This paper examines inverse Cauchy problems that are governed by a kind of elliptic partial differential equation. The inverse problems involve recovering the missing data on an inaccessible boundary from the measured data on an accessible boundary, which is severely ill-posed. By using the coupled complex boundary method (CCBM), which integrates both Dirichlet and Neumann data into a single Robin boundary condition, we reformulate the underlying problem into an operator equation. Based on this new formulation, we study the solution existence issue of the reduced problem with noisy data. A Golub-Kahan bidiagonalization (GKB) process together with Givens rotation is employed for iteratively solving the proposed operator equation. The regularizing property of the developed method, called CCBM-GKB, and its convergence rate results are proved under a posteriori stopping rule. Finally, a linear finite element method is used for the numerical realization of CCBM-GKB. Various numerical experiments demonstrate that CCBM-GKB is a kind of accelerated iterative regularization method, as it is much faster than the classic Landweber method.

翻译：本文研究由一类椭圆型偏微分方程控制的柯西反问题。这些问题涉及从可观测边界上的测量数据恢复不可达边界上的缺失数据，属于严重不适定问题。通过采用耦合复边界法（CCBM），该方法将Dirichlet和Neumann数据整合为单一Robin边界条件，我们将原问题重新表述为算子方程。基于这一新表述，我们研究了含噪声数据简化问题的解存在性。采用Golub-Kahan双对角化（GKB）过程结合Givens旋转迭代求解所提出的算子方程。给出了所开发方法（称为CCBM-GKB）的正则化性质及其在后验停止准则下的收敛率结果。最后，采用线性有限元方法实现CCBM-GKB的数值计算。各种数值实验表明，CCBM-GKB是一种加速迭代正则化方法，其计算速度远快于经典Landweber方法。