一种基于CCBM的广义GKB迭代正则化算法用于反柯西问题
A CCBM-based generalized GKB iterative regularization algorithm for inverse Cauchy problems
AI总结：
本文针对反柯西问题，提出基于耦合复边界法的CCBM-GKB迭代正则化算法，结合GKB双对角化与Givens旋转求解算子方程，并证明其正则化性质与收敛率，数值实验表明其远快于Landweber方法。
摘要：
本文研究由一类椭圆型偏微分方程控制的反柯西问题。该反问题涉及从可及边界上的测量数据恢复不可及边界上的缺失数据，具有严重的病态性。通过使用耦合复边界法（CCBM），将Dirichlet数据和Neumann数据整合到单一的Robin边界条件中，我们将原问题重新表述为一个算子方程。基于这一新表述，我们研究了含噪声数据下约化问题的解存在性问题。采用Golub-Kahan双对角化（GKB）过程结合Givens旋转来迭代求解所提出的算子方程。在后验停止准则下，证明了所发展方法（称为CCBM-GKB）的正则化性质及其收敛率结果。最后，使用线性有限元方法对CCBM-GKB进行数值实现。多种数值实验表明，CCBM-GKB是一种加速的迭代正则化方法，其速度远快于经典的Landweber方法。
英文摘要：
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.
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