A Preliminary Comparison of MLFMA and $\mathcal{H}$-matrix Acceleration of Locally Corrected Nyström Solutions to Scattering Problems

Fast solutions of scattering problems with Multi-Level Fast Multipole Method (MLFMM) and $\mathcal{H}$ -matrix accelerated Locally Corrected Nyström (LCN) discretization of the Electric Field Integral Equation (EFIE), Magnetic Field Integral Equation (MFIE), and Combined Field Integral Equation (CFIE), are discussed and compared in terms of their CPU time usage. Fast iterative solutions of the matrix equations resulting from LCN discretizations of the EFIE and MFIE enabled with MLFMM acceleration of the matrix-vector products have been observed to require heavy preconditioning for realistic targets due to poor conditioning of the pertinent matrices. The application of sufficiently effective near field preconditioning can be comparable in complexity to solving the original matrix equation. $\mathcal{H}$ -matrix acceleration scheme enables direct solution of the pertinent matrix equation but exhibit a high cost associated with $\mathcal{H}-\text{LU}$ decomposition. Preliminary comparisons of these two iterative and non-iterative acceleration strategies is provided in this work.