The method of fundamental solutions for the inverse time-dependent perfusion coefficient problem

Abstract

This paper deals with an inverse problem associated with the bio-heat equation in living tissue in the human body. The inverse problem consists of the identification of time-dependent perfusion coefficient when the exact and noisy measurements of temperature at a fixed space point x* are specified. The numerical method for the retrieval of the unknown perfusion coefficient is based on the method of fundamental solutions (MFS). By introducing the fundamental solution of the heat equation and theoretical properties of these solutions, the MFS is used in conjunction with the Tikhonov regularization method. The choice of the regularization parameter is based on L-curve criteria to obtain a stable solution. Our numerical approach for numerical differentiation of discrete noisy data is focused on the iterated Tikhonov method due to ill-posedness of problem. Numerical results show the efficiency and applicability of the proposed algorithm in approximation of unknown perfusion coefficient.