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Computational Methods in Applied Mathematics

Editor-in-Chief: Carstensen, Carsten

Managing Editor: Matus, Piotr

4 Issues per year


IMPACT FACTOR 2016: 1.097

CiteScore 2016: 1.09

SCImago Journal Rank (SJR) 2016: 0.872
Source Normalized Impact per Paper (SNIP) 2016: 0.887

Mathematical Citation Quotient (MCQ) 2016: 0.75

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1609-9389
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Volume 10, Issue 4

Issues

Quantics-TT Collocation Approximation of Parameter-Dependent and Stochastic Elliptic PDEs

B.N. Khoromskij
I. Oseledets

Abstract

We investigate the convergence rate of the quantics-TT (QTT) stochas- tic collocation tensor approximations to solutions of multiparametric elliptic PDEs and construct efficient iterative methods for solving arising high-dimensional parameter- dependent algebraic systems of equations. Such PDEs arise, for example, in the para- metric, deterministic reformulation of elliptic PDEs with random field inputs, based, for example, on the M-term truncated Karhunen-Loève expansion. We consider both the case of additive and log-additive dependence on the multivariate parameter. The local-global versions of the QTT-rank estimates for the system matrix in terms of the parameter space dimension is proven. Similar rank bounds are observed in numerics for the solutions of the discrete linear system. We propose QTT-truncated iteration based on the construction of solution-adaptive preconditioner that provides robust conver- gence in both additive and log-additive cases. Various numerical tests indicate that the numerical complexity scales almost linearly in the dimension of parametric space M.

Keywords: elliptic equations; stochastic PDEs; the Karhunen-Loève expansion; separable approximation; quantics-TT tensors; preconditioners; tensor-truncated iteration

About the article

Received: 2010-10-02

Revised: 2010-10-17

Accepted: 2010-11-10

Published in Print:


Citation Information: Computational Methods in Applied Mathematics Comput. Methods Appl. Math., Volume 10, Issue 4, Pages 376–394, ISSN (Online) 1609-9389, ISSN (Print) 1609-4840, DOI: https://doi.org/10.2478/cmam-2010-0023.

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© Institute of Mathematics, NAS of Belarus. This article is distributed under the terms of the Creative Commons Attribution Non-Commercial License, which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited. BY-NC-ND 4.0

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