  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.
