An Adaptive Method Based on Local Dynamic Mode Decomposition for Parametric Dynamical Systems (Communications in Computational Physics, 2024)
Parametric dynamical systems are widely used to model physical systems,but their numerical simulation can be computationally demanding due to nonlinearity,long-time simulation, and multi-query requirements. Model reduction methods aimto reduce computation complexity and improve simulation efficiency. However, traditional model reduction methods are inefficient for parametric dynamical systems withnonlinear structures. To address this challenge, we propose an adaptive method basedon local dynamic mode decomposition (DMD) to construct an efficient and reliablesurrogate model. We propose an improved greedy algorithm to generate the atoms set $Θ$ based on a sequence of relatively small training sets, which could reduce the effect oflarge training set. At each enrichment step, we construct a local sub-surrogate modelusing the Taylor expansion and DMD, resulting in the ability to predict the state at anytime without solving the original dynamical system. Moreover, our method providesthe best approximation almost everywhere over the parameter domain with certainsmoothness assumptions, thanks to the gradient information. At last, three concreteexamples are presented to illustrate the effectiveness of the proposed method.