Piecewise DMD for oscillatory and Turing spatio-temporal dynamics

TL;DR

This paper introduces a piecewise DMD method to improve the approximation of oscillatory and Turing pattern datasets, demonstrating its effectiveness on reaction-diffusion systems.

Contribution

The paper proposes a novel piecewise DMD approach that enhances accuracy for oscillatory and Turing dynamics by partitioning datasets and applying DMD locally.

Findings

pDMD achieves highly accurate reconstructions of reaction-diffusion datasets.

Numerical experiments validate pDMD's effectiveness on models like FitzHugh-Nagumo and lambda-omega.

Error indicators help evaluate performance as dataset partitioning increases.

Abstract

Dynamic Mode Decomposition (DMD) is an equation-free method that aims at reconstructing the best linear fit from temporal datasets. In this paper, we show that DMD does not provide accurate approximation for datasets describing oscillatory dynamics, like spiral waves and relaxation oscillations, or spatio-temporal Turing instability. Inspired from the classical "divide and conquer" approach, we propose a piecewise version of DMD (pDMD) to overcome this problem. The main idea is to split the original dataset in N submatrices and then apply the exact (randomized) DMD method in each subset of the obtained partition. We describe the pDMD algorithm in detail and we introduce some error indicators to evaluate its performance when N is increased. Numerical experiments show that very accurate reconstructions are obtained by pDMD for datasets arising from time snapshots of some reaction-diffusion PDE systems, like the FitzHugh-Nagumo model, the lambda-omega system and the DIB morpho-chemical system for battery modeling.