Low-rank approximation of linear parabolic equations by space-time tensor Galerkin methods
This work provides a novel numerical method for efficiently solving parabolic evolution problems, which is important for computational scientists and engineers dealing with time-dependent PDEs.
The paper introduces a space-time tensor Galerkin method for low-rank approximation of linear parabolic equations, achieving stable and accurate solutions with reduced computational cost compared to standard methods.
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.