论文标题
基于CCBM的广义GKB迭代式正则化算法，用于逆向Cauchy问题
A CCBM-based generalized GKB iterative regularization algorithm for inverse Cauchy problems
论文作者
论文摘要
本文研究了由一种椭圆形偏微分方程控制的cauchy问题。反问题涉及从无法访问的边界上的测量数据中恢复无法访问的边界上的丢失数据，该数据严重不足。通过使用将Dirichlet和Neumann数据集成到单个Robin边界条件中的耦合复杂边界方法（CCBM），我们将基本问题重新加密到操作员方程。基于这种新的公式，我们研究了解决方案存在的问题，这些问题与嘈杂的数据减少了问题。 Golub-kahan bidiagonalization（GKB）过程与givens旋转一起迭代求解了所提出的操作员方程。所开发方法的正规化属性（称为CCBM-GKB）及其收敛率结果在后验停止规则下证明。最后，将线性有限元方法用于CCBM-GKB的数值实现。各种数值实验表明，CCBM-GKB是一种加速的迭代正则化方法，因为它比经典的Landweber方法快得多。
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.
