基于分散测量的逆电导率问题误差分析
Error Analysis of the Inverse Conductivity Problem with Scattered Measurements
AI总结：
本文针对从含噪分散测量恢复椭圆方程电导率系数的逆问题，提出带W^{1,4}(Ω)惩罚的正则化最小二乘数值格式，给出误差界并通过实验验证。
AI中文摘要：
本工作研究从区域Ω内有限个确定性分散点采集的含随机噪声的测量数据，恢复椭圆方程中电导率系数的逆问题。受正则性分析启发，提出基于W^{1,4}(Ω)惩罚项的正则化最小二乘公式的数值格式，采用连续分段线性单元的Galerkin有限元法离散该正则化问题。在问题数据的适当假设下，对正则化解及其Galerkin近似进行误差分析，建立高概率意义下的L^2(Ω)误差界，该界显式依赖于正则化参数γ、数据点数量n和网格尺寸h。还通过数值实验验证理论结果。
英文摘要：
In this work, we investigate the inverse problem of recovering the conductivity coefficient in an elliptic equation from noisy measurements collected at finitely many deterministic scattered points in the domain $Ω$, and corrupted by random noise. Inspired by the regularity analysis, we propose a numerical scheme based on the regularized least-squares formulation with a $W^{1,4}(Ω)$ penalty, and discretize the regularized problem using the Galerkin finite element method with continuous piecewise linear elements. Under suitable assumptions on the problem data, we provide an error analysis of the regularized solution and its Galerkin approximation. We establish $L^2(Ω)$ error bounds in a high-probability sense, which depend explicitly on the regularization parameter $γ$, the number $n$ of data points and the mesh size $h$. We also present numerical experiments to illustrate the theoretical findings.
