Multilinear POD-DEIM model reduction for 2D and 3D semilinear systems of differential equations

TL;DR

This paper introduces a multilinear POD-DEIM model reduction approach for efficiently solving 2D and 3D semilinear PDE systems by exploiting Kronecker structures and tensor representations, improving computational performance.

Contribution

It develops a novel multilinear model reduction strategy directly applied to matrix and tensor ODE systems, enhancing efficiency for high-dimensional PDEs.

Findings

Significant reduction in computational time compared to existing methods

Effective handling of tensor-valued linear systems at each timestep

Validated on benchmark problems like 2D and 3D Burgers equations

Abstract

We are interested in the numerical solution of coupled nonlinear partial differential equations (PDEs) in two and three dimensions. Under certain assumptions on the domain, we take advantage of the Kronecker structure arising in standard space discretizations of the differential operators and illustrate how the resulting system of ordinary differential equations (ODEs) can be treated directly in matrix or tensor form. Moreover, in the framework of the proper orthogonal decomposition (POD) and the discrete empirical interpolation method (DEIM) we derive a two- and three-sided model order reduction strategy that is applied directly to the ODE system in matrix and tensor form respectively. We discuss how to integrate the reduced order model and, in particular, how to solve the tensor-valued linear system arising at each timestep of a semi-implicit time discretization scheme. We illustrate the efficiency of the proposed method through a comparison to existing techniques on classical benchmark problems such as the two- and three-dimensional Burgers equation.