Preconditioning techniques for the method of auxiliary sources applied to geometries characterized as perturbations of a circle

Two versions of a preconditioning procedure are described for a class of two-dimensional scattering problems, when the latter are solved via the Method of Auxiliary Sources (MAS). The method is useful for the case of shrinking auxiliary surfaces, i.e. when the linear system may become ill-conditioned. If the cross sectional boundary can be regarded as a deformation of a circle, the MAS matrix is shown to be quasi-circulant, as expected intuitively. By utilizing the circulant properties of the matrix pertaining to the original circular configuration, the perturbed system is transformed to a quasi-diagonal one, whose inversion is a numerically stable operation.