ncts

The Cauchy problem of the Laplace equation often arises in inverse problems. It is known as a typical ill-posed problem in the sense of Hadamard and has been extensively studied in the literature. Focusing on the two-dimensional case, we propose a numerical method based on Cauchy's integral formula and the jump relation. The core of our approach is a singular integral equation involving the Cauchy kernel. The piecewise constant approximation leads to a simultaneous linear equations whose condition number increases at most linearly with respect to the discretization number. This means that the proposed procedure is reliable in terms of stability without any explicit regularizations. We also demonstrate that the error growth remains within a range that can be processed in the standard floating-point arithmetic in digital computers.