Total variation regularization for recovering the spatial source term in a time-fractional diffusion equation
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866929674430447616 |
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| author | Fan, Bin |
| author_facet | Fan, Bin |
| contents | In this paper, we consider an inverse space-dependent source problem for a time-fractional diffusion equation. To deal with the ill-posedness of the problem, we transform the problem into an optimal control problem with total variational (TV) regularization. In contrast to the classical Tikhonov model incorporating $L^2$ penalty terms, the inclusion of a TV term proves advantageous in reconstructing solutions that exhibit discontinuities or piecewise constancy. The control problem is approximated by a fully discrete scheme, and convergence results are provided within this framework. Furthermore, a lineraed primal-dual iterative algorithm is proposed to solve the discrete control model based on an equivalent saddle-point reformulation, and several numerical experiments are presented to demonstrate the efficiency of the algorithm. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_12029 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Total variation regularization for recovering the spatial source term in a time-fractional diffusion equation Fan, Bin Optimization and Control Numerical Analysis In this paper, we consider an inverse space-dependent source problem for a time-fractional diffusion equation. To deal with the ill-posedness of the problem, we transform the problem into an optimal control problem with total variational (TV) regularization. In contrast to the classical Tikhonov model incorporating $L^2$ penalty terms, the inclusion of a TV term proves advantageous in reconstructing solutions that exhibit discontinuities or piecewise constancy. The control problem is approximated by a fully discrete scheme, and convergence results are provided within this framework. Furthermore, a lineraed primal-dual iterative algorithm is proposed to solve the discrete control model based on an equivalent saddle-point reformulation, and several numerical experiments are presented to demonstrate the efficiency of the algorithm. |
| title | Total variation regularization for recovering the spatial source term in a time-fractional diffusion equation |
| topic | Optimization and Control Numerical Analysis |
| url | https://arxiv.org/abs/2310.12029 |