# Randomized Dynamic Mode Decomposition

## TL;DR

This paper introduces a randomized algorithm for efficiently computing low-rank dynamic mode decomposition, enabling scalable analysis of large data sets by approximating the dominant dynamic modes with controlled accuracy.

## Contribution

The paper develops a probabilistic framework for randomized DMD that reduces computational cost and scales with data rank, not dimension, improving efficiency over traditional methods.

## Key findings

- Accurately extracts dynamic modes from large datasets.
- Demonstrates efficiency and scalability on benchmark examples.
- Provides controllable approximation quality via oversampling and power iterations.

## Abstract

This paper presents a randomized algorithm for computing the near-optimal low-rank dynamic mode decomposition (DMD). Randomized algorithms are emerging techniques to compute low-rank matrix approximations at a fraction of the cost of deterministic algorithms, easing the computational challenges arising in the area of `big data'. The idea is to derive a small matrix from the high-dimensional data, which is then used to efficiently compute the dynamic modes and eigenvalues. The algorithm is presented in a modular probabilistic framework, and the approximation quality can be controlled via oversampling and power iterations. The effectiveness of the resulting randomized DMD algorithm is demonstrated on several benchmark examples of increasing complexity, providing an accurate and efficient approach to extract spatiotemporal coherent structures from big data in a framework that scales with the intrinsic rank of the data, rather than the ambient measurement dimension. For this work we assume that the dynamics of the problem under consideration is evolving on a low-dimensional subspace that is well characterized by a fast decaying singular value spectrum.