an efficient algorithm for the generalized foldy–lax formulation

Journal of Computational Physics 234 (2013) 376–398

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Journal of Computational Physics

journal homepage: www.elsevier .com/locate / jcp

An efficient algorithm for the generalized Foldy–Lax formulation

Kai Huang a, Peijun Li b,⇑,1, Hongkai Zhao c,2

a Department of Mathematics, Florida International University, Miami, FL 33199, United Statesb Department of Mathematics, Purdue University, West Lafayette, IN 47907, United Statesc Department of Mathematics, University of California at Irvine, Irvine, CA 92697, United States

a r t i c l e i n f o

Article history:Received 19 November 2011Received in revised form 15 August 2012Accepted 24 September 2012Available online 23 October 2012

Keywords:Foldy–Lax formulationPoint scatterersObstacle scatteringBoundary integral equationHelmholtz equation

0021-9991/$ - see front matter Published by Elsevihttp://dx.doi.org/10.1016/j.jcp.2012.09.027

⇑ Corresponding author.E-mail addresses: [email protected] (K. Huang), lip

1 The research was supported in part by NSF Gran2 The research was supported in part by NSF Gran

a b s t r a c t

Consider the scattering of a time-harmonic plane wave incident on a two-scale heteroge-neous medium, which consists of scatterers that are much smaller than the wavelengthand extended scatterers that are comparable to the wavelength. In this work we treat thosesmall scatterers as isotropic point scatterers and use a generalized Foldy–Lax formulationto model wave propagation and capture multiple scattering among point scatterers andextended scatterers. Our formulation is given as a coupled system, which combines theoriginal Foldy–Lax formulation for the point scatterers and the regular boundary integralequation for the extended obstacle scatterers. The existence and uniqueness of the solutionfor the formulation is established in terms of physical parameters such as the scatteringcoefficient and the separation distances. Computationally, an efficient physically motivatedGauss–Seidel iterative method is proposed to solve the coupled system, where only a linearsystem of algebraic equations for point scatterers or a boundary integral equation for asingle extended obstacle scatterer is required to solve at each step of iteration. The conver-gence of the iterative method is also characterized in terms of physical parameters. Numer-ical tests for the far-field patterns of scattered fields arising from uniformly or randomlydistributed point scatterers and single or multiple extended obstacle scatterers arepresented.

Published by Elsevier Inc.

1. Introduction

Scattering problems are concerned with how an inhomogeneous medium scatters an incident field. The direct scatteringproblem is to determine the scattered field from the incident field and the differential equation governing the wave prop-agation; the inverse scattering problem is to determine the nature of the inhomogeneity, such as location, geometry, andmaterial property, from a knowledge of the scattered field. These problems have played a fundamental role in diversescientific areas such as radar and sonar, geophysical exploration, medical imaging, near-field and nano-optics. This workis devoted to the modeling and computation of a direct scattering problem, where the inhomogeneous medium is assumedto be of two-scale nature: it is composed of a group of isotropic point scatterers and an extended obstacle scatterer withpossible multiple disjoint components. ‘‘Isotropic point scatterer’’ is a simple but effective model for a scatterer whose sizeis much smaller than the wavelength of the incident wave so that the scatterer can be represented by a source point withinit; ‘‘extended’’ means that the scale of the obstacle scatterer is comparable with the wavelength of the incident field. Pre-cisely, we consider the scattering of a time-harmonic plane wave incident on a heterogeneous medium with both point

er Inc.

[email protected] (P. Li), [email protected] (H. Zhao).ts DMS-0914595, DMS-1042958, and DMS-1151308.t DMS-1115698 and ONR Grant N00014-11-1-0602.

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Fig. 1. Schematic of problem geometry. The scattered wave w is generated from the incidence of a plane wave /inc on the heterogeneous medium consistinga group of point scatterers centered at rj and an extended obstacle scatterer with possible multiple disjoint components represented by the domain Dk withboundary Ck .

K. Huang et al. / Journal of Computational Physics 234 (2013) 376–398 377

and extended scatterers, and intend to study the wave propagation and the far-field pattern of the scattered field arisingfrom the interactions between the incident field and all the scatterers, as seen in Fig. 1.

The original Foldy–Lax formulation was proposed by Foldy [4] and Lax [8] to describe the multiple scattering of acousticwaves by a group of small isotropic scatterers. This formulation effectively gives the wave fields by solving a self-consistentlinear system of algebraic equations for the idealized situation, where the medium is viewed as a collection of isotropic pointscatterers. As the size of the scatterers increases to be comparable with the wavelength of the incident field, the originalFoldy–Lax formulation will no longer be appropriate to accurately describe the multiple scattering among these scatterers.It is required to consider the effect of the shape of these scatterers on the scattered field. As pointed out in the book by Coltonand Kress [3], one of the two basic problems in classical scattering theory is the scattering of time-harmonic acoustic or elec-tromagnetic waves by bounded impenetrable obstacles, which is known as the obstacle scattering problem. Integral equa-tion methods have played an important role in the study of exterior boundary value problems associated with the obstaclescattering. The major advantage of the use of boundary integral equation methods lies in the fact that this approach reducesa problem defined over an unbounded domain to one defined on a bounded domain of lower dimension, i.e., the boundary ofthe scattering obstacle. We refer to the monograph by Colton and Kress [2] for a comprehensive account of the boundaryintegral equation methods for solving the obstacle scattering problem.

