A Mann iterative regularization method for elliptic Cauchy problems

TL;DR

This paper introduces a Mann iterative regularization method for solving ill-posed elliptic Cauchy problems, combining segmentation and fixed point techniques to improve solution reconstruction.

Contribution

It generalizes previous iterative methods by integrating Mann iteration with fixed point equations, enhancing regularization and convergence analysis.

Findings

The method effectively reconstructs boundary traces in elliptic Cauchy problems.

Theoretical analysis confirms regularization properties.

Numerical experiments demonstrate convergence and stability.

Abstract

We investigate the Cauchy problem for linear elliptic operators with $C^\infty$-coefficients at a regular set $\Omega \subset R^2$, which is a classical example of an ill-posed problem. The Cauchy data are given at the manifold $\Gamma \subset \partial\Omega$ and our goal is to reconstruct the trace of the $H^1(\Omega)$ solution of an elliptic equation at $\partial \Omega / \Gamma$. The method proposed here composes the segmenting Mann iteration with a fixed point equation associated with the elliptic Cauchy problem. Our algorithm generalizes the iterative method developed by Maz'ya et al., who proposed a method based on solving successive well-posed mixed boundary value problems. We analyze the regularizing and convergence properties both theoretically and numerically.