On the Green's functions technique for the numerical solution of inverse boundary problems

We consider two types of inverse boundary problems related to the Laplace equation in planar double connected domains. The first one consists in the determining the Cauchy data on an inclusion from given Cauchy data on an accessible exterior boundary. On this accessible part the function (or the normal derivative) is known and, additionally, on a portion of this exterior boundary the normal derivative (or the function) is also given. For this linear inverse problem we propose a direct boundary integral approach in combination with Tikhonov regularization for stable determination of the Cauchy data on the inclusion. To determine these quantities on the inclusion, boundary integral equations are derived using Green's functions. We outline an effective way of discretizing these boundary integral equations based on the Nyström method and trigonomtric approximations. The second set of inverse problems is the reconstruction of the interior curve from the given Cauchy data of a harmonic function on the exterior boundary. With the help of Green's function and potential theory this non-linear boundary problem is reduced to the system of non-linear boundary integral equations. We develop three iterative algorithms for its numerical solution. Full discretization of the linearized systems is realized by a trigonometric quadrature method. Due to the inherited ill-possedness in the obtained system of linear equations we apply the Tikhonov regularization. Numerical results show that the proposed approach gives good accuracy of reconstructions with economical computational cost.