In many situations it is desirable to develop a model for the simulation of wave propagation in a cluttered medium withembedded extended scatterers, such as in the application of imaging a target in a cluttered environment [10]. Thus it isimportant to resolve not only the geometry of the extended scatterers but also the effect of the surrounding cluttered med-ium on the scattered field. Multiple scattering in a multiscale medium presents both mathematical and numerical chal-lenges. In this work, the medium clutter is modeled as a group of isotropic point scatterers and the target is modeled byan extended obstacle scatterer with possible multiple disjoint components. When a time-harmonic plane wave is incidenton the heterogeneous medium, the scattered waves will be generated from the point scatterers, the extended obstacles andmultiple scattering among them. The original Foldy–Lax formulation provides an effective approach to solve the scatteringproblem from a collection of point scatterers; the method of boundary integral equation has been considered as a desirablechoice to solve the obstacle scattering problem. The objective of this work is combine these two methods such that the mul-tiple scattering among all the scatterers can be fully taken into account, without increasing the computational complexityrelative to the separated scattering problems especially for multiple extended obstacle scatterers.

Recently, based on the extended-boundary condition method [1], a generalized Foldy–Lax formulation was developed byHuang et al. [6] for solving the two-dimensional Helmholtz equation in the heterogeneous medium, where the formulation isrepresented as a series in terms of spherical harmonics and scattering operators. Due to the restriction of the extended-boundary condition method, the formulation in [6] is limited to a single obstacle scatterer. It also assumes that the pointscatterers and the obstacle scatterer can be separated by a circle: all the point scatterers are located in the exterior of thecircle; the obstacle scatterer is located in the interior of the circle. To overcome the limitations, Huang and Li [5] studiedthe two-scale multiple scattering problem and proposed a boundary integral equation based generalized Foldy–Lax formu-lation for the three-dimensional Helmholtz equation for arbitrarily distributed point scatterers. In principle, the formulationin [5] can be used to deal with not only an obstacle scatterer with general geometry but also multiple disjoint obstacle scat-terers. However, no discussion is given on how to efficiently solve the generalized Foldy–Lax formulation when the extendedobstacle scatterer has more than one component.

To be able to handle more general obstacle scatterers including multiple disjoint components, we re-examine the general-ized Foldy–Lax formulation given in [5] and develop an improved formulation, which is more stable with respect to thewavenumber since a uniquely solvable boundary integral equation is incorporated into the coupled system. Sufficient con-ditions for the existence and uniqueness for the solution of the improved formulation are established in terms of the physicalparameters such as the scattering coefficient of the point scatterers and the separation distance between the point scatterersand the extended obstacle scatterers. Computationally, an efficient physically based Gauss–Seidel iterative method is pro-posed to solve the coupled system, where only a linear system of algebraic equations or a boundary integral equation for

378 K. Huang et al. / Journal of Computational Physics 234 (2013) 376–398

a single obstacle is required to solve at each step of iteration. The convergence of the iterative method is also characterized interms of the physical parameters such as the scattering coefficients and the separation distances.

Although all discussion in this work uses the two-dimensional Helmholtz equation, all the results can be straightfor-wardly extended to the three-dimensional Helmholtz equation which has a even simpler Green’s function with a fasterdecay in distance.

The outline of the paper is as follows. In Section 2, the original Foldy–Lax formulation is briefly reviewed for the scatteredfield from the interaction among a group of point scatterers. Based on combined single- and double-layer potential repre-sentations, a uniquely solvable boundary integral equation is introduced for the sound-soft obstacle scattering problem inSection 3. Section 4 is devoted to the improved generalized Foldy–Lax formulation: a coupled scattering system is derivedfrom the boundary integral equation approach; the existence and uniqueness for the solution of the improved formulation isestablished; a physically motivated block Gauss–Seidel iterative method is proposed, and the convergence of the iteration isdiscussed. In Section 5, the far-field patterns of the scattered fields from original and generalized Foldy–Lax formulationsare described. Numerical techniques, particularly the implementation of the uniquely solvable boundary integral equation,are presented in Section 6. Numerical results are shown in Section 7 for various examples of the point scatterers and theextended obstacle scatterers. The paper is concluded with comments and directions for future work in Section 8.

2. Foldy–Lax formulation for point scatterers

In this section, we briefly introduce the original Foldy–Lax formulation to model the scattering of isotropic point scatter-ers. We refer to the book by Martin [9] for detailed discussions on the Foldy–Lax formulation.

Consider a collection of m separated isotropic point scatterers, which can be represented by point sources located atr1; . . . ; rm. Let /inc be the plane incident wave, given explicitly

/incðrÞ ¼ eijr�d in R2; ð2:1Þ

where i is the imaginary unit, j is the wavenumber, and d ¼ ðcos a; sin aÞ 2 S1 is the propagation direction defined on theunit circle and a 2 ½0;2p� is the incident angle. The incident field satisfies the Helmholtz equation

D/inc þ j2/inc ¼ 0 in R2: ð2:2Þ

The total field is represented as the sum of the incident field and the scattered field

/ðrÞ ¼ /incðrÞ þXm

j¼1

rj/jGðr; rjÞ; ð2:3Þ

where rj > 0 is referred to as the scattering coefficient for the jth point scatterer, and G the free space Green’s function givenas

Gðr; r0Þ ¼ i4

Hð1Þ0 ðjjr� r0jÞ:

Here Hð1Þ0 is the Hankel function of first kind with order zero.Evaluating (2.3) at ri and excluding the self-interaction yield a linear system of algebraic equation for /j; j ¼ 1; . . . ;m:

/i ¼ /incðriÞ þX

j¼1j–i

mrj/jGðri; rjÞ; ð2:4Þ

which is known as the Foldy–Lax formulation. Once (2.4) is solved, the scattered field can be computed as

wðrÞ ¼Xm

j¼1

rj/jGðr; rjÞ: ð2:5Þ

The Foldy–Lax formulation (2.4) can be written in the matrix form

A/ ¼ /inc; ð2:6Þ

where / ¼ ð/1; . . . ;/mÞT;/inc ¼ ð/incðr1Þ; . . . ;/incðrmÞÞT, and the m�m coefficient matrix A is

A ¼

1 �r2Gðr1; r2Þ � � � �rmGðr1; rmÞ

�r1Gðr2; r1Þ 1 � � � �rmGðr2; rmÞ

..

