A Priori Analysis of a Tensor ROM for Parameter Dependent Parabolic Problems

Abstract

SIAM Journal on Numerical Analysis, Volume 63, Issue 1, Page 239-261, February 2025.
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 [math]-widths of the parameterized manifold of solutions. Theoretical results are illustrated with several numerical experiments.