A priori analysis of a tensor ROM for parameter dependent parabolic problems

TL;DR

This paper provides an a priori error analysis for tensor-based reduced order models of parametric parabolic PDEs, demonstrating uniform error bounds and efficiency improvements over traditional methods.

Contribution

It introduces a rigorous a priori error estimate for tensor Galerkin LRTD-ROMs that is uniform across parameters and extends to unseen parameter values.

Findings

Error estimate depends on discretization, sampling, and tensor accuracy.

The estimate is uniform over parameters, including untrained values.

Numerical experiments validate theoretical results.

Abstract

A space-time-parameters structure of parametric parabolic PDEs motivates the application of tensor methods to define reduced order models (ROMs). Within a tensor-based ROM framework, the matrix SVD - a traditional dimension reduction technique - yields to a low-rank tensor decomposition (LRTD). Such tensor extension of the Galerkin proper orthogonal decomposition ROMs (POD-ROMs) benefits both the practical efficiency of the ROM and its amenability for rigorous error analysis when applied to parametric PDEs. The paper addresses the error analysis of the Galerkin LRTD-ROM for an abstract linear parabolic problem that depends on multiple physical parameters. An error estimate for the LRTD-ROM solution is proved, which is uniform with respect to problem parameters and extends to parameter values not in a sampling/training set. The estimate is given in terms of discretization and sampling mesh properties, and LRTD accuracy. The estimate depends on the local smoothness rather than on the Kolmogorov n-widths of the parameterized manifold of solutions. Theoretical results are illustrated with several numerical experiments.