A Dual Reciprocity Boundary Element Method for the Regularized Numerical Solution of the Inverse Source Problem Associated to the Poisson Equation

Abstract

This article revisits the application of the Dual Reciprocity Method to a class of inverse problems governed by the Poisson equation in a thorough and careful manner. Here the term inverse refers to the fact that the non-homogenous part of the Poisson equation is unknown, i.e. the governing equation of the problem is unknown and has to be determined from Cauchy data at the boundary. We show that, although the inverse problem does not have a unique solution, by employing the Tikhonov regularization method we can recover the minimal norm solution. This is usually the solution of most practical interest from the many solutions of the ill-posed problem of source identification. Other different, more complex solutions might be recovered if estimates of these solutions are available at some particular points inside the solution domain.

Keywords:

• Dual Reciprocity Method
• Inverse Source Problem
• Poisson Equation
• Tikhonov Regularization
• Discrepancy Principle