Matrix oriented reduction of space-time Petrov-Galerkin variational problems

TL;DR

This paper introduces matrix-oriented reduction techniques for space-time Petrov-Galerkin variational problems, significantly improving computational efficiency in solving high-dimensional PDEs by enabling adaptive and reduced models.

Contribution

It presents a novel matrix-oriented approach that reduces computational time for space-time variational problems, outperforming traditional time-stepping and solver methods.

Findings

Matrix-oriented techniques significantly reduce computational timings.

The approach outperforms traditional time-stepping schemes.

Enables efficient adaptivity and model reduction in space-time PDE solutions.

Abstract

Variational formulations of time-dependent PDEs in space and time yield $(d+1)$-dimensional problems to be solved numerically. This increases the number of unknowns as well as the storage amount. On the other hand, this approach enables adaptivity in space and time as well as model reduction w.r.t. both type of variables. In this paper, we show that matrix oriented techniques can significantly reduce the computational timings for solving the arising linear systems outperforming both time-stepping schemes and other solvers.