Reconstruction of Inner Boundaries Subjected to Generalized Impedance Boundary Conditions for the Modified Helmholtz Equation

The reconstruction of complex and irregular targets buried in a surrounding medium from a finite set of non-destructive Cauchy data pairs is an important practical problem in tomographic imaging. The unknown target to be identified may represent a breast tumour or a landmine, to give just a couple of examples of the utmost significance of the present study. The main originality consists in developing the numerical solution, based on the method of fundamental solutions, for reconstructing the unknown interior defects, subjected to impedance boundary conditions, and possibly their generalized impedance functions, from five pairs of boundary Cauchy data. The governing partial differential equation is given by the modified Helmholtz equation which governs fundamental phenomena in heat and bio-heat steady-state reaction-diffusion. The resulting least-squares functional estimating the gap between the measured and the computed data is regularized and minimized using the lsqnonlin toolbox routine in Matlab.