@Article{CiCP-35-1,
author = {Qiuqi, Li and Liu, Chang and Mengnan, Li and Zhang, Pingwen},
title = {An Adaptive Method Based on Local Dynamic Mode Decomposition for Parametric Dynamical Systems},
journal = {Communications in Computational Physics},
year = {2024},
volume = {35},
number = {1},
pages = {38--69},
abstract = {<p style="text-align: justify;">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 aim
to reduce computation complexity and improve simulation efficiency. However, traditional model reduction methods are inefficient for parametric dynamical systems with
nonlinear structures. To address this challenge, we propose an adaptive method based
on local dynamic mode decomposition (DMD) to construct an efficient and reliable
surrogate model. We propose an improved greedy algorithm to generate the atoms set $\Theta$ based on a sequence of relatively small training sets, which could reduce the effect of
large training set. At each enrichment step, we construct a local sub-surrogate model
using the Taylor expansion and DMD, resulting in the ability to predict the state at any
time without solving the original dynamical system. Moreover, our method provides
the best approximation almost everywhere over the parameter domain with certain
smoothness assumptions, thanks to the gradient information. At last, three concrete
examples are presented to illustrate the effectiveness of the proposed method.</p>},
issn = {1991-7120},
doi = {https://doi.org/10.4208/cicp.OA-2023-0163},
url = {https://global-sci.com/article/90917/an-adaptive-method-based-on-local-dynamic-mode-decomposition-for-parametric-dynamical-systems}
}