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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
51/2012

Direct tensor-product solution of one-dimensional elliptic equations with parameter-dependent coefficients

Sergey Dolgov, Vladimir A. Kazeev and Boris N. Khoromskij

Abstract

We consider a one-dimensional second-order elliptic equation with a high-dimensional parameter in a hypercube as a parametric domain. Such a problem arises, for example, from the Karhunen-Loève expansion of a stochastic PDE posed in a one-dimensional physical domain. For the discretization in the parametric domain we use the collocation on a tensor-product grid. The paper is focused on the tensor-structured solution of the resulting multiparametric problem, which allows to avoid the curse of dimensionality owing to the use of the separation of parametric variables in the tensor~train and quantized tensor train formats.

We suggest an efficient tensor-structured preconditioning of the entire multiparametric family of one-dimensional elliptic problems and arrive at a direct solution formula. We compare this method to a tensor-structured preconditioned GMRES solver in a series of numerical experiments.

Received:
Aug 15, 2012
Published:
Aug 16, 2012
MSC Codes:
35J15, 15A69, 65F10, 34B08, 60H15
Keywords:
elliptic equations, parametric problems, iterative methods, Tensor formats, Sherman-Morrison correction, preconditioning

Related publications

inJournal
2018 Repository Open Access
Sergey Dolgov, Vladimir A. Kazeev and Boris N. Khoromskij

Direct tensor-product solution of one-dimensional elliptic equations with parameter-dependent coefficients

In: Mathematics and computers in simulation : transactions of IMACS, 145 (2018), pp. 136-155