Solution of the Cauchy problem using iterated Tikhonov regularization (Q2738853)

The boundary data completion problem for harmonic functions in the case of an open bounded set \(\Omega\) in \(\mathbb{R}^2\) or \(\mathbb{R}^3\) with a smooth boundary \(\Gamma-\overline\Gamma_d\cup\overline\Gamma_u\) is considered. The authors are interested in recovering Dirichlet and Neumann data on \(\Gamma_u\) from the knowledge of Cauchy data on the other part \(\Gamma_d\) of the boundary. As is well known since Hadamard, this problem gets severely ill-posed and a regularization cannot be avoided if noisy data are handled. It is shown that a convergent fixed point iteration solves the reconstruction problem. The associated iteration process can be interpreted as iterated Tikhonov regularization, where the choice of the regularization parameter is not so crucial. Using a Hilbert space setting the analytic background and a numerical implementation are presented. Numerical case studies illustrate the theoretical considerations. After discretization the solution process shows a surprising robustness with respect to highly noisy data. The numerical experiments indicate that the chances for recovering Dirichlet data on the remaining part of the boundary are much better than the chances for finding the normal derivatives.