A CCBM-based generalized GKB iterative regularization algorithm for inverse Cauchy problems

TL;DR

This paper introduces a novel iterative regularization algorithm, CCBM-GKB, for solving severely ill-posed inverse Cauchy problems governed by elliptic PDEs, demonstrating faster convergence than traditional methods.

Contribution

It develops a new regularization algorithm combining CCBM and GKB, with proven convergence and efficiency improvements for inverse boundary data problems.

Findings

CCBM-GKB accelerates convergence compared to Landweber method.

The method effectively handles noisy data in inverse problems.

Numerical experiments confirm the method's efficiency and stability.

Abstract

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.