# A Hierarchical Butterfly LU Preconditioner for Two-Dimensional Electromagnetic Scattering Problems Involving Open Surfaces

## TL;DR

This paper presents a hierarchical butterfly LU preconditioner that significantly accelerates the solution of large-scale 2D electromagnetic scattering problems involving open surfaces, with scalable efficiency and high accuracy.

## Contribution

It introduces the H-IDBF-LU preconditioner using interpolative decomposition butterfly factorization for efficient iterative solutions of EFIEs on open surfaces.

## Key findings

- Memory and computational cost scale as O(N log^2 N) per iteration.
- Number of iterations remains bounded or grows slowly with problem size.
- Successfully applied to problems with up to 100 million unknowns.

## Abstract

This paper introduces a hierarchical interpolative decomposition butterfly-LU factorization (H-IDBF-LU) preconditioner for solving two-dimensional electric-field integral equations (EFIEs) in electromagnetic scattering problems of perfect electrically conducting objects with open surfaces. H-IDBF-LU leverages the interpolative decomposition butterfly factorization (IDBF) to compress dense blocks of the discretized EFIE operator to expedite its application; this compressed operator also serves as an approximate LU factorization of the EFIE operator leading to an efficient preconditioner in iterative solvers. Both the memory requirement and computational cost of the H-IDBF-LU solver scale as $O(N\log^2 N)$ in one iteration; the total number of iterations required for a reasonably good accuracy scales as $O(1)$ to $O(\log^2N)$ in all of our numerical tests. The efficacy and accuracy of the proposed preconditioned iterative solver are demonstrated via its application to a broad range of scatterers involving up to $100$ million unknowns.