Dynamic Mode Decomposition in Adaptive Mesh Refinement and Coarsening Simulations

TL;DR

This paper extends Dynamic Mode Decomposition (DMD) to handle adaptive mesh refinement/coarsening simulations by projecting adaptive snapshots onto a common reference space, enabling analysis of complex, variable-resolution data.

Contribution

It introduces a novel strategy to apply DMD to simulations with different mesh topologies, allowing extraction of coherent structures from adaptive mesh data.

Findings

Successfully applied to epidemiological, fluid dynamics, and bubble rising simulations.

Demonstrated DMD's ability to reconstruct and extrapolate dynamics.

Validated the method's effectiveness in complex adaptive mesh scenarios.

Abstract

Dynamic Mode Decomposition (DMD) is a powerful data-driven method used to extract spatio-temporal coherent structures that dictate a given dynamical system. The method consists of stacking collected temporal snapshots into a matrix and mapping the nonlinear dynamics using a linear operator. The standard procedure considers that snapshots possess the same dimensionality for all the observable data. However, this often does not occur in numerical simulations with adaptive mesh refinement/coarsening schemes (AMR/C). This paper proposes a strategy to enable DMD to extract features from observations with different mesh topologies and dimensions, such as those found in AMR/C simulations. For this purpose, the adaptive snapshots are projected onto the same reference function space, enabling the use of snapshot-based methods such as DMD. The present strategy is applied to challenging AMR/C simulations: a continuous diffusion-reaction epidemiological model for COVID-19, a density-driven gravity current simulation, and a bubble rising problem. We also evaluate the DMD efficiency to reconstruct the dynamics and some relevant quantities of interest. In particular, for the SEIRD model and the bubble rising problem, we evaluate DMD's ability to extrapolate in time (short-time future estimates).