发表机构
Saarland University; Lappeenranta-Lahti University of Technology（萨尔大学; 拉彭兰塔-拉赫蒂工业大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对分数阶电导率反问题，提出正则化有限元方法，从外部噪声测量重构电导率并确定内部值，证明收敛性及误差估计，数值实验验证准确性。
AI 中文摘要
我们针对分数阶电导率方程开发了一种正则化有限元方法，从含噪的Dirichlet-to-Neumann测量中恢复外部观测区域上的电导率，并利用该重构结果确定内部电导率。网格局部化输入和集中质量Tikhonov正则化对Covi--Railo--Zimmermann（2026, Calc. Var. PDE）的外部确定原理进行离散化。我们证明了外部\\(L^2\\)-和\\(L^\infty\\)-收敛性及误差估计，包括临界Hölder指数处的对数修正。外部重构使得Liouville约化可用于分数阶Schrödinger方程的Calderón问题。分离测量恢复内部势，之后重叠测量确定外部势尾部。结合恢复的外部电导率，该尾部为Tikhonov正则化有限元最小二乘延拓提供外部Cauchy数据。我们证明了变换电导率的\\(H^s\\)-收敛性和内部电导率的\\(L^2\\)-收敛性，以及全观测空间的对数状态偏差和电导率误差估计。可容许的重构保持一致正且有界。我们展示了一维和二维数值实验，说明了外部和内部重构的准确性及其对测量噪声和离散化的敏感性。
英文摘要
We develop a regularized finite element method for the fractional conductivity equation, recovering the conductivity on an exterior observation region from noisy Dirichlet-to-Neumann measurements and using this reconstruction to determine the interior conductivity. Mesh-localized inputs and mass-lumped Tikhonov regularization discretize the exterior determination principle of Covi--Railo--Zimmermann (2026, Calc. Var. PDE). We prove exterior \(L^2\)- and \(L^\infty\)-convergence with error estimates, including the logarithmic correction at the critical Hölder exponent. The exterior reconstruction enables the Liouville reduction to the Calderón problem for the fractional Schrödinger equation. Separated measurements recover the interior potential, after which overlapping measurements determine the exterior potential tail. Together with the recovered exterior conductivity, the tail supplies the exterior Cauchy data for a Tikhonov-regularized finite element least-squares continuation. We prove \(H^s\)-convergence of the transformed conductivity and \(L^2\)-convergence of the interior conductivity, together with logarithmic state-bias and conductivity error estimates for full observation spaces. The admissible reconstructions remain uniformly positive and bounded. We present one- and two-dimensional numerical experiments illustrating the accuracy of the exterior and interior reconstructions and their sensitivity to measurement noise and discretization.
