Stochastic Convergence Analysis of Inverse Potential Problem
| Autor: | Jin, Bangti, Quan, Qimeng, Zhang, Wenlong |
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| Rok vydání: | 2024 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | In this work, we investigate the inverse problem of recovering a potential in an elliptic problem from random pointwise observations in the domain. We employ a regularized output-least squares formulation with an $H^1(\Omega)$ penalty for the numerical reconstruction, and the Galerkin finite element method for the spatial discretization. Under mild regularity assumptions on the problem data, we provide a stochastic $L^2(\Omega)$ convergence analysis on the regularized solution and the finite element approximation in a high probability sense. The obtained error bounds depend explicitly on the regularization parameter $\gamma$, the number $n$ of observation points and the mesh size $h$. These estimates provide a useful guideline for choosing relevant algorithmic parameters. Furthermore, we develop a monotonically convergent adaptive algorithm for determining a suitable regularization parameter in the absence of \textit{a priori} knowledge. Numerical experiments are provided to complement the theoretical results. Comment: 21 pages, 3 figures |
| Databáze: | arXiv |
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