The fast multipole method (FMM) and multilevel fast multipole algorithm (MLFMA) are reviewed. The number of modes required, block-diagonal preconditioner, near singularity extraction, and the choice of initial guesses are discussed to apply the MLFMA to calculating electromagnetic scattering by large complex objects. Using these techniques, we can solve the problem of electromagnetic scattering by large complex three-dimensional (3-D) objects such as an aircraft (VFY218) on a small computer.

The fast multipole method (FMM) has been implemented to speed up the matrix‐vector multiply when an iterative method is used to solve the combined field integral equation (CFIE). FMM reduces the complexity from O(N2) to O(N1.5). With a multilevel fast multipole algorithm (MLFMA), it is further reduced to O(N log N). A 110, 592‐unknown problem can be solved within 24 h on a SUN Sparc 10. © 1995 John Wiley & Sons, Inc.

In this paper, we present an accurate method of moments (MoM) solution of the combined field integral equation (CFIE) using the multilevel fast multipole algorithm (MLFMA) for scattering by large, three-dimensional (3-D), arbitrarily shaped, homogeneous objects. We first investigate several different MoM formulations of CFIE and propose a new formulation, which is both accurate and free of interior resonances. We then employ MLFMA to significantly reduce the memory requirement and computational complexity of the MoM solution. Numerical results are presented to demonstrate the accuracy and capability of the proposed method. The method can be extended in a straightforward manner to scatterers composed of different homogeneous dielectric and conducting objects.

ABSTRACT|-FISC (Fast Illinois Solver Code) is designed to compute RCS of a target described by a triangular facet le. The problem is formulated by the method of moments (MoM), where the RWG (Rao, Wilton, and Glisson) basis functions are used. The resultant matrix equation is solved iteratively by the conjugate gradient (CG) method. The multilevel fast multipole algorithm (MLFMA) is used to speed up the matrix-vector multiply in CG. Both complexities for the CPU time per iteration and memory requirements are of O(N logN), where N is the number of unknowns.

The fast multipole method is used to solve the electromagnetic scattering from three‐dimensional conducting bodies of arbitrary shape. The electric field integral equation is discretized by the method of moments. Instead of directly computing the matrix‐vector multiplication, which needs N2 multiplications, this approach reduces the complexity to O(N1.5). © 1994 John Wiley & Sons, Inc.

The numerical solution of wave scattering from large objects or from a large cluster of scatterers requires excessive computational resources and it becomes necessary to use approximatebut fast-methods such as the fast multipole method; however, since these methods are only approximate, it is important to have an estimate for the error introduced in such calculations. An analysis of the error for the fast multipole method is presented and estimates for truncation and numerical integration errors are obtained. The error caused by polynomial interpolation in a multilevel fast multipole algorithm is also analyzed. The total error introduced in a multilevel implementation is also investigated numerically.Key words. fast multipole method, multilevel fast multipole algorithm, error analysis, truncation error, integration error, interpolation error AMS subject classifications. 65G99, 78A40PII. S00361429973281111. Introduction. The major computational task in scattering problems is the solution of dense systems of linear equations. When iterative methods are employed to solve these systems, the computation of the product of the coefficient matrix with a trial vector is the basic part of the algorithm and would require O(N 2 ) operations with traditional methods. The matrix vector product can be considered as the evaluation of all pairwise interactions between scatterers or parts of a scatterer. The fast multipole method (FMM) was proposed by Greengard and Rokhlin [1] for particle simulations and later extended for acoustic and electromagnetic scattering calculations [2,3,4,5,6,7]. By using a multilevel fast multipole algorithm (MLFMA), all pairwise interactions can be evaluated with O(N ) complexity for a volume distribution of scatterers.The FMM/MLFMA algorithm is based on the diagonal forms of the spherical addition theorem. These diagonal forms are given as integrals of truncated infinite sums. In the numerical evaluation of these expressions, two kinds of errors are introduced. First, the infinite expansions are truncated at some finite value. Second, the integrals are evaluated by a numerical quadrature. Therefore, it is imperative to derive estimates for these errors in terms of the basic parameters of the algorithm. In a multilevel implementation of the fast multipole algorithm, one resorts to interpolating the far field expansions in order to achieve O(N ) complexity resulting in a third kind of error.

The fast multipole method-fast Fourier transform (FMM-FFT) method is developed to compute the scattering of an electromagnetic wave from a two-dimensional rough surface. The resulting algorithm computes a matrixvector multiply in O(NlogN) operations. This algorithm is shown to be more e cient than another O(NlogN) algorithm, the multi-level fast multipole algorithm (MLFMA), for surfaces of small height. For surfaces with larger roughness, the MLFMA is found to be more e cient. Using the MLFMA, Monte Carlo simulations are carried out to compute the statistical properties of the electromagnetic scattering from two-dimensional random rough surfaces using a workstation. For the rougher surface, backscattering enhancement is clearly observable as a pronounced peak, in the backscattering direction, of the computed bistatic scattering coe cient. For the smoother surface, the Monte Carlo results compare well with the results of the approximate Kirchho theory.

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