POD/DEIM Reduced-Order Modeling of Time-Fractional Partial Differential Equations with Applications in Parameter Identification

摘要

In this paper, a reduced-order model (ROM) based on the proper orthogonaldecomposition and the discrete empirical interpolation method is proposed forefficiently simulating time-fractional partial differential equations (TFPDEs).Both linear and nonlinear equations are considered. We demonstrate theeffectiveness of the ROM by several numerical examples, in which the ROMachieves the same accuracy of the full-order model (FOM) over a long-termsimulation while greatly reducing the computational cost. The proposed ROM isthen regarded as a surrogate of FOM and is applied to an inverse problem foridentifying the order of the time-fractional derivative of the TFPDE model.Based on the Levenberg--Marquardt regularization iterative method with theArmijo rule, we develop a ROM-based algorithm for solving the inverse problem.For cases in which the observation data is either uncontaminated orcontaminated by random noise, the proposed approach is able to achieve accurateparameter estimation efficiently.