Sparse Inverse Preconditioning of Multilevel Fast Multipole Algorithm for Hybrid Integral Equations in Electromagnetics

Abstract

Abstract-In computational electromagnetics, the multilevel fast multipole algorithm (MLFMA) is used to reduce the computational complexity of the matrix vector product operations. In iteratively solving the dense linear systems arising from discretized hybrid integral equations, the sparse approximate inverse (SAI) preconditioning technique is employed to accelerate the convergence rate of the Krylov iterations. We show that a good quality SAI preconditioner can be constructed by using the near part matrix numerically generated in the MLFMA. The main purpose of this study is to show that this class of the SAI preconditioners are effective with the MLFMA and can reduce the number of Krylov iterations substantially. Our experimental results indicate that the SAI preconditioned MLFMA maintains the computational complexity of the MLFMA, but converges a lot faster, thus effectively reduces the overall simulation time.Index Terms-Electromagnetic scattering, Krylov methods, multilevel fast multipole algorithm (MLFMA), sparse approximate inverse (SAI) preconditioning.

A Calderón multiplicative preconditioner for the electromagnetic Poincaré–Steklov operator of a heterogeneous domain with scattering applications

“…Consider the electric field wave equation in the heterogeneous domain (25), which follows directly from (2), and its weak formulation (26).…”

“…If e ∈ V h satisfies (25), then e and h = j ωµ0 ∇ × e satisfy (15), implying thatn × ( j ωµ0 ∇ × e) = P(n × e) = P(n × e b ). By (33) and (31) we get…”

“…A wide class of preconditioners for hybrid FEM-BEM formulations exists [24][25][26][27][28], with an important subset of them relying on domain decomposition methods [29,30]. For the first time, a Calderón multiplicative preconditioner is applied to a reduced hybrid FEM-BEM formulation for electromagnetic scattering at a heterogeneous obstacle, and is compatible with existing matrix vector product acceleration schemes, such as the multilevel fast multipole algorithm (MLFMA) [31][32][33][34].…”

AN OVERVIEW OF THE EVOLUTION OF METHOD OF MOMENTS TECHNIQUES IN MODERN EM SIMULATORS (Invited Paper)

“…Obviously, with a higher number of elements the resulting factors involve a closer estimation of [Z n ] −1 , at the expense of higher CPU-time and storage requirements. A third commonly used choice in modern simulators is the Sparse Approximate Inverse (SAI) preconditioner [38][39][40][41], whose goal is the generation of a sparse preconditioning matrix that resembles the inverse matrix [Z n ] −1 , but using a predefined (or dynamically defined in some cases) sparsity pattern. This preconditioning technique was presented in [38] for dense matrices and has recently received wide attention in the EM community.…”

Geometry Based Preconditioner for Radiation Problems Involving Wire and Surface Basis Functions

“…Some of the most common techniques are those based on the sparse approximate inverse (SAI) preconditioning [10][11][12] and the incomplete LU (ILU) factorization type preconditioning [8,13,14].…”