The balancing principle for the regularization of elliptic Cauchy problems

A classical ill-posed elliptic Cauchy problem is discussed. We consider the reconstruction of the Dirichlet trace of the solution at the part of a boundary where no data are available. By natural linearization we transform it into a linear ill-posed operator equation. Discretization is applied as a regularization method (also known as a self-regularization) to obtain a stable approximate solution. The balancing principle as an adaptive strategy is studied to choose an appropriate discretization level without quantitative knowledge of a convergent rate or stability. Numerical tests illustrate theoretical results.