A HSS Matrix-Inspired Butterfly-Based Direct Solver for Analyzing   Scattering from Two-dimensional Objects
  A butterfly-based fast direct integral equation solver for analyzing high-frequency scattering from two-dimensional objects is presented. The solver leverages a randomized butterfly scheme to compress blocks corresponding to near- and far-field interactions in the discretized forward and inverse electric field integral operators. The observed memory requirements and computational cost of the proposed solver scale as O(Nlog^2N) and O(N^1.5 logN), respectively. The solver is applied to the analysis of scattering from electrically large objects spanning over ten thousand of wavelengths and modeled in terms of five million unknowns. 
Yang Liu, Han Guo, and Eric Michielssen
math.NA physics.comp-ph