Quantitative estimates for a nonlinear inverse source problem in a coupled diffusion equations with uncertain measurements
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866915262435950592 |
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| author | Sun, Chunlong Zhang, Wenlong Zhang, Zhidong |
| author_facet | Sun, Chunlong Zhang, Wenlong Zhang, Zhidong |
| contents | This work considers a nonlinear inverse source problem in a coupled diffusion equation from the terminal observation. Theoretically, under some conditions on problem data, we build the uniqueness theorem for this inverse problem and show two Lipschitz-type stability results in $L^2$ and $(H^1(\cdot))^*$ norms, respectively. However, in practice, we could only observe the measurements at discrete sensors, which contain the noise. Hence, this work further investigates the recovery of the unknown source from the discrete noisy measurements. We propose a stable inversion scheme and provide probabilistic convergence estimates between the reconstructions and exact solution in two cases: convergence respect to expectation and convergence with an exponential tail. We provide several numerical experiments to illustrate and complement our theoretical analysis. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_19421 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Quantitative estimates for a nonlinear inverse source problem in a coupled diffusion equations with uncertain measurements Sun, Chunlong Zhang, Wenlong Zhang, Zhidong Numerical Analysis This work considers a nonlinear inverse source problem in a coupled diffusion equation from the terminal observation. Theoretically, under some conditions on problem data, we build the uniqueness theorem for this inverse problem and show two Lipschitz-type stability results in $L^2$ and $(H^1(\cdot))^*$ norms, respectively. However, in practice, we could only observe the measurements at discrete sensors, which contain the noise. Hence, this work further investigates the recovery of the unknown source from the discrete noisy measurements. We propose a stable inversion scheme and provide probabilistic convergence estimates between the reconstructions and exact solution in two cases: convergence respect to expectation and convergence with an exponential tail. We provide several numerical experiments to illustrate and complement our theoretical analysis. |
| title | Quantitative estimates for a nonlinear inverse source problem in a coupled diffusion equations with uncertain measurements |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2504.19421 |