Fast Direct Solvers With Arbitrary Admissibility Using Generalized Source Integral Equations

A hierarchical- (or $\mathcal {H}$ -) matrix fast direct solver, utilizing a generalized source integral equation (IE) for essentially convex scatterers, that is suitable for both strong and weak admissibility settings, is presented. The IE assigns to each elemental source auxiliary sources that significantly attenuate its radiation into the scatterer. Choosing the auxiliary sources to effectively form impenetrable convex reflective “shields” in front of each source forces most interactions to be via creeping waves. This reduces the effective dimensionality of weak and strong admissibility interactions between source and observer regions that are smaller or comparable in size to the shield and enhances the low-rank (LR) compressibility of the corresponding moment matrix blocks. For strong admissibility interactions between larger subdomains, the singular values (SVs) are greatly attenuated but decay similar to their conventional IE counterparts. The enhancement of their LR compressibility is achieved by “dynamically” adjusting the truncation threshold to account for the attenuation, compared to the conventional IE counterparts, without increasing the solver error. A fast method is also proposed for avoiding the costly repeated computation of the modified IE kernel. It uses coarse and tailored sampling and interpolation of phase- and amplitude-compensated representations of the entire kernel or its components, in accordance with their ray-asymptotic behavior, in a region dependent manner. The results validate the proposed IE formulation and demonstrate the associated LR compressibility and fast direct solver performance enhancement.