Dynamic mode decomposition (DMD) is an efficient tool for decomposing
spatio-temporal data into a set of low-dimensional modes, yielding the
oscillation frequencies and the growth rates of physically significant modes.
In this paper, we propose a novel DMD model that can be used for dynamical
systems affected by multiplicative noise. We first derive a maximum a
posteriori (MAP) estimator for the data-based model decomposition of a linear
dynamical system corrupted by certain multiplicative noise. Applying penalty
relaxation to the MAP estimator, we obtain the proposed DMD model whose
epigraphical limits are the MAP estimator and the conventional optimized DMD
model. We also propose an efficient alternating gradient descent method for
solving the proposed DMD model, and analyze its convergence behavior. The
proposed model is demonstrated on both the synthetic data and the numerically
generated one-dimensional combustor data, and is shown to have superior
reconstruction properties compared to state-of-the-art DMD models. Considering
that multiplicative noise is ubiquitous in numerous dynamical systems, the
proposed DMD model opens up new possibilities for accurate data-based modal
decomposition.Comment: 35 pages, 10 figure