POD/DEIM Reduced-Order Modeling of Time-Fractional Partial Differential Equations with Applications in Parameter Identification
For researchers working on time-fractional PDEs and inverse problems, this work provides a computationally efficient surrogate model that maintains accuracy, though it is an incremental application of existing ROM techniques to a specific class of equations.
The paper proposes a reduced-order model (ROM) using POD and DEIM to efficiently simulate time-fractional PDEs, achieving the same accuracy as the full-order model with reduced computational cost. It applies the ROM to an inverse problem for identifying the fractional derivative order, achieving accurate parameter estimation even with noisy data.
In this paper, a reduced-order model (ROM) based on the proper orthogonal decomposition and the discrete empirical interpolation method is proposed for efficiently simulating time-fractional partial differential equations (TFPDEs). Both linear and nonlinear equations are considered. We demonstrate the effectiveness of the ROM by several numerical examples, in which the ROM achieves the same accuracy of the full-order model (FOM) over a long-term simulation while greatly reducing the computational cost. The proposed ROM is then regarded as a surrogate of FOM and is applied to an inverse problem for identifying the order of the time-fractional derivative of the TFPDE model. Based on the Levenberg--Marquardt regularization iterative method with the Armijo rule, we develop a ROM-based algorithm for solving the inverse problem. For cases in which the observation data is either uncontaminated or contaminated by random noise, the proposed approach is able to achieve accurate parameter estimation efficiently.