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.

翻译：动态模态分解（DMD）是一种无方程方法，旨在从时间数据集中重建最优线性拟合。本文表明，DMD无法为描述振荡型动力学（如螺旋波、松弛振荡）或时空图灵不稳定性的数据集提供精确近似。受经典"分而治之"策略启发，我们提出一种分段DMD方法（pDMD）来解决该问题。核心思想是将原始数据集划分为N个子矩阵，然后对每个子集应用精确（随机化）DMD方法。我们详细描述了pDMD算法，并引入若干误差指标以评估其性能随N增大的表现。数值实验表明，针对反应扩散偏微分方程组（如FitzHugh-Nagumo模型、lambda-omega系统及电池建模中的DIB形态化学系统）的时间快照数据集，pDMD可获得高精度重建结果。