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

TL;DR

This paper introduces a reduced-order modeling approach using POD and DEIM for efficiently simulating time-fractional PDEs, enabling accurate parameter identification with reduced computational cost in both linear and nonlinear cases.

Contribution

It develops a novel ROM framework for TFPDEs that effectively approximates full models and facilitates inverse problem solving for fractional order identification.

Findings

ROM achieves similar accuracy to FOM over long-term simulations

Significant reduction in computational cost

Accurate parameter estimation even with noisy data

Abstract

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.