On level set type methods for elliptic Cauchy problems

TL;DR

This paper introduces two level set methods for solving elliptic Cauchy problems, providing convergence proofs, stability analysis, and numerical experiments demonstrating their effectiveness in handling this ill-posed problem.

Contribution

The paper proposes novel level set based approaches for elliptic Cauchy problems, with proven convergence, stability, and demonstrated numerical efficiency.

Findings

Both methods are proven to converge and are stable as regularization techniques.

Numerical experiments confirm the methods' efficiency and validate theoretical convergence.

The approaches effectively address the ill-posedness of the elliptic Cauchy problem.

Abstract

Two methods of level set type are proposed for solving the Cauchy problem for an elliptic equation. Convergence and stability results for both methods are proven, characterizing the iterative methods as regularization methods for this ill-posed problem. Some numerical experiments are presented, showing the efficiency of our approaches and verifying the convergence results.