncertainty Quantification; Low-Rank

MSC 2010: 35R60; 47B80; 60H35; 65C20; 65N12; 65N22; 65J10

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About the article

Received: 2017-10-18

Revised: 2018-02-08

Accepted: 2018-05-02

Published Online: 2018-07-25

Published in Print: 2019-01-01


Funding Source: Deutsche Forschungsgemeinschaft

Award identifier / Grant number: Matheon project SE13

Award identifier / Grant number: Matheon project SE10

Research of Johannes Neumann was funded in part by the DFG Matheon project SE13. Research of Sebastian Wolf was funded in part by the DFG Matheon project SE10.


Citation Information: Computational Methods in Applied Mathematics, Volume 19, Issue 1, Pages 39–53, ISSN (Online) 1609-9389, ISSN (Print) 1609-4840, DOI: https://doi.org/10.1515/cmam-2018-0028.

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