Direct Solve of Electrically Large Integral Equations for Problem Sizes to 1 M Unknowns

When unknowns are spatially grouped for electrically large bodies, the blocked method of moments (MoM) Z matrix and its LU factors are comprised of blocks which can be well approximated by matrices of low rank expressed as the outer product of a column matrix times a row matrix. This approximation is called matrix compression and can save significant memory and operations count. And for monostatic scattering, where there are many right-hand sides (RHS) for electrically large bodies, the blocked RHS V and current solution J are also well approximated by this low rank outer product form. The blocked system equation ZJ = LUJ = V is LU factored and solved using this compressed block form where each outer product approximant is computed using the adaptive cross approximation (ACA). This approach has been applied to a frequency domain EFIE RWG Galerkin integral equation for 3-D PEC surfaces. Preliminary results for this compressed block LU factor and solve technique are presented and compared to measured data, BOR code predictions, and the same code without matrix compression. Complexity of this approach is studied numerically for an open pipe geometry with specific error tolerances, using a PC Workstation, with unknowns N ranging from 2592 to 1 025 109 and the number of RHS varying from 310 to 6162. Matrix fill time scales as N^1.34, LU matrix memory storage scales as N^1.5, LU factor time scales as N^2.0, and the time per RHS solve scales as N^1.67. Complete solution wall time scales as N^1.8.