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| Main Authors: | , , , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2507.11130 |
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| _version_ | 1866916844591382528 |
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| author | Kartmann, Michael Klein, Benedikt Ohlberger, Mario Schuster, Thomas Volkwein, Stefan |
| author_facet | Kartmann, Michael Klein, Benedikt Ohlberger, Mario Schuster, Thomas Volkwein, Stefan |
| contents | We consider nonlinear inverse problems arising in the context of parameter identification for parabolic partial differential equations (PDEs). For stable reconstructions, regularization methods such as the iteratively regularized Gauss-Newton method (IRGNM) are commonly used, but their application is computationally demanding due to the high-dimensional nature of PDE discretizations. To address this bottleneck, we propose a reduced-order modeling approach that accelerates both the state and adjoint evaluations required for derivative-based optimization. Our method builds on the recent contribution [Kartmann et al. Adaptive reduced basis trust region methods for parameter identification problems. Comput. Sci. Eng. 1, 3 (2024)] for elliptic forward operators and constructs the reduced forward operator adaptively in an online fashion, combining both parameter and state space reduction. To ensure reliability, we embed the IRGNM iteration within an adaptive, error-aware trust-region framework that certifies the accuracy of the reduced-order approximations. We demonstrate the effectiveness of the proposed approach through numerical results for both time-dependent and time-independent parameter identification problems in dynamic reaction-diffusion systems. The implementation is made available for reproducibility and further use. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_11130 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Adaptive Reduced Basis Trust Region Methods for Parabolic Inverse Problems Kartmann, Michael Klein, Benedikt Ohlberger, Mario Schuster, Thomas Volkwein, Stefan Numerical Analysis Optimization and Control 35R30, 35K90, 65M32, 35K57 We consider nonlinear inverse problems arising in the context of parameter identification for parabolic partial differential equations (PDEs). For stable reconstructions, regularization methods such as the iteratively regularized Gauss-Newton method (IRGNM) are commonly used, but their application is computationally demanding due to the high-dimensional nature of PDE discretizations. To address this bottleneck, we propose a reduced-order modeling approach that accelerates both the state and adjoint evaluations required for derivative-based optimization. Our method builds on the recent contribution [Kartmann et al. Adaptive reduced basis trust region methods for parameter identification problems. Comput. Sci. Eng. 1, 3 (2024)] for elliptic forward operators and constructs the reduced forward operator adaptively in an online fashion, combining both parameter and state space reduction. To ensure reliability, we embed the IRGNM iteration within an adaptive, error-aware trust-region framework that certifies the accuracy of the reduced-order approximations. We demonstrate the effectiveness of the proposed approach through numerical results for both time-dependent and time-independent parameter identification problems in dynamic reaction-diffusion systems. The implementation is made available for reproducibility and further use. |
| title | Adaptive Reduced Basis Trust Region Methods for Parabolic Inverse Problems |
| topic | Numerical Analysis Optimization and Control 35R30, 35K90, 65M32, 35K57 |
| url | https://arxiv.org/abs/2507.11130 |