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arXiv · 2311.07883

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

Alexander V. Mamonov·Maxim A. Olshanskii
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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.

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Alexander V. Mamonov, Maxim A. Olshanskii. 2023-11-14. A priori analysis of a tensor ROM for parameter dependent parabolic problems. https://arxiv.org/abs/2311.07883

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