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

Thomas Boiveau (1, 2) , Virginie Ehrlacher (1, 3) , Alexandre Ern (1, 2) , Anthony Nouy (4, 5)

1 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique
2 SERENA - Simulation for the Environment: Reliable and Efficient Numerical Algorithms
3 MATHERIALS - MATHematics for MatERIALS
4 LMJL - Laboratoire de Mathématiques Jean Leray
5 ECN - École Centrale de Nantes

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.

Mots clés: Parabolic equations; Tensor methods; Proper Generalised Decomposition; Greedy algorithm

Domaines: Analyse numérique [math.NA]

Thomas Boiveau, Virginie Ehrlacher, Alexandre Ern, Anthony Nouy. Low-rank approximation of linear parabolic equations by space-time tensor Galerkin methods. 2017. ⟨hal-01668316v1⟩