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We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint
37/2010

Quantics-TT collocation approximation of parameter-dependent and stochastic elliptic PDEs

Boris N. Khoromskij and Ivan V. Oseledets

Abstract

We investigate the convergence rate of QTT stochastic collocation tensor approximations to solutions of multi-parametric 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 parametric, 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. Various numerical tests indicate that the numerical complexity scales almost linearly in the dimension of parametric space, and the adaptive preconditioner provides robust convergence in both additive and log-additive cases.

Received:
Jul 28, 2010
Published:
Jul 30, 2010
MSC Codes:
65F30, 65F50, 65N35
Keywords:
stochastic elliptic PDEs, separable approximation, quintics-TT tensors, preconditioners, tensor-truncated iteration, quantics-TT tensors

Related publications

inJournal
2010 Journal Open Access
Boris N. Khoromskij and Ivan V. Oseledets

Quantics-TT collocation approximation of parameter-dependent and stochastic elliptic PDEs

In: Computational methods in applied mathematics, 10 (2010) 4, pp. 376-394