Fast Direct Solver for Essentially Convex Scatterers Using Multilevel Non-Uniform Grids

A fast algorithm for the direct solution of the method of moments (MoM) systems of equations describing scattering from essentially convex bodies is presented. The algorithm reveals the ranks of interactions between subdomains and compresses the system to that of interacting unknowns only. The procedure is facilitated by representing the interactions via non-uniform sampling grids (NGs). In a multilevel procedure, the interactions' "skeletons," revealed at each level of the subdomain hierarchy, are aggregated and recompressed. The algorithm is demonstrated here for the generalized equivalence integral equation (GEIE). This recently introduced integral representation, relying on a generalized equivalence theorem, is highly compressible for convex scatterers. The algorithm is detailed, including the treatment of computational bottlenecks by using NG-approach schemes that are tailored to the GEIE formulation. For the essentially circular case, compression to O(1) unknowns at an O(N log N) computational complexity with O(N) storage is demonstrated.