The method of fundamental solutions for the identification of a scatterer with impedance boundary condition in interior inverse acoustic scattering

Abstract

We employ the method of fundamental solutions (MFS) for detecting a scatterer surrounding a host acoustic homogeneous medium D due to a given point source inside it. On the boundary of the unknown scatterer (assumed to be star-shaped), allowing for the normal velocity to be proportional to the excess pressure, a Robin impedance boundary condition is considered. The coupling Robin function λ may or may not be known. The additional information which is supplied in order to compensate for the lack of knowledge of the boundary ∂D of the interior scatterer D and/or the function λ is given by the measurement of the scattered field (generated by the interior point source) on a curve inside D. These measurements may be contaminated with noise so their inversion requires regularization. This is enforced by minimizing a penalised least-squares functional containing various regularization parameters to be prescribed. In the MFS, the unknown scattered field us is approximated with a linear combination of fundamental solutions of the Helmholtz operator with their singularities excluded from the solution domain D and this yields the discrete version of the objective functional. Physical constraints are added and the resulting constrained minimization problem is solved using the MATLAB© toolbox routine lsqnonlin. Numerical results are presented and discussed.