Sketching Methods for Dynamic Mode Decomposition in Spherical Shallow Water Equations

TL;DR

This paper introduces a sketching-based framework to accelerate dynamic mode decomposition for high-dimensional systems, demonstrated on spherical shallow water equations, achieving significant computational efficiency improvements.

Contribution

The paper develops a novel sketching method to reduce computational costs in DMD, enabling faster analysis of high-dimensional geophysical flow data.

Findings

Sketching methods effectively approximate the data matrix in DMD.

Significant computational speedups achieved compared to classical DMD.

Framework applicable to high-dimensional geophysical models.

Abstract

Dynamic mode decomposition (DMD) is an emerging methodology that has recently attracted computational scientists working on nonintrusive reduced order modeling. One of the major strengths that DMD possesses is having ground theoretical roots from the Koopman approximation theory. Indeed, DMD may be viewed as the data-driven realization of the famous Koopman operator. Nonetheless, the stable implementation of DMD incurs computing the singular value decomposition of the input data matrix. This, in turn, makes the process computationally demanding for high dimensional systems. In order to alleviate this burden, we develop a framework based on sketching methods, wherein a sketch of a matrix is simply another matrix which is significantly smaller, but still sufficiently approximates the original system. Such sketching or embedding is performed by applying random transformations, with certain properties, on the input matrix to yield a compressed version of the initial system. Hence, many of the expensive computations can be carried out on the smaller matrix, thereby accelerating the solution of the original problem. We conduct numerical experiments conducted using the spherical shallow water equations as a prototypical model in the context of geophysical flows. The performance of several sketching approaches is evaluated for capturing the range and co-range of the data matrix. The proposed sketching-based framework can accelerate various portions of the DMD algorithm, compared to classical methods that operate directly on the raw input data. This eventually leads to substantial computational gains that are vital for digital twinning of high dimensional systems.