A study of Cauchy problem of the Helmholtz equation based on a relaxation model: Regularization and analysis

In this paper, we consider a Cauchy problem of the Helmholtz equation of recovering both missing voltage and current on inaccessible boundary from Cauchy data measured on the remaining accessible boundary. With an introduction of a relaxation parameter, the Dirichlet boundary conditions are approximated by two Robin ones. Associated with two mixed boundary value problems, a regularized Kohn-Vogelius formulation is proposed. Compared to the existing work, weaker regularity is required on the Dirichlet data and no Dirichlet BVPs needs to be solved. This makes the proposed model simpler and more efficient in computation. The well-posedness analysis about the relaxation model and error estimates of the corresponding inverse problem are obtained. A series of theoretical results are established for the new reconstruction model. Several numerical examples are provided to show feasibility and effectiveness of the proposed method.