Fast Integral Equation Solvers based on the Randomized Cross Approximation

This paper outlines a fast and fully algebraic integral equation method for the solution of electromagnetic scattering and radiation of perfect electric conducting structures via the Rao-Wilton-Glisson (RWG) Boundary Element Method (BEM) discretization of the Electric Field Integral Equation (EFIE). The present single-level grouping implementation accelerates computations by exploiting the well documented rank deficiency of non-neighboring group BEM interactions. The rank-revealing computational kernel of the proposed method resembles that of the popular Adaptive Cross Approximation (ACA), but bypasses altogether the time-consuming adaptive partial-pivoting of rows and columns using a much simpler randomized scheme for selecting crosses, i.e. row and column pivots, and leveraging blocked linear-algebra operations. More importantly, the method borrows an essential, yet often understated, ingredient from the Fast Multiple Method (FMM), i.e. working with the Green's function instead of the fully assembled BEM matrix, thus strives to numerically factorize the integral equation kernel (Green's function). As a result, the proposed Randomized Cross Approximation (RCA) method delivers considerably lower ranks than ACA, and same memory and set-up time and matrix-vector multiplication time complexities as the single-level FMM while avoiding the low-frequency breakdown of FMM.