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

TL;DR

This work introduces a POD/DEIM reduced-order framework for time-fractional diffusion–reaction equations governed by the Caputo derivative ${}_0^C D_t^{\beta}$. The authors construct an efficient ROM via proper orthogonal decomposition and discrete empirical interpolation to handle nonlinearities, yielding a low-dimensional system with substantial speedups while preserving accuracy over long-time simulations. They couple the ROM with a Levenberg–Marquardt regularization scheme, augmented by an Armijo line search, to solve an inverse problem for identifying the fractional order $\beta$ from observations; Gauss-Newton steps are used within both the full and reduced systems. Numerical experiments in 1D and 2D, for linear and nonlinear cases, show that the ROM matches FOM accuracy but dramatically reduces online computation time, even under data noise, demonstrating the method’s practical viability for parameter identification and large-scale simulations.

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.

Figures (16)

• Figure 1: The first four POD basis functions in Test I.
• Figure 2: The first four POD basis functions (left) and the first four nonlinear POD basis functions with DEIM points in Test II.
• Figure 3: $\beta^*=0.75$ for uncontaminated observation data in Example 1.
• Figure 4: $\beta^*=0.75$ for 1%-level noise contaminated observation data in Example 1.
• Figure 5: $\beta^*=0.75$ for uncontaminated observation data in Example 2.

Theorems & Definitions (1)

• remark thmcounterremark