. ... . .

. ...

�r1Gðrm; r1Þ �r2Gðrm; r2Þ � � � 1

2666666664

3777777775:

Given a vector x ¼ ðx1; x2; . . . ; xmÞT 2 Cm and a matrix B ¼ ½bij�m�m 2 Cm�m, introduce the maximum norms


kxk1 ¼ max16j6m

jxij and kBk1 ¼ max16i6m

Xm

j¼1

jbijj:

Define the following two physical parameters, maximum scattering strength and the least separation distance, for the groupof point scatterers,

rmax ¼ max16j6m

rj and qP ¼ min16i<j6m

jri � rjj:

They will be used to characterize the criterion for the well-posedness of the Foldy–Lax formulation (2.6).

Theorem 2.1. If rmax q�1=2P is sufficiently small, then the coefficient matrix A is invertible and its inverse matrix has the estimate

kA�1k1 6 c1; ð2:7Þ

where the constant c1 is independent of rj and rj for j ¼ 1; . . . ;m.

Proof. Split the matrix A ¼ I �M, where

M ¼

0 r2Gðr1; r2Þ � � � rmGðr1; rmÞr1Gðr2; r1Þ 0 � � � rmGðr2; rmÞ

..

. ... . .

. ...

r1Gðrm; r1Þ r2Gðrm; r2Þ � � � 0

266664

377775:

Recalling for large argument, we have the following asymptotic behavior for the Hankel function of first kind with order l:

Hð1Þl ðzÞ ¼ffiffiffiffiffiffi2pz

rei z�p

4�lp2ð Þ 1þ Oðz�1Þ� �

: ð2:8Þ

Using (2.8), we have for well-separated point scatterers that

jGðri; rjÞj ¼14jHð1Þ0 ðjjri � rjjÞj 6 c0jri � rjj�1=2

6 c0q�1=2P ; ð2:9Þ

where c0 is a positive constant. Following the matrix norm and (2.9) yield

kMk1 6 rmax

Xj¼1j–i

mjGðri; rjÞj 6 mc0rmax q�1=2P < 1

for sufficiently small rmaxq�1=2P . The above estimate shows that the matrix A ¼ I �M is invertible, and furthermore leads to

the estimate

kA�1k1 ¼ kðI �MÞ�1k1 6 ð1� kMk1Þ�16 1�mc0rmax q�1=2

P

� ��1¼ c1: �

Remark 2.1. As seen from Theorem 2.1, given the number of the point scatterers m, if the maximum scattering coefficientrmax is sufficiently small or the separation distance among the point scatterers qP is sufficiently large, then the Foldy–Laxformulation (2.6) has a unique solution. Furthermore, we have from (2.4) that the multiple scattering among the pointscatterers is weak and the scattered field can be approximated by the simple summation

wðrÞ �Xm

j¼1

rj/incðrjÞGðr; rjÞ; ð2:10Þ

which is known as the Born approximation.

3. Boundary integral equations for extended obstacle scatterers

This section is concerned with a brief introduction of the boundary integral equation method for solving the obstacle scat-tering problem. We assume that the obstacle is represented by the domain D with boundary C, which is the open comple-ment of an unbounded domain of class C2, i.e., scattering from more than one component is included in our analysis.

Consider the two-dimensional Helmholtz equation

D/þ j2/ ¼ 0 in R2 n D; ð3:1Þ

along with the sound-soft boundary condition

/ ¼ 0 on C; ð3:2Þ


where / is the total field and j is the wavenumber.The obstacle is illuminated by the plane incident wave (2.1). The total field / consists of the incident field /inc and the

scattered field w:

/ ¼ /inc þ w: ð3:3Þ

It follows from (2.2) and (3.1)–(3.3) that the scattered field satisfies

Dwþ j2w ¼ 0 in R2 n D; ð3:4Þ

together with the boundary condition

w ¼ �/inc on C: ð3:5Þ

In addition, the scattered field is required to satisfy the Sommerfeld radiation condition

limq!1

ffiffiffiffiqp @w

@q� ijw

� �¼ 0; q ¼ jrj; ð3:6Þ

uniformly for all directions r̂ ¼ r=jrj.Based on the Green’s representation theorem, it can be shown that the scattered field has the integral representation in

terms of the normal derivative of the total field:

wðrÞ ¼ �Z

CGðr; r0Þ@n0/ðr0Þdsðr0Þ; r 2 D: ð3:7Þ

To compute the scattered field, it is required to determine @/ on C. We adopt the following uniquely solvable integralequation

12@n/ðrÞ þ

ZC@nGðr; r0Þ � igGðr; r0Þ½ �@n0/ðr0Þdsðr0Þ ¼ ð@n � igÞ/incðrÞ; ð3:8Þ

where n is the unit outward normal with respect to the variable r and g is a nonzero real number and is called the couplingparameter. We refer to Kress [7] for an investigation on the proper choice of the coupling parameter g with respect to thecondition number of the coefficient matrix for the integral equation.

