MLMDA-based direct integral equation solver for dielectric scatterers

Fast direct techniques for solving integral equations are rapidly gaining traction as viable alternatives to iterative schemes. Indeed, iterative methods converge slowly when applied to (intrinsically) ill-conditioned problem and grind to a halt when applied to problems involving many right-hand sides. Our group recently developed direct electric and combined field integral equation solvers that use the multilevel matrix decomposition algorithm (MLMDA) (a.k.a. butterfly schemes) to compress blocks of LU-factorized impedance matrices to attain O(N log² N) storage and CPU costs when applied to the analysis of scattering from 3D perfect electrically conducting surfaces (H. Guo, J. Hu and E. Michielssen, IEEE Radio Sci. Meeting, 2013 USNC-URSI, pp.91, Jul. 2013). Even though a rigorous proof of the method´s storage and CPU requirements remains elusive, all experimental results obtained to date indicate that these solvers´ memory and CPU requirements scale far more favorably than those of low-rank or adaptive cross approximation based solvers, which scale as O(N^α log N) (α = 1.3 ∼ 2.0) and O(N^α log^β N) (α = 2.0 ∼ 3.0, β ≥ 1.0), respectively.