Model order reduction with novel discrete empirical interpolation methods in space-time

TL;DR

This paper introduces new space-time reduced basis methods with discrete empirical interpolation techniques to efficiently simulate unsteady PDEs, maintaining accuracy and reducing computational costs.

Contribution

It develops novel hyper-reduction techniques tailored for unsteady problems within a space-time reduced basis framework, including error bounds and numerical validation.

Findings

Methods achieve high accuracy compared to high-fidelity simulations

Significant reduction in computational time for unsteady PDEs

Effective handling of nonaffine parameterizations in space-time models

Abstract

This work proposes novel techniques for the efficient numerical simulation of parameterized, unsteady partial differential equations. Projection-based reduced order models (ROMs) such as the reduced basis method employ a (Petrov-)Galerkin projection onto a linear low-dimensional subspace. In unsteady applications, space-time reduced basis (ST-RB) methods have been developed to achieve a dimension reduction both in space and time, eliminating the computational burden of time marching schemes. However, nonaffine parameterizations dilute any computational speedup achievable by traditional ROMs. Computational efficiency can be recovered by linearizing the nonaffine operators via hyper-reduction, such as the empirical interpolation method in matrix form. In this work, we implement new hyper-reduction techniques explicitly tailored to deal with unsteady problems and embed them in a ST-RB framework. For each of the proposed methods, we develop a posteriori error bounds. We run numerical tests to compare the performance of the proposed ROMs against high-fidelity simulations, in which we combine the finite element method for space discretization on 3D geometries and the Backward Euler time integrator. In particular, we consider a heat equation and an unsteady Stokes equation. The numerical experiments demonstrate the accuracy and computational efficiency our methods retain with respect to the high-fidelity simulations.