Given a domain X, define

kuk0;1;X ¼ supr2XjuðrÞj:

For a positive integer l, introduce a standard Sobolev space

Wl;1ðXÞ ¼ fu 2 L1locðXÞ : kukl;1;X 61g;

where the Sobolev norm is

kukl;1;X ¼maxjkj6lkDkuk0;1;X:

Due to the unique solvability of the boundary integral Eq. (3.8), we have the following stability result. The proof may befound in [5].

Theorem 3.1. The unique solution to the boundary integral Eq. (3.8) depends continuously on the incident field, i.e.,

k@n/k0;1;C 6 c2k/inck0;1;C; ð3:9Þ

where the constant c2 depends on j;g, and C.

Remark 3.1. If the extended obstacle scatterer is consisted of n disjoint components, i.e., C ¼ C1 [ � � � [ Cn; Ck \ Ck0 ¼ ;; k – k0,then the boundary integral Eq. (3.8) can be written as

12uðrÞ þ

Xn

k¼1

ZCk

@nGðr; r0Þ � igGðr; r0Þ½ �ukðr0Þdsðr0Þ ¼ ð@n � igÞ/incðrÞ; ð3:10Þ

where uðrÞ ¼ @n/ðrÞ and ukðrÞ ¼ uðrÞjr2Ck.


4. Generalized Foldy–Lax formulation

This section is devoted to an improved generalized Foldy–Lax formulation for the multiple scattering involving extendedobstacle scatterers and a group of point scatterers. This new coupled system is different from that in [5] since a uniquelysolvable boundary integral equation is incorporated, which make it more stable with respect to the wavenumber. Thewell-posedness of the new formulation is discussed, an efficient block Gauss–Seidel iterative method is proposed for solvingthe coupled system, and the convergence criteria of the iteration are given in terms of physical parameters.

4.1. Coupled scattering system

Viewing the external field acting on the point scatterers as point sources for the obstacle, we consider the Helmholtzequation

D/þ j2/ ¼ �Xm

j¼1

rj/jdðr� rjÞ in R2 n D; ð4:1Þ

along with the sound-soft boundary condition

/ ¼ 0 on C; ð4:2Þ

where /j is the external field acting on the jth point scatterer. Subtracting the incident field from the total field, we mayobtain the equation for the scattered field

Dwþ j2w ¼ �Xm

j¼1

rj/jdðr� rjÞ inR2 n D: ð4:3Þ

The scattered field is also required to satisfy the sound-soft boundary condition (3.5) and the Sommerfeld radiation condition(3.6).

It follows from Green’s theorem and the Helmholtz Eq. (4.3) that the scattered field satisfies

wðrÞ ¼Xm

j¼1

rj/jGðr; rjÞ þZ

C@n0Gðr; r0Þwðr0Þdsðr0Þ �

ZC

Gðr; r0Þ@n0wðr0Þdsðr0Þ; r 2 R2 n D ð4:4Þ

and

0 ¼Z

C@n0Gðr; r0Þwðr0Þdsðr0Þ �

ZC

Gðr; r0Þ@n0wðr0Þdsðr0Þ; r 2 D: ð4:5Þ

Similarly, the incident field satisfies

/incðrÞ ¼Z

CGðr; r0Þ@n0/incðr0Þdsðr0Þ �

ZC@n0Gðr; r0Þ/incðr0Þdsðr0Þ; r 2 D ð4:6Þ

and

0 ¼Z

C@n0Gðr; r0Þ/incðr0Þdsðr0Þ �

ZC

Gðr; r0Þ@n0/incðr0Þdsðr0Þ; r 2 R2 n D: ð4:7Þ

Adding (4.4) and (4.7) and using the sound-soft boundary condition (3.5) gives the representation for the scattered field interms of the total field and the external fields acting on the point scatterers:

wðrÞ ¼Xm

j¼1

rj/jGðr; rjÞ �Z

CGðr; r0Þ@n0/ðr0Þdsðr0Þ; r 2 R2 n D: ð4:8Þ

Adding the incident field on both sides of (4.8) yields

/ðrÞ ¼ /incðrÞ þXm

j¼1

rj/jGðr; rjÞ �Z

CGðr; r0Þ@n0/ðr0Þdsðr0Þ; r 2 R2 n D: ð4:9Þ

To compute the scattered field, it is required to compute the normal derivative of the total field, @n/, and the external fielddue to point scatterers, /j; j ¼ 1; . . . ;m.

Evaluating (4.9) on both sides at ri and excluding the self-interaction for the point scatterers give



j¼1j–i

mrj/jGðri; rjÞ �

ZC

Gðri; r0Þ@n0/ðr0Þdsðr0Þ: ð4:10Þ

Using the jump relation for the single-layer and double-layer potentials and taking the normal derivative, we obtain aboundary integral equation on C:

12@n/ðrÞ ¼ @n � igð Þ/incðrÞ þ

Xm

j¼1

rj/j@nGðr; rjÞ �Z

C@nGðr; r0Þ � igGðr; r0Þ½ �@n0/ðr0Þdsðr0Þ: ð4:11Þ

It is noticed that the boundary integral over C vanishes in (4.10) if the extended obstacle is not present, which reduces to theoriginal Foldy–Lax formulation (2.4); the summation over the number of point scatterers vanishes in (4.11) if the point scat-terers are not present, which reduces to the regular boundary integral Eq. (3.8) for solving the obstacle scattering problem.Thus, Eqs. (4.10) and (4.11) form the self-consistent generalized Foldy–Lax formulation to compute the interaction betweenthe extended obstacle and a set of point scatterers.

