Piecewise DMD for oscillatory and Turing spatio-temporal dynamics

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, relaxation oscillations and spatio-temporal Turing instability. Inspired by 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 certain reaction-diffusion PDE systems, like the FitzHugh-Nagumo model, a λ-ω system and the DIB morpho-chemical system for battery modeling. Finally, a discussion about the overall computational load and the future prediction features of the new algorithm is also provided.