Incomplete LU Preconditioning with the Multilevel Fast Multipole Algorithm for Electromagnetic Scattering

Abstract

Iterative solution of large-scale scattering problems in computational electromagnetics with the multilevel fast multipole algorithm (MLFMA) requires strong preconditioners, especially for the electric-field integral equation (EFIE) formulation. Incomplete LU (ILU) preconditioners are widely used and available in several solver packages. However, they lack robustness due to potential instability problems. In this study, we consider various ILU-class preconditioners and investigate the parameters that render them safely applicable to common surface integral formulations without increasing the O(n log n) complexity of MLFMA. We conclude that the no-fill ILU(0) preconditioner is an optimal choice for the combined-field integral equation (CFIE). For EFIE, we establish the need to resort to methods depending on drop tolerance and apply pivoting for problems with high condition estimate. We propose a strategy for the selection of the parameters so that the preconditioner can be used as a black-box method. Robustness and efficiency of the employed preconditioners are demonstrated over several test problems. Introduction.A popular approach in studying wave scattering phenomena in computational electromagnetics (CEM) is to solve discretized surface integral equations, which give rise to large, dense, complex systems in the form of A · x = b. Two kinds of surface integral equations are commonly used. The electric-field integral equation (EFIE) can be used for both open and closed geometries, but it results in poorly conditioned systems, especially when the geometry is large in terms of the wavelength. On the other hand, the combined-field integral equation (CFIE) produces well-conditioned systems, but it is applicable to closed geometries only [21].For the solution of such dense systems, direct methods based on Gaussian elimination are still widely used due to their robustness [34]. However, the large problem sizes confronted in computational electromagnetics prohibit the use of these methods which have O(n 2 ) memory and O(n 3 ) computational complexity for n unknowns. On the other hand, by making use of the multilevel fast multipole algorithm (MLFMA) [12], the dense matrix-vector products required at least once in each step of the iterative solvers can be performed in O(n log n) time and by storing only the sparse near-field matrix elements, rendering these solvers very attractive for large problems.However, the iterative solver may not converge, or convergence may require too many iterations. We need to have a suitable preconditioner to reach convergence in

Solutions of Large-Scale Electromagnetics Problems Using an Iterative Inner-Outer Scheme With Ordinary and Approximate Multilevel Fast Multipole Algorithms

“…Recently, we showed that an ILU preconditioner without a threshold provides an inexpensive and good approximation to the solution of the near-field matrix for CFIE, hence it reduces the iteration counts and solution times substantially [18]. For ill-conditioned EFIE matrices, however, ILUT (i.e., threshold-based ILU) with pivoting [19] is required to prevent the potential instability.…”

Computational analysis of complicated metamaterial structures using MLFMA and nested preconditioners

“…To obtain a convergence in a reasonable number of iterations, we employ enhanced preconditioning techniques that are also convenient for MLFMA implementations. For example, we consider preconditioners based on incomplete LU (ILU) factorization [10] or sparse approximate inverse (SAI) of the near-field matrix [1]. Although these preconditioners provide efficient solutions for ordinary frequencies, formulations at resonance frequencies lead to very ill-conditioned matrix equations, which need further information than that provided by the near-field matrix for a convergence.…”

The Efficient Analysis of Electromagnetic Scattering From Composite Structures Using Hybrid Cfie-Iefie

“…In this paper, we propose a novel technique denoted as the hybrid CFIE-IEFIE to accelerate the convergence rate, which replaces the EFIE part in the hybrid CFIE-EFIE techniques with the improved electric field integral equation (IEFIE) [17]. To further accelerate the solution, the incomplete LU preconditioning technique in [18,19] is proposed to solve the hybrid CFIE-IEFIE.…”

“…There are two popular drop strategies for ILU factorization: the level based drop strategy and the threshold based drop strategy, the former is denoted ILU(p), where p ≥ 0 is an integer denoted as the level of fill-in, and the latter is ILUT. The ILU-class preconditioners have been tested for electromagnetic problems in [19]. It can be found that ILU(0) preconditioner produces highly unstable and hence useless factorizations for the EFIE as the number of unknowns increases.…”