Computation of scattering from N spheres using multipole reexpansion

Gumerov, Nail A.; Duraiswami, Ramani

doi:10.1121/1.1517253

Cited by 71 publications

(65 citation statements)

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“…¡ r ¢ in a speci ed range of indices n; l; m; and s: Below we represent one of the possible algorithms, which we used for computation of the Sj R-translation matrix in multiple scattering problem [24]. In this problem we needed a truncated matrix, where the indices lie in the range ¡N 6 m; s 6 N ; 0 6 n; l 6 N , where N is the truncation number.…”

Section: Computation Of Translation Coef Cients a Variety Of Recurrementioning

confidence: 99%

“…Here the complexity of the translation expressions on the one hand, and the numerical accuracy achievable on the other, are key barriers to use of these methods to more complicated problems that are of interest, and these are thus an area of active research. Other scienti c computing areas where there is a need for such translation theorems are in the solution of boundary value problems of scatterings from many spheres [24], and in the use of the T-matrix method for solution of scattering problems from many scatterers [14]. Note that in some multipole methods (e.g.…”

mentioning

confidence: 99%

“…Note that in some multipole methods (e.g. [24]) computation of each entry of the translation matrix is needed. In this case the recursive computation of the matrix elements provides the algorithm with theoretical minimum of asymptotic complexity.…”

mentioning

confidence: 99%

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Recursions for the Computation of Multipole Translation and Rotation Coefficients for the 3-D Helmholtz Equation

Gumerov¹,

Duraiswami²

2004

SIAM J. Sci. Comput.

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Abstract. We develop exact expressions for the coef cients of series representations of translations and rotations of local and multipole fundamental solutions of the Helmholtz equation in spherical coordinates. These expressions are based on the derivation of recurrence relations, some of which, to our knowledge are presented here for the rst time. The symmetry and other properties of the coef cients are also examined, and based on these, ef cient procedures for calculating them are presented. Our expressions are direct, and do not use the Clebsch-Gordan coef cients or the Wigner 3-symbols, though we compare our results with methods that use these, to prove their accuracy. For evaluating a term truncation of the translated series (involving multipoles), compared to previous exact expressions that require operations, our expressions require evaluations.Key words. Helmholtz equation, multipole solutions, translation and rotation coef cients, fast evaluation.AMS subject classi cations. 33C55, 33C10, 35J05, 65N38, 65N99, 65Y201. Introduction. In several scienti c computing applications, the solution to the Helmholtz or Maxwell Equations is expressed in terms of the singular (multipole) and regular solutions of the Helmholtz equation in spherical coordinates, centered at various points. Series of such solutions (see Eq. (2.16)) in one coordinate system must be expressed, in terms of series of singular or regular solutions in another coordinate system. Such expressions are guaranteed to exist by the completeness of the functions on a sphere. Addition theorems [5], [15] provide the expressions for the coef cients of the series in the shifted coordinates, in terms of the original coef cients. The paper by Epton and Dembart [8] provides an introduction to expressions of the coef cients. Chew [22] applied differentiation theorems for spherical functions similar to those in this paper, to obtain recursions for the translation coef cients.One important scienti c computing area where there is a need for such expressions is in the Fast Multipole Method (FMM) solution of the Helmholtz and Maxwell equations [9,27,10]. The FMM algorithm was referred in [1] as one of the top algorithms of the 20th century. Here the complexity of the translation expressions on the one hand, and the numerical accuracy achievable on the other, are key barriers to use of these methods to more complicated problems that are of interest, and these are thus an area of active research. Other scienti c computing areas where there is a need for such translation theorems are in the solution of boundary value problems of scatterings from many spheres [24], and in the use of the T-matrix method for solution of scattering problems from many scatterers [14]. Note that in some multipole methods (e.g. [24]) computation of each entry of the translation matrix is needed. In this case the recursive computation of the matrix elements provides the algorithm with theoretical minimum of asymptotic complexity. For speci c applications we refer the reader to these papers....

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Section: Computation Of Translation Coef Cients a Variety Of Recurrementioning

confidence: 99%

mentioning

confidence: 99%

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Recursions for the Computation of Multipole Translation and Rotation Coefficients for the 3-D Helmholtz Equation

Gumerov¹,

Duraiswami²

2004

SIAM J. Sci. Comput.

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“…To solve the multiple scattering problem, we use the multipole reexpansion technique developed by Gumerov and Duraiswami [7]. Now, we consider the qth scatterer.…”

Section: Multipole Reexpansionmentioning

confidence: 99%

“…In this letter, we extend a semi-analytical method to solve multiple acoustic scattering problems from many scatterers by introducing the multipole reexpansion technique [7]. There are two purposes to formulating this semi-analytical method.…”

Section: Introductionmentioning

confidence: 99%