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arXiv · 1712.07256

Low-rank approximation of linear parabolic equations by space-time tensor Galerkin methods

Thomas Boiveau·Virginie Ehrlacher·Alexandre Ern·Anthony Nouy
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Abstract

We devise a space-time tensor method for the low-rank approximation of linear parabolic evolution equations. The proposed method is a stable Galerkin method, uniformly 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 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 performances 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.

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Thomas Boiveau, Virginie Ehrlacher, Alexandre Ern, Anthony Nouy. 2017-12-19. Low-rank approximation of linear parabolic equations by space-time tensor Galerkin methods. https://doi.org/10.1051/m2an/2018073

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