Remark 4.1. If there are multiple extended scatterers, then the generalized Foldy–Lax formulation can be written as thefollowing coupled system


j¼1j–i

mrj/jGðri; rjÞ �

Xn

k¼1

ZCk

Gðri; r0Þukðr0Þdsðr0Þ; ð4:12Þ

12uðrÞ ¼ @n � igð Þ/incðrÞ þ

Xm

j¼1

rj/j@nGðr; rjÞ �Xn

k¼1

ZCk

@nGðr; r0Þ � igGðr; r0Þ½ �ukðr0Þdsðr0Þ; ð4:13Þ

where uðrÞ ¼ @n/ðrÞ and ukðrÞ ¼ uðrÞjr2Ck.

4.2. Well-posedness of the formulation

This section is concerned with the well-posedness of the generalized Foldy–Lax formulation 4.10 and 4.11. An iterativeapproach is proposed to deduce criteria which assure the convergence of the iteration, and thus the existence and unique-ness of a solution to the generalized Foldy–Lax formulation.

Theorem 4.1. Given the number of point scatterers m and the arclength of the extended obstacle scatterer jCj, the generalizedFoldy–Lax formulation (4.10) and (4.11) has a unique solution if one of the following two conditions holds: (i) rmax is sufficientlysmall and (ii) qP and qPE are sufficiently large.Proof. Define the separation distance parameter between the point scatterers and the extended obstacle scatterer

qPE ¼ min16j6m

minr2Cjrj � rj:

Define two linear operators that characterizes wave field scattered by the extended scatterers and the point scatterersrespectively

Mu ¼Z

CGðr1; r0Þuðr0Þdsðr0Þ; . . . ;

ZC

Gðrm; r0Þuðr0Þdsðr0Þ� �T

and

ðN/ÞðrÞ ¼ �Xm

j¼1

rj/jGðr; rjÞ:

The generalized Foldy–Lax formulation (4.10) and (4.11) can be written as the operator form

A M

N 12 I þ K 0 � igS

" #/

u

¼

/inc

f

; ð4:14Þ

where f ¼ ð@n � igÞ/incðrÞ, and S;K 0 are the single layer and the adjoint double layer potential operators defined as

ðSuÞðrÞ ¼Z

CGðr; r0Þuðr0Þdsðr0Þ;

ðK 0uÞðrÞ ¼Z

C@nGðr; r0Þuðr0Þdsðr0Þ:

Consider the following iteration: let /ð0Þ ¼ ð0; . . . ;0ÞT and uð0ÞðrÞ ¼ 0, define /ðmÞ and uðmÞ for m P 1 by the solutions of thefollowing system of equations


A/ðmÞ ¼ /inc �Muðm�1Þ ð4:15Þ

and

12

I þ K 0 � igS� �

uðmÞ ¼ f � N/ðmÞ: ð4:16Þ

Clearly, the vector /ðmÞ and the function uðmÞ are well-defined due to the invertibility of the matrix A from Theorem 2.1 andthe operator 1

2 I þ K 0 � igS from Theorem 3.1. After mth iteration, waves that have been scattered fewer or equal to m timesbetween points scatterers and the extended obstacle have been captured.

To prove the convergence of the iteration (4.15) and (4.16), we define the differences of two consecutive approximations

d/ðmÞ ¼ /ðmÞ � /ðm�1Þ and duðmÞ ¼ uðmÞ �uðm�1Þ:

It follows from (4.15) and (4.16) that we have the error equations

Ad/ðmÞ ¼ �Mduðm�1Þ ð4:17Þ

and

12

I þ K 0 � igS� �

duðmÞ ¼ �Nd/ðmÞ: ð4:18Þ

An application of Theorem 2.1 to (4.17) yields

kd/ðmÞk1 6 c1kMduðm�1Þk1:

Using the asymptotic behavior of the Hankel function (2.8) and the definition of the operator M, we have

kd/ðmÞk1 6 c0c1jCjq�1=2PE kduðm�1Þk0;1;C; ð4:19Þ

where jCj is the arclength of the boundary C for the extended obstacle scatterer. An application of Theorem 3.1 to (4.18)gives

kduðmÞk0;1;C 6 c2kNd/ðmÞk0;1;C

It follows from the asymptotic behavior of the Hankel function (2.8) and the definition of the operator N that

kduðmÞk0;1;C 6 mc0c2rmax q�1=2PE kd/ðmÞk1: ð4:20Þ

Combining (4.19) and (4.20) gives

kduðmÞk0;1;C 6 nkduðm�1Þk0;1;C and kd/ðmÞk1 6 nkd/ðm�1Þk1; ð4:21Þ

where

n ¼ mc20c1c2jCjrmax q�1

PE < 1

for sufficiently small rmax or sufficiently large qPE.Following the definition of uðmÞ, we obtain

uðmÞ ¼ duð1Þ þ duð2Þ þ � � � þ duðmÞ;

which gives after using (4.21)

kuðmÞk0;1;C 6 ð1þ nþ n2 þ � � � þ nmÞkuð1Þk0;1;C 6 ð1� nÞ�1kuð1Þk0;1;C:

Obviously, it follows from the well-posedness of the Foldy–Lax formulation (2.6) and the boundary integral Eq. (3.8) thatkuð1Þk0;1;C <1. Hence, the above estimate indicate that the sequence fuðmÞg is a bounded sequence, and thus there existsa subsequence, which will be still denoted by fuðmÞg, converge to a limit u.

