A novel shape optimization approach for source identification in elliptic equations
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866909240827838464 |
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| author | Gong, Wei Zhang, Ziyi |
| author_facet | Gong, Wei Zhang, Ziyi |
| contents | In this paper, we propose a novel shape optimization approach for the source identification of elliptic equations. This identification problem arises from two application backgrounds: actuator placement in PDE-constrained optimal controls and the regularized least-squares formulation of source identifications. The optimization problem seeks both the source strength and its support. By eliminating the variable associated with the source strength, we reduce the problem to a shape optimization problem for a coupled elliptic system, known as the first-order optimality system. As a model problem, we derive the shape derivative for the regularized least-squares formulation of the inverse source problem and propose a gradient descent shape optimization algorithm, implemented using the level-set method. Several numerical experiments are presented to demonstrate the efficiency of our proposed algorithms. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_02909 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A novel shape optimization approach for source identification in elliptic equations Gong, Wei Zhang, Ziyi Optimization and Control Numerical Analysis In this paper, we propose a novel shape optimization approach for the source identification of elliptic equations. This identification problem arises from two application backgrounds: actuator placement in PDE-constrained optimal controls and the regularized least-squares formulation of source identifications. The optimization problem seeks both the source strength and its support. By eliminating the variable associated with the source strength, we reduce the problem to a shape optimization problem for a coupled elliptic system, known as the first-order optimality system. As a model problem, we derive the shape derivative for the regularized least-squares formulation of the inverse source problem and propose a gradient descent shape optimization algorithm, implemented using the level-set method. Several numerical experiments are presented to demonstrate the efficiency of our proposed algorithms. |
| title | A novel shape optimization approach for source identification in elliptic equations |
| topic | Optimization and Control Numerical Analysis |
| url | https://arxiv.org/abs/2407.02909 |