Transport reversal for model reduction of hyperbolic partial differential equations

TL;DR

This paper introduces an iterative method that decomposes snapshot matrices of hyperbolic PDE solutions into shifting profiles to improve model reduction, addressing slow singular value decay issues.

Contribution

It extends symmetry reduction techniques to hyperbolic PDEs by proposing a novel shift-based decomposition algorithm with geometric interpretation.

Findings

Effective in numerical examples for hyperbolic problems

Improves singular value decay for better model reduction

Flexible extensions to shift operators considered

Abstract

Snapshot matrices built from solutions to hyperbolic partial differential equations exhibit slow decay in singular values, whereas fast decay is crucial for the success of projection- based model reduction methods. To overcome this problem, we build on previous work in symmetry reduction [Rowley and Marsden, Physica D (2000), pp. 1-19] and propose an iterative algorithm that decomposes the snapshot matrix into multiple shifting profiles, each with a corresponding speed. Its applicability to typical hyperbolic problems is demonstrated through numerical examples, and other natural extensions that modify the shift operator are considered. Finally, we give a geometric interpretation of the algorithm.