POD/DEIM reduced-order modeling of time-fractional partial differential equations with applications in parameter identification. (English) Zbl 1404.65140
Summary: 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.
MSC:
| 65M32 | Numerical methods for inverse problems for initial value and initial-boundary value problems involving PDEs |
| 35R11 | Fractional partial differential equations |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 65M22 | Numerical solution of discretized equations for initial value and initial-boundary value problems involving PDEs |
| 65K05 | Numerical mathematical programming methods |
| 90C30 | Nonlinear programming |
Keywords:
time-fractional partial differential equations; proper orthogonal decomposition; discrete empirical interpolation method; reduced-order model; parameter identificationSoftware:
rbMITReferences:
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