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.

翻译：本文研究从域内随机点观测数据中恢复椭圆问题势函数的逆问题。我们采用带 $H^1(\Omega)$ 罚项的正则化输出最小二乘格式进行数值重构，并利用 Galerkin 有限元法进行空间离散化。在问题数据满足温和正则性假设的条件下，我们以高概率意义对正则化解及有限元逼近进行了随机 $L^2(\Omega)$ 收敛性分析。所得误差界显式依赖于正则化参数 $\gamma$、观测点数量 $n$ 和网格尺寸 $h$。这些估计为选择相关算法参数提供了实用指导。此外，我们提出了一种单调收敛的自适应算法，用于在缺乏先验知识时确定合适的正则化参数。数值实验对理论结果进行了补充验证。