Volume 53, Number 2, March-April 2019
|Page(s)||635 - 658|
|Published online||01 May 2019|
Low-rank approximation of linear parabolic equations by space-time tensor Galerkin methods⋆
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: email@example.com
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.
Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.