Similarly, we have from the definition of d/ that

/ðmÞ ¼ d/ð1Þ þ d/ð2Þ þ � � � þ d/ðmÞ;

which gives after using (4.21)

k/ðmÞk1 6 ð1þ nþ n2 þ � � � þ nmÞk/ð1Þk1 6 ð1� nÞ�1k/ð1Þk1:

Since k/ð1Þk1 <1, there exists a subsequence, which will be still denoted by f/ðmÞg, converge to a limit /.Next is to prove the uniqueness of the limits. If /0 ¼ ð/01; . . . ;/0mÞ

T and u0 are also limits of subsequences f/ðmÞg and fuðmÞg,then we have


/0i ¼ /incðriÞ þX

j¼1j–i

mrj/

0jGðri; rjÞ �

ZC

Gðri; r0Þu0ðr0Þdsðr0Þ; ð4:22Þ

12u0ðrÞ ¼ @n � igð Þ/incðrÞ þ

Xm

j¼1

rj/0j@nGðr; rjÞ �

ZC@nGðr; r0Þ � igGðr; r0Þ½ �u0ðr0Þdsðr0Þ: ð4:23Þ

Subtracting (4.22) and (4.23) from (4.10) and (4.11), respectively, and denoting d/ ¼ /� /0 and du ¼ u�u0, we obtain

d/i �Xm

j ¼ 1j–i

rjd/jGðri; rjÞ ¼ �Z

CGðri; r0Þduðr0Þdsðr0Þ; ð4:24Þ

12

duðrÞ þZ

C@nGðr; r0Þ � igGðr; r0Þ½ �duðr0Þdsðr0Þ ¼

Xm

j¼1

rjd/j@nGðr; rjÞ: ð4:25Þ

Regarding (4.24), it follows from Theorem 2.1 that we have

kd/k1 6 c0c1jCjq�1=2PE kduk0;1;C:

Regarding (4.25), using Theorem 3.1 yields

kduk0;1;C 6 mc0c2rmax q�1=2PE kd/k1:

Combining above two estimates gives

kd/k1 6 nkd/k1 and kduk0;1;C 6 nkduk0;1;C:

The uniqueness of the solutions follow from d/ ¼ 0 and du ¼ 0 since n < 1 for either sufficiently small rmax or sufficientlylarge qPE. h

Remark 4.2. The conditions in Theorem (4.1) are sufficient but may not be necessary for the well-posedness of the general-ized Foldy–Lax formulation (4.10) and (4.11).

4.3. Block Gauss–Seidel iterative method

In this section, we propose an efficient block Gauss–Seidel iterative method to numerically solve the generalized Foldy–Lax formulation (4.12) and (4.13), especially when there are multiple obstacles, and deduce some criteria to characterize theconvergence of the proposed iterative method.

Theorem 4.2. Given the number of point scatterers m and the arclength of the extended obstacle jCj, the block Gauss–Seideliteration (4.27) and (4.28) is convergent if one of the following two conditions holds: (i) rmax is sufficiently small and qE issufficiently large; (ii) qP;qPE, and qE are sufficiently large.

Proof. Denote two linear operators over the boundary Ck:

Mku ¼Z

Ck

Gðr1; r0Þuðr0Þdsðr0Þ; . . . ;

ZCk

Gðrm; r0Þuðr0Þdsðr0Þ !T

;

and

ðNk/ÞðrÞ ¼ �Xm

j¼1

rj/jGðr; rjÞjr2Ck:

Recall the single-layer and adjoint double-layer potential operators defined on the boundary Ck:

ðSkuÞðrÞ ¼Z

Ck

Gðr; r0Þuðr0Þdsðr0Þ;

ðK 0kuÞðrÞ ¼Z

Ck

@nGðr; r0Þuðr0Þdsðr0Þ:

The generalized Foldy–Lax formulation 4.12 and 4.13 can be written as


A M1 M2 � � � Mn

N112 I þ K 01 � igS1 K 02 � igS2 � � � K 0n � igSn

N2 K 01 � igS112 I þ K 02 � igS2 � � � K 0n � igSn

..

. ... ..

. . .. ..

.

Nn K 01 � igS1 K 02 � igS2...

12 I þ K 0n � igSn

2666666666664

3777777777775

/

u1

u2

..

.

un

266666666664

377777777775¼

/inc

f1

f2

..

.

fn

266666666664

377777777775; ð4:26Þ

where fkðrÞ ¼ ð@n � igÞ/incðrÞjr2Ck¼ f ðrÞjr2Ck

; k ¼ 1; . . . ;n. A block Gauss–Seidel method for the solution of (4.26) reads: Let/ð0Þ ¼ ð0; . . . ;0ÞT and uð0Þk ðrÞ ¼ 0 for k ¼ 1; . . . ;n, define /ðmÞ and uðmÞk for m P 1 by the solutions of the following system ofequations

A/ðmÞ ¼ /inc �Xn

k¼1

Mkuðm�1Þk ð4:27Þ

and

12

I þ K 0k � igSk

� �uðmÞk ¼ fk � Nk/

ðmÞ �Xk�1

j¼1

ðK 0j � igSjÞuðmÞj �Xn

j¼kþ1

ðK 0j � igSjÞuðm�1Þj : ð4:28Þ

Next we analyze the convergence of the block Gauss–Seidel iteration (4.27) and (4.28). Let / and uk for k ¼ 1; . . . ;n be thesolution of the generalized Foldy–Lax formulation (4.12) and (4.13). Denote by

d/ðmÞ ¼ /� /ðm�1Þ and duðmÞk ¼ uk �uðm�1Þk

the error between the exact solution and the iterative solution at the step m. Subtracting (4.27) and (4.28) from (4.12) and(4.13), respectively, we obtain

