Direct method for solution of inverse boundary value problem of the laplace equation

Abstract

This chapter considers a boundary inverse problem for the Laplace equation in two dimensions. The Laplace equation is considered in the domain enclosed by a smooth boundary, on which information about Dirichlet or Neumann data is incompletely specified so that the defined problem is not well posed. By introducing a convex functional to be minimized, the solution of the inverse problem is understood as the minimizer of the functional. The necessary condition for the functional to attain the minimum is paraphrased by the primary and adjoint boundary value problems of the Laplace equation. The boundary element method is applied to obtain a numerical solution of the problems, yielding an augmented system of linear algebraic equations. The linear system of equations can be solved directly. Four test examples presented in this chapter suggest the validity of this direct method for the inverse boundary value problem.