Model order reduction of parametric dynamical systems by slice sampling tensor completion

TL;DR

This paper introduces a novel tensor completion method using slice sampling and hybrid tensor train format to enable efficient reduced order modeling of parametric dynamical systems with many parameters.

Contribution

It proposes a low-rank tensor completion approach with slice sampling for high-dimensional parameter spaces in tensor-based reduced order models.

Findings

Effective tensor completion with slice sampling demonstrated on PDE systems.

Reduced computational cost for high-dimensional parametric systems.

Improved accuracy of reduced order models with sparse sampling.

Abstract

Recent studies have demonstrated the great potential of reduced order modeling for parametric dynamical systems using low-rank tensor decompositions (LRTD). In particular, within the framework of interpolatory tensorial reduced order models (ROM), LRTD is computed for tensors composed of snapshots of the system's solutions, where each parameter corresponds to a distinct tensor mode. This approach requires full sampling of the parameter domain on a tensor product grid, which suffers from the curse of dimensionality, making it practical only for systems with a small number of parameters. To overcome this limitation, we propose a sparse sampling of the parameter domain, followed by a low-rank tensor completion. The resulting specialized tensor completion problem is formulated for a tensor of order $C + D$, where $C$ fully sampled modes correspond to the snapshot degrees of freedom, and $D$ partially sampled modes correspond to the system's parameters. To address this non-standard tensor completion problem, we introduce a low-rank tensor format called the hybrid tensor train. Completion in this format is then integrated into an interpolatory tensorial ROM. We demonstrate the effectiveness of both the completion method and the ROM on several examples of dynamical systems derived from finite element discretizations of parabolic partial differential equations with parameter-dependent coefficients or boundary conditions.