A Generalized Source Integral Equation for Enhanced Compression in Three Dimensions

An integral equation formulation, using generalized directional sources, for 3-D scattering by impenetrable and essentially-convex bodies, is presented. This formulation increases the rank deficiency of moment matrix blocks representing interactions between nonoverlapping source and observer regions. Thereby, it enables enhanced low-rank approximation-based matrix compression and the development of corresponding fast direct solvers. The directional sources are constructed by augmenting the conventional basis functions with spherical absorbing 3-D shields, on which auxiliary source distributions are defined. The bottleneck arising from the need to integrate over these distributions upon computing the modified Green's function (MGF) is removed by using efficient nonuniform sampling and tabulation of the MGF or its components, in a region-dependent manner. The formulation is studied and its favorable compressibility is demonstrated, for two fundamental types of compression strategies. The nonuniform sampling approach is employed also to facilitate the rank-revealing analysis of large off-diagonal blocks.