A Butterfly-Based Direct Integral Equation Solver Using Hierarchical LU   Factorization for Analyzing Scattering from Electrically Large Conducting   Objects
  A butterfly-based direct combined-field integral equation (CFIE) solver for analyzing scattering from electrically large, perfect electrically conducting objects is presented. The proposed solver leverages the butterfly scheme to compress blocks of the hierarchical LU-factorized discretized CFIE operator and uses randomized butterfly reconstruction schemes to expedite the factorization. The memory requirements and computational cost of the direct butterfly-CFIE solver scale as $O(N\mathrm{log}^2N)$ and $O(N^{1.5}\mathrm{log}N)$, respectively. These scaling estimates permit significant memory and CPU savings when compared to those realized by low-rank (LR) decomposition-based solvers. The efficacy and accuracy of the proposed solver are demonstrated through its application to the analysis of scattering from canonical and realistic objects involving up to 14 million unknowns. 
Han Guo, Yang Liu, Jun Hu, and Eric Michielssen
math.NA