Abstract
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.
Highlights
The goal of this work is to devise a space-time tensor method for the low-rank approximation of the solution to linear parabolic evolution equations
Proper Generalized Decomposition (PGD) approximations based on a discrete MinRes formulation measured in the space-time Euclidean norm of the components in a basis of Xhk have been devised and evaluated numerically in [30], obtaining promising results on various model parabolic evolution problems
We specify the functional setting for parabolic evolution equations and the MinRes formulation. This formulation is the basis for the discrete MinRes Galerkin formulation devised in Section 3, where one key idea is the use of a space semi-discrete test space to measure the dual norm of the residual
Summary
The goal of this work is to devise a space-time tensor method for the low-rank approximation of the solution to linear parabolic evolution equations. PGD approximations based on a discrete MinRes formulation measured in the space-time Euclidean norm of the components in a basis of Xhk have been devised and evaluated numerically in [30], obtaining promising results on various model parabolic evolution problems. This formulation is the basis for the discrete MinRes Galerkin formulation devised, where one key idea is the use of a space semi-discrete test space to measure the dual norm of the residual.
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