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Numerical Reconstruction of a Discontinuous Diffusive Coefficient in Variable-Order Time-Fractional Subdiffusion

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Abstract

We consider a discontinuous coefficient reconstruction problem associated with a variable-order time-fractional subdiffusion equation. Both interface identification and reconstruction of piecewise constant coefficient values are considered. We show existence of a minimizer of the regularized inverse problem. Shape sensitivity analysis is performed to propose a shape gradient optimization algorithm allowing deformations. Moreover, an algorithm allowing shape and topological changes is proposed by a phase-field method with sensitivity analysis. Numerical examples are presented to demonstrate effectiveness of the two algorithms for recovering both subdiffusion interface and the two subdiffusion constants.

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Funding

This work was supported in part by the National Key Basic Research Program under Grant 2022YFA1004402, the Science and Technology Commission of Shanghai Municipality (Nos. 21JC1402500, 22ZR1421900, and 22DZ2229014), and the National Natural Science Foundation of China under Grant (No. 12071149).

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Authors and Affiliations

  1. Department of Mathematics, East China Normal University, Shanghai, 200241, China

    Wei Fan, Xindi Hu & Shengfeng Zhu

  2. Shanghai Key Laboratory of Pure Mathematics and Mathematical Practice, East China Normal University, Shanghai, 200241, China

    Shengfeng Zhu

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  1. Wei Fan
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  2. Xindi Hu
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All authors contributed to the study conception and design. All authors read and approved the final manuscript.

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Correspondence to Shengfeng Zhu.

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Fan, W., Hu, X. & Zhu, S. Numerical Reconstruction of a Discontinuous Diffusive Coefficient in Variable-Order Time-Fractional Subdiffusion. J Sci Comput 96, 13 (2023). https://doi.org/10.1007/s10915-023-02237-y

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