De-Biasing the Dynamic Mode Decomposition for Applied Koopman Spectral Analysis

TL;DR

This paper introduces a noise-aware total DMD (TDMD) method that corrects systematic bias in standard DMD caused by snapshot errors, improving the accuracy of Koopman spectral analysis in fluid flow data.

Contribution

The paper proposes a novel total DMD approach that removes bias errors by incorporating an augmented snapshot matrix, enhancing robustness against measurement noise.

Findings

TDMD reduces bias errors in DMD when snapshot data are noisy.

Numerical and experimental fluid flow examples demonstrate improved accuracy with TDMD.

TDMD generalizes to other subspace-based methods for bias correction.

Abstract

The Dynamic Mode Decomposition (DMD)---a popular method for performing data-driven Koopman spectral analysis---has gained increased adoption as a technique for extracting dynamically meaningful spatio-temporal descriptions of fluid flows from snapshot measurements. Often times, DMD descriptions can be used for predictive purposes as well, which enables informed decision-making based on DMD model-forecasts. Despite its widespread use and utility, DMD regularly fails to yield accurate dynamical descriptions when the measured snapshot data are imprecise due to, e.g., sensor noise. Here, we express DMD as a two-stage algorithm in order to isolate a source of systematic error. We show that DMD's first stage, a subspace projection step, systematically introduces bias errors by processing snapshots asymmetrically. To remove this systematic error, we propose utilizing an augmented snapshot matrix in a subspace projection step, as in problems of total least-squares, in order to account for the error present in all snapshots. The resulting unbiased and noise-aware total DMD (TDMD) formulation reduces to standard DMD in the absence of snapshot errors, while the two-stage perspective generalizes the de-biasing framework to other related methods as well. TDMD's performance is demonstrated in numerical and experimental fluids examples.