| Issue |
ESAIM: M2AN
Volume 53, Number 2, March-April 2019
|
|
|---|---|---|
| Page(s) | 635 - 658 | |
| DOI | https://doi.org/10.1051/m2an/2018073 | |
| Published online | 01 May 2019 | |
Low-rank approximation of linear parabolic equations by space-time tensor Galerkin methods⋆
1
Université Paris-Est, CERMICS (ENPC), 77455 Marne-la-Vallée 2, France
Inria Paris, 75589 Paris, France
2
Centrale Nantes, LMJL - UMR CNRS 6629, 1 rue de la Noë, 44321 Nantes, France
* Corresponding author: ern@cermics.enpc.fr
Received:
19
December
2017
Accepted:
29
November
2018
We devise a space-time tensor method for the low-rank approximation of linear parabolic evolution equations. The proposed method is a Galerkin method, uniformly stable in the discretization parameters, based on a Minimal Residual formulation of the evolution problem in Hilbert–Bochner spaces. The discrete solution is sought in a linear trial space composed of tensors of discrete functions in space and in time and is characterized as the unique minimizer of a discrete functional where the dual norm of the residual is evaluated in a space semi-discrete test space. The resulting global space-time linear system is solved iteratively by a greedy algorithm. Numerical results are presented to illustrate the performance of the proposed method on test cases including non-selfadjoint and time-dependent differential operators in space. The results are also compared to those obtained using a fully discrete Petrov–Galerkin setting to evaluate the dual residual norm.
Mathematics Subject Classification: 65M12 / 65M22 / 35K20
Key words: Parabolic equations / tensor methods / proper generalized decomposition / greedy algorithm
© The authors. Published by EDP Sciences, SMAI 2019
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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