Error Analysis for the Numerical Evaluation of the Diagonal Forms of the Scalar Spherical Addition Theorem

Abstract

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: Numerical Implementation

“…A particularly crucial point of the implementation is the interpolation and anterpolation steps. We here presented a variant of the scheme proposed by Chew and Lu in [41]. Its complexity is O(K ), if K is the number of samples in S 2 .…”

“…We will discuss here the scheme due to Alpert and Jakob-Chien [5] and the one due to Song, Lu, and Chew [40,41,53].…”

Error estimates in the Fast Multipole Method for scattering problems Part 2: Truncation of the Gegenbauer series

“…Greengard and Rokhlin were the first authors to provide empirical laws for the truncation integer L that achieves a given precision, at least when v is not too large. Those formulas have been fixed and improved by Chew and Song [17], (see also Chew [8]), but with no precise analytical error estimates. On the other hand, Rahola [21] then Darve [11], gave precise results, i.e.…”

“…Rahola [21] showed that the series was bounded by a geometrical series, Darve [11] analyzed more precisely the absolute error, and Koc et al [17] gave some elements of analysis, and mentioned the formula…”

“…where W (x) is the Lambert function defined in (17). Since x (v) is not necessarily in the form v k+ 1 2 with k an integer, we take…”