Ad/ðmÞ ¼ �Xn

k¼1

Mkduðm�1Þk ð4:29Þ

and

12

I þ K 0k � igSk

� �duðmÞk ¼ �Nkd/ðmÞ �

Xk�1

j¼1

ðK 0j � igSjÞduðmÞj �Xn

j¼kþ1

ðK 0j � igSjÞduðm�1Þj : ð4:30Þ

Regarding (4.29), an application of Theorem 2.1 yields

kd/ðmÞk1 6 c1

Xn

k¼1

kMkduðm�1Þk k1:

Using the asymptotic behavior of the Hankel function (2.8) and the definition of the operator Mk, we have

kd/ðmÞk1 6 c0c1 jCjq�1=2PE kduðm�1Þk0;1;C; ð4:31Þ

where

kduðm�1Þk0;1;C ¼ max16k6n

kduðm�1Þk k0;1;Ck

:

Using Theorem 3.1 for (4.30) gives

kduðmÞk k0;1;Ck6 c2 kNkd/

ðmÞk0;1;CkþXk�1

j¼1

kðK 0j � igSjÞduðmÞj k0;1;CjþXn

j¼kþ1

kðK 0j � igSjÞduðm�1Þj k0;1;Cj

!: ð4:32Þ

Using the definition of the linear operator Nk gives

kNkd/ðmÞk0;1;Ck

6

Xm

j¼1

rjd/ðmÞj Gðr; rjÞjr2Ck

��

��

0;1;Ck

6 mc0 rmax q�1=2PE kd/ðmÞk1: ð4:33Þ

Following the asymptotic behavior of the Hankel functions and the definitions of Sj and K 0j gives

Xk�1

j¼1

kðK 0j � igSjÞduðmÞj k0;1;Cj6 ð1þ gÞc0 q�1=2

E

Xk�1

j¼1

jCjjkduðmÞj k0;1;Cj;6 ð1þ gÞc0 jCjq

�1=2E kduðmÞk0;1;C ð4:34Þ

and


Xn

j¼kþ1

kðK 0j � igSjÞduðm�1Þj k0;1;Cj

6 ð1þ gÞc0 q�1=2E

Xn

j¼kþ1

jCjjkduðm�1Þj k0;1;Cj

6 ð1þ gÞc0 jCjq�1=2E kduðm�1Þk0;1;C; ð4:35Þ

where jCkj is the arclength of the boundary curve Ck and the parameter

qE ¼ minr2Ck ; r02C0

k16k<k06n

jr� r0j

is the separation distance among the different boundary components Ck for the extended obstacle.Combining (4.32)–(4.35) leads to

kduðmÞk0;1;C 6 mc0c2rmax q�1=2PE kd/ðmÞk1 þ ð1þ gÞc0c2jCjq

�1=2E kduðmÞk0;1;C þ ð1þ gÞc0c2jCjq

�1=2E kduðm�1Þk0;1;C: ð4:36Þ

For a sufficiently large separation distance qE, we have ð1þ gÞc0c2jCjq�1=2E < 1. Let

c3 ¼ 1� ð1þ gÞc0c2jCjq�1=2E

h i�1:

We obtain from (4.36) that

kduðmÞk0;1;C 6 mc0c2c3rmax q�1=2PE kd/

ðmÞk1 þ ð1þ gÞc0c2c3jCjq�1=2E kduðm�1Þk0;1;C: ð4:37Þ

Combing the estimates (4.31) and (4.37) yield

kduðmÞk0;1;C 6 f1kduðm�1Þk0;1;C; ð4:38Þ

where

f1 ¼ mc0c1rmax q�1PE þ ð1þ gÞq�1=2

E

h ic0c2c3jCj < 1

for sufficiently small rmax and sufficiently large qE; or sufficiently large qPE and qE. Therefore, we have from (4.38) that

kduðmÞk0;1;C 6 fm�11 kuð1Þk0;1;C ! 0 as m!1:

Here we have used the fact that kuð1Þk0;1;C <1 due to the well-posedness of the Foldy–Lax formulation (2.6) and the bound-ary integral Eq. (3.8).

Similarly, we have from (4.31) and (4.37) that

kd/ðmÞk1 6 f2kd/ðm�1Þk1 þ f3kduðm�2Þk0;1;C; ð4:39Þ

where

f2 ¼ mc20c1c2c3rmax q�1

PE jCj and f3 ¼ ð1þ gÞc20c1c2c3 q�1=2

PE q�1=2E jCj:

An application of the recurrence relation (4.38) to the estimate (4.39) yields

kd/ðmÞk1 6 f2kd/ðm�1Þk1 þ f3fm�31 kuð1Þk0;1;C: ð4:40Þ

For either sufficiently small rmax or sufficiently large qPE, we have f2 < 1. We have from the recurrence relation (4.40) that

kd/ðmÞk1 6 fm�12 k/

ð1Þk1 þ f3ðfm�31 þ f2f

m�41 þ � � � þ fm�4

2 f1 þ fm�32 Þkuð1Þk0;1;C

6 fm�12 k/

ð1Þk1 þ f3ðf1 þ f2Þm�3kuð1Þk0;1;C ! 0 asm! 0: ð4:41Þ

Here we assume that f1 þ f2 < 1 for sufficiently small rmax and sufficiently large qE; or sufficiently large qPE and qE, and havealso used the fact that k/ð1Þk1 <1 and kuð1Þk0;1;C <1 due to the well-posedness of the Foldy–Lax formulation (2.6) and theboundary integral Eq. (3.8). h

Remark 4.3. Theorem 4.2 gives a sufficient condition in terms of the physical parameters. In particular if the separation dis-tance among point scatterers and extended scatterers are sufficiently large, the physically motivated block Gauss–Seidel iter-ation is convergent. In practice, we find that the iteration still converges when separation distance among extendedscatterers is much smaller than a wavelength, as long as the discretization of the boundary curves can resolve the separationdistance, i.e., the discrete sampling can distinguish points on different obstacles, see numerical results in Section 7.

