Butterfly Matrices and Tensors: New Tools for High-Frequency Integral and Differential Equation Solvers, and High-Dimensional Transforms

In this work, I will overview the recent development of butterfly algorithms for constructing fast iterative and direct solvers for high-frequency integral and differential equations in electromagnetics, as well as large-scale integral transforms. The topics will include (1) matrix butterfly-based direct and iterative solvers for surface and volume integral equations, (2) matrix butterfly-based direct solvers for differential equation-based formulations, (3) tensor butterfly algorithms for 3D volume integral equations and high-dimensional integral transforms, and (4) the parallel implementation of butterfly algorithms and their availability in the open-source community software ButterflyPACK.