A parallel and streaming Dynamic Mode Decomposition algorithm with finite precision error analysis for large data

TL;DR

This paper introduces a parallel, streaming Dynamic Mode Decomposition algorithm based on a modified Full Orthogonalization Arnoldi method, optimized for large datasets and finite precision error control, with theoretical and numerical validation.

Contribution

A novel FOA-based DMD algorithm that avoids SVD, supports streaming data, reduces memory and computation, and includes finite precision error analysis for large-scale applications.

Findings

Finite precision error is proportional to $\epsilon_m\kappa_2(X)$.

Algorithm requires only one snapshot at a time, enabling streaming.

Compared to existing methods, it reduces computational cost and memory usage.

Abstract

A novel technique based on the Full Orthogonalization Arnoldi (FOA) is proposed to perform Dynamic Mode Decomposition (DMD) for a sequence of snapshots. A modification to FOA is presented for situations where the matrix $A$ is unknown, but the set of vectors $\{A^{i-1}v_1\}_{i=1}^{N-1}$ are known. The modified FOA is the kernel for the proposed projected DMD algorithm termed, FOA based DMD. The proposed algorithm to compute DMD modes and eigenvalues i) does not require Singular Value Decomposition (SVD) for snapshot matrices $X$ with $\kappa_2(X) \ll 1/\epsilon_m$, where $\kappa_2(X)$ is the 2-norm condition number of the snapshot matrix and $\epsilon_m$ is the relative round-off error or machine epsilon, ii) has an optional rank truncation step motivated by round off error analysis for snapshot matrices $X$ with $\kappa_2(X) \approx 1/\epsilon_m$, iii) requires only one snapshot at a time, thus making it a 'streaming' method even with the optional rank truncation step, iv) consumes less memory and requires less floating point operations to obtain the projected matrix than existing projected DMD methods and v) lends itself to easy parallelism as the main computational kernel involves only vector additions, dot products and matrix vector products. The new technique is therefore well-suited for DMD of large datasets on parallel computing platforms. We show both theoretically and using numerical examples that for FOA based DMD without rank truncation, the finite precision error in the computed projection of the linear mapping is $O(\epsilon_m\kappa_2(X))$. The proposed method is also compared to existing projected DMD methods for computational cost, memory consumption and relative round off error. Error indicators are presented that are useful to decide when to stop acquiring new snapshots. The proposed method is applied to several examples of numerical simulations of fluid flow.