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 problem of determining the field scattered by a cluster of scatterers when they are insonified by a known acoustical field is addressed. The problem is formulated by using the T-matrix method and the resulting system of linear equations is solved by using the multilevel fast multipole algorithm (MLFMA) and the fast multipole method–fast Fourier transform (FMMFFT) method, and the efficiency of the two methods is compared. It was observed that, in general, the MLFMA performs better than the FMMFFT algorithm. However, when the scatterers are distributed uniformly on a rectangular grid, the FMMFFT algorithm performs as good as the MLFMA. The accuracy of the methods is evaluated by modeling a spherical scatterer as composed of many small spheres.

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The importance of expanding Green's functions, particularly free-space Green's functions, in terms of orthogonal wave functions is practically self-evident when frequency domain scattering problems are of interest. With the relatively recent and widespread interest in time domain scattering problems, similar expansions of Green's functions are expected to be useful in the time domain. In this paper, two alternative expressions, expanded in terms of orthogonal spherical wave functions, for the free-space time domain scalar Green's functions are presented. Although the two expressions are equivalent, one of them is seen to be more convenient for the calculation of the scattered field for a known equivalent source density, whereas the second expression is more suitable for setting up an integral equation for the equivalent source density. Such an integral equation may be setup, for example, by the application of a time domain equivalent of the T-matrix ͑extended boundary condition͒ method.

A broadband (1.71–2.17 GHz), high isolation, and dual‐polarized suspended plate antenna (SPA) is presented. Modified L‐probe feed is used for broadband matching and capacitive coupling. Port isolation between two orthogonally polarized channels is improved by making extensions on suspended plates (SPs) and inserting a vertical ungrounded square tube under the first SP (SP1). The prototyped SPA resulted in 8 ± 0.5 dBi gain and 67° ± 6° symmetric E‐plane and H‐plane radiation patterns with low cross‐polarization levels. Return losses ≤−15 dB and port isolation ≤−30 dB are achieved in the entire band. © 2015 Wiley Periodicals, Inc. Microwave Opt Technol Lett 57:603–607, 2015

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The detection and estimation problems with large dimensional vectors frequently appear in the phased array radar systems equipped with, possibly, several hundreds of receiving elements. For such systems, a preprocessing stage reducing the large dimensional input to a manageable dimension is required. The present work shows that the subspace spanned by the generalized eigenvectors of signal and noise covariance matrices is the optimal subspace to this aim from several different viewpoints. Numerical results on the subspace selection for the radar target detection problem is provided to examine the performance of detectors with reduced dimensions.