RBF Approximation of the Lippmann-Schwinger Equation

We consider the direct scattering problem that consists of the computation of the scattered wave generated by an incident plane wave and an inhomogeneous object defined in terms of the refractive index. From some suitable physical and geometrical hypotheses, this is formulated as a boundary value problem for the Helmholtz equation and, in turn, as the Lippman-Schwinger equation. For the numerical solution of this integral equation, we propose an approximation approach by using Radial Basis Functions (RBF), which allows a relevant reduction in the computational cost of the numerical procedure. This new method is described in full detail and its performance is shown by using a wide numerical experiment for the approximate solution of the Lippman-Schwinger equation with different approaches.