Bi-level iterative regularization for inverse problems in nonlinear PDEs
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arXiv
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866914704119562240 |
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| author | Nguyen, Tram Thi Ngoc |
| author_facet | Nguyen, Tram Thi Ngoc |
| contents | We investigate the ill-posed inverse problem of recovering unknown spatially dependent parameters in nonlinear evolution PDEs. We propose a bi-level Landweber scheme, where the upper-level parameter reconstruction embeds a lower-level state approximation. This can be seen as combining the classical reduced setting and the newer all-at-once setting, allowing us to, respectively, utilize well-posedness of the parameter-to-state map, and to bypass having to solve nonlinear PDEs exactly. Using this, we derive stopping rules for lower- and upper-level iterations and convergence of the bi-level method. We discuss application to parameter identification for the Landau-Lifshitz-Gilbert equation in magnetic particle imaging. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_16617 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Bi-level iterative regularization for inverse problems in nonlinear PDEs Nguyen, Tram Thi Ngoc Numerical Analysis Optimization and Control 65M32, 65J22, 35R30 We investigate the ill-posed inverse problem of recovering unknown spatially dependent parameters in nonlinear evolution PDEs. We propose a bi-level Landweber scheme, where the upper-level parameter reconstruction embeds a lower-level state approximation. This can be seen as combining the classical reduced setting and the newer all-at-once setting, allowing us to, respectively, utilize well-posedness of the parameter-to-state map, and to bypass having to solve nonlinear PDEs exactly. Using this, we derive stopping rules for lower- and upper-level iterations and convergence of the bi-level method. We discuss application to parameter identification for the Landau-Lifshitz-Gilbert equation in magnetic particle imaging. |
| title | Bi-level iterative regularization for inverse problems in nonlinear PDEs |
| topic | Numerical Analysis Optimization and Control 65M32, 65J22, 35R30 |
| url | https://arxiv.org/abs/2308.16617 |