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Multilevel tensor approximation of PDEs with random data

Ballani, Jonas and Kressner, Daniel and Peters, Michael. (2016) Multilevel tensor approximation of PDEs with random data. Preprints Fachbereich Mathematik, 2016 (12).

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Official URL: https://edoc.unibas.ch/69964/

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Abstract

In this paper, we introduce and analyze a new low-rank multilevel strategy for the solution of random diffusion problems. Using a standard stochastic collocation scheme, we first approximate the infinite dimensional random problem by a deterministic parameter-dependent problem on a high-dimensional parameter domain. Given a hierarchy of finite element discretizations for the spatial approximation, we make use of a multilevel framework in which we consider the differences of the solution on two consecutive finite element levels in the collocation points. We then address the approximation of these high-dimensional differences by adaptive low-rank tensor techniques. This allows to equilibrate the error on all levels by exploiting analytic and algebraic properties of the solution at the same time. We arrive at an explicit representation in a low-rank tensor format of the approximate solution on the entire parameter domain, which can be used for, e.g., the direct and cheap computation of statistics. Numerical results are provided in order to illustrate the approach.
Faculties and Departments:05 Faculty of Science > Departement Mathematik und Informatik > Mathematik > Computational Mathematics (Harbrecht)
12 Special Collections > Preprints Fachbereich Mathematik
UniBasel Contributors:Peters, Michael
Item Type:Preprint
Publisher:Universität Basel
Language:English
Last Modified:20 Apr 2019 21:52
Deposited On:28 Mar 2019 09:51

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