Numerical Analysis of Simultaneous Reconstruction of Initial Condition and Potential in Subdiffusion
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| Format: | Preprint |
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2025
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| _version_ | 1866914115888349184 |
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| author | Wu, Xu Yang, Jiang Zhou, Zhi |
| author_facet | Wu, Xu Yang, Jiang Zhou, Zhi |
| contents | This paper investigates the simultaneous identification of a spatially dependent potential and the initial condition in a subdiffusion model based on two terminal observations. The existence, uniqueness, and conditional stability of the inverse problem are established under weak regularity assumptions through a constructive fixed-point iteration approach. The theoretical analysis further inspires the development of an easy-to-implement iterative algorithm. A fully discrete scheme is then proposed, combining the finite element method for spatial discretization, convolution quadrature for temporal discretization, and the quasi-boundary value method to handle the ill-posedness of recovering the initial condition. Inspired by the conditional stability estimate, we demonstrate the linear convergence of the iterative algorithm and provide a detailed error analysis for the reconstructed initial condition and potential. The derived \textsl{a priori} error estimate offers a practical guide for selecting regularization parameters and discretization mesh sizes based on the noise level. Numerical experiments are provided to illustrate and support our theoretical findings. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_15633 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Numerical Analysis of Simultaneous Reconstruction of Initial Condition and Potential in Subdiffusion Wu, Xu Yang, Jiang Zhou, Zhi Numerical Analysis This paper investigates the simultaneous identification of a spatially dependent potential and the initial condition in a subdiffusion model based on two terminal observations. The existence, uniqueness, and conditional stability of the inverse problem are established under weak regularity assumptions through a constructive fixed-point iteration approach. The theoretical analysis further inspires the development of an easy-to-implement iterative algorithm. A fully discrete scheme is then proposed, combining the finite element method for spatial discretization, convolution quadrature for temporal discretization, and the quasi-boundary value method to handle the ill-posedness of recovering the initial condition. Inspired by the conditional stability estimate, we demonstrate the linear convergence of the iterative algorithm and provide a detailed error analysis for the reconstructed initial condition and potential. The derived \textsl{a priori} error estimate offers a practical guide for selecting regularization parameters and discretization mesh sizes based on the noise level. Numerical experiments are provided to illustrate and support our theoretical findings. |
| title | Numerical Analysis of Simultaneous Reconstruction of Initial Condition and Potential in Subdiffusion |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2509.15633 |