Interpolatory tensorial reduced order models for parametric dynamical systems

TL;DR

This paper presents a tensor-based reduced order modeling approach for parametric dynamical systems that efficiently captures parameter variations and reduces online computational costs, outperforming traditional methods in high-dimensional settings.

Contribution

It introduces a tensorial ROM framework that leverages compressed tensor formats to efficiently handle high-dimensional parameter spaces and improve online phase performance.

Findings

Tensorial ROM achieves lower online computational costs.

The method effectively manages high-dimensional parameter spaces.

Numerical experiments show improved accuracy over conventional POD-ROM.

Abstract

The paper introduces a reduced order model (ROM) for numerical integration of a dynamical system which depends on multiple parameters. The ROM is a projection of the dynamical system on a low dimensional space that is both problem-dependent and parameter-specific. The ROM exploits compressed tensor formats to find a low rank representation for a sample of high-fidelity snapshots of the system state. This tensorial representation provides ROM with an orthogonal basis in a universal space of all snapshots and encodes information about the state variation in parameter domain. During the online phase and for any incoming parameter, this information is used to find a reduced basis that spans a parameter-specific subspace in the universal space. The computational cost of the online phase then depends only on tensor compression ranks, but not on space or time resolution of high-fidelity computations. Moreover, certain compressed tensor formats enable to avoid the adverse effect of parameter space dimension on the online costs (known as the curse of dimension). The analysis of the approach includes an estimate for the representation power of the acquired ROM basis. We illustrate the performance and prediction properties of the ROM with several numerical experiments, where tensorial ROM's complexity and accuracy is compared to those of conventional POD-ROM.