A Generalized Source Integral Equation for Enhanced Compression in Three Dimensions

Abstract

An integral equation formulation, using generalized directional sources, for three-dimensional scattering by impenetrable and essentially convex bodies, is presented. This formulation increases the rank-deficiency of moment matrix blocks representing interactions between non-overlapping 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 three-dimensional 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 is removed by using efficient non-uniform sampling and tabulation of the modified Green’s function 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 non-uniform sampling approach is employed also to facilitate the rank-revealing analysis of large off-diagonal blocks.