Dynamic mode decomposition with core sketch

Abstract

With the increase in collected data volumes, either from experimental measurements or high fidelity simulations, there is an ever-growing need to develop computationally efficient tools to process, analyze, and interpret these data sets. Modal analysis techniques have gained great interest due to their ability to identify patterns in the data and extract valuable information about the system being considered. Dynamic mode decomposition (DMD) relies on elements of Koopman approximation theory to compute a set of modes, each associated with a fixed oscillation frequency and decay/growth rate. Extracting these details from large data sets can be computationally expensive due to the need to implement singular value decomposition of the input data matrix. Sketching algorithms have become popular in numerical linear algebra where statistical theoretic approaches are utilized to reduce the cost of major operations. A sketch of a matrix is another matrix which is significantly smaller, but still sufficiently approximates the original system. We put forth an efficient DMD framework, SketchyDMD, based on a core sketching algorithm that captures information about the range and co-range (an their mutual relationship) of input data. The proposed sketching-based framework can accelerate various portions of the DMD routines, compared to classical methods that operate directly on the raw input data. We conduct numerical experiments using the spherical shallow water equations as a prototypical model in the context of geophysical flows. We show that the proposed SketchyDMD is superior to existing randomized DMD methods that are based on capturing only the range of the input data.

Residual dynamic mode decomposition: robust and verified Koopmanism

“…One can also simultaneously sketch the range and corange of X. This method, known as sketchy DMD, was proposed by Ahmed et al (2022).…”