Trimmed multilevel fast multipole algorithm for D-type volume integral equations in electromagnetic scattering problems

Abstract

The Multilevel Fast Multipole Algorithm (MLFMA) is a state of the art com-putational method that requires O(NlogN) memory and computational complex-ity for N unknowns. Despite the low memory and computational complexity, the conventional MLFMA has still some challenges due to prolonging iterations for large-scale electromagnetic scattering problems, especially for volumetric prob-lems. We present a novel application of trimmed MLFMA (T-MLFMA) to D-type volume integral equations. In T-MLFMA, thresholding and machine learning (ML) techniques are performed to eliminate the unneeded interactions as the MLFMA iterations proceed. Particularly, the converged current coefficients are determined via a Fully Connected Neural Network (FCNN), and the tree structure is systematically modified and becomes sparser. Therefore, the convergence of the problem is accelerated while matrix-vector multiplication time per iteration is also reduced. The training of the network is performed with only a small size of homogeneous dielectric spheres with different permittivity values. Then, we attack the scattering problem of homogeneous and inhomogeneous relatively more complex dielectric geometries, such as spherical shell layer, cone, torus, cylinder, cube, etc. As a result, significant speed-up is achieved with a controllable and limited error with respect to the conventional MLFMA in all simulations.