Reduced-order modeling for Koopman operators of nonautonomous dynamic systems in multiscale media

In this article, we present reduced-order modeling for Koopman operators of nonautonomous dynamic systems in multiscale media. Koopman operators can transform the nonlinear dynamic systems into linear systems through acting on observation function spaces. Different from the case of autonomous dynamic systems, the Koopman operator family of nonautonomous dynamic systems significantly depend on a time pair. In order to effectively estimate the time-dependent Koopman operators, a moving time window is used to decompose the snapshot data, and the extended dynamic mode decomposition method is applied to computing the Koopman operators in each local temporal domain. To accurately construct the models of dynamic systems in multiscale media, we may use high spatial dimension of observation data. It is challenging to compute the Koopman operators using the high dimensional data. Thus, the strategy of reduced-order modeling is proposed to treat the difficulty. The proposed reduced-order modeling includes two stages: offline stage and online stage. In offline stage, a block-wise low rank decomposition is used to reduce the spatial dimension of initial snapshot data. For the nonautonomous dynamic systems, real-time observation data may be required to update the Koopman operators. An online reduced-order modeling is proposed to correct the offline reduced-order modeling.Three methods are developed for the online reduced-order modeling: fully online, semi-online and adaptive online. The adaptive online method automatically selects the fully online or semi-online and can achieve a good trade-off between modeling accuracy and efficiency. A few numerical examples are presented to illustrate the performance of the different reduced-order modeling methods.