Remark 4.4. Though the discussion is focused on the two-dimensional Helmholtz equation, all the results on the well-posedness of the generalized Foldy–Lax formulation and the convergence of the block Gauss–Seidel iteration can be straight-forwardly extended to the three-dimensional Helmholtz equation with the power changed to �1 for the separation distanceparameters.


Remark 4.5. In addition to the block Gauss–Seidel iteration, the block Jacobi iteration may also be used to solve the general-ized Foldy–Lax formulation (4.12) and (4.13). The block Jacobi iteration is equivalent to the series solution formulation,which is presented in [5] and proceeds as follows: Let /ð0Þ ¼ ð0; . . . ;0ÞT and uð0Þk ðrÞ ¼ 0 for k ¼ 1; . . . ;n, define /ðmÞ and uðmÞk

for m P 1 by the solutions of the following system of equations

A/ðmÞ ¼ /inc �Xn

k¼1

Mkuðm�1Þk ð4:42Þ

and

12

I þ K 0k � igSk

� �uðmÞk ¼ fk � Nk/

ðm�1Þ �Xk�1

j¼1

ðK 0j � igSjÞuðm�1Þj �

Xn

j¼kþ1

ðK 0j � igSjÞuðm�1Þj : ð4:43Þ

5. Far-field pattern

The far-field pattern of scattered wave plays a fundamental role in the inverse scattering theory since it contains impor-tant geometrical and physical information, e.g., location, shape, and the impedance of the boundary, of the scattering object.More specifically, given an incident field with incident direction d, if w is the scattered field, then w has the asymptoticbehavior

wðr;dÞ ¼ eijjrjffiffiffiffiffijrj

p w1ðr̂;dÞ þ OðjrjÞ�1h i

as jrj ! 1 ð5:1Þ

uniformly in all directions r̂ ¼ r=jrj, where the function w1 is called as the far-field pattern of the scattered field w, andr̂ ¼ ðcos b; sin bÞ is known as the observation direction and b is the observation direction.

Recalling the following asymptotic behavior for the Hankel function (2.8) for large arguments and comparing with (5.1),we obtain from (2.5) and the following identity

jr� r0j ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffijrj2 � 2r � r0 þ jr0j2

q¼ jrj � r̂ � r0 þ Oðjrj�1Þ as jrj ! 1

that the far-field pattern of the scattered field for the scattering problem of a set of m point scatterers is

w1; FLðr̂;dÞ ¼eip4ffiffiffiffiffiffiffiffiffiffi8pjp

Xm

j¼1

rj/jðdÞe�ijr̂�rj ; ð5:2Þ

which is called the far-field pattern of the Foldy–Lax formulation. Under the Born approximation (2.10), the far-field patternof the scattered field can be approximated by

w1; Bðr̂;dÞ ¼eip4ffiffiffiffiffiffiffiffiffiffi8pjp

Xm

j¼1

rjeijðd�r̂Þ�rj ; ð5:3Þ

which will be called the far-field pattern of the Born approximation.It follows from the integral representation of the scattered field (3.7) and the asymptotic expansion of Green’s function

that the far-field pattern for the extended obstacle is given by

w1; Eðr̂;dÞ ¼ �eip4ffiffiffiffiffiffiffiffiffiffi8pjp

ZC@n0/ðr0; dÞe�ijr̂�r0dsðr0Þ: ð5:4Þ

Based on the integral equation representation (4.8) for the multiple scattering problem and the asymptotic behavior forGreen’s function, we may also obtain the far-field pattern of the scattered field for the generalized Foldy–Lax formulation

w1;GFLðr̂;dÞ ¼eip4ffiffiffiffiffiffiffiffiffiffi8pjp

Xm

j¼1

rj/jðdÞe�ijr̂�rj �Z

C@n0/ðr0; dÞe�ijr̂�r0dsðr0Þ

" #: ð5:5Þ

Remark 5.1. For the extended obstacle scatterer with more than one component, the far-field patterns for the extendedobstacle and the generalized Foldy–Lax formulation can be written as

w1; Eðr̂;dÞ ¼ �eip4ffiffiffiffiffiffiffiffiffiffi8pjp

Xn

k¼1

ZCk

@n0/ðr0; dÞe�ijr̂�r0dsðr0Þ:

and


w1;GFLðr̂;dÞ ¼eip4ffiffiffiffiffiffiffiffiffiffi8pjp

Xm

j¼1

rj/jðdÞe�ijr̂�rj �Xn

k¼1

ZCk

@n0/ðr0; dÞe�ijr̂�r0dsðr0Þ" #

:

Remark 5.2. The formulas of the far-field patterns can be extended to the three-dimensional Helmholtz equation with theconstant changed to 1=4p in front of the summation.

6. Numerical techniques

To compute the far-field patterns, we describe the numerical implementation of the Block Gauss–Seidel iterative methodfor the generalized Foldy–Lax formulation. As seen in (4.27) and (4.28), each step of the iteration essentially consists of solv-ing two decoupled systems: a linear system of algebraic Eqs. (2.4) for the point scatterers and a boundary integral Eq. (3.8)for a single extended obstacle scatterer. The linear system (2.4) can be easily solved by using a direct method such as the LUdecomposition. In the following, we describe the Nyström method to solve the boundary integral Eq. (3.8). The method issimilar to that in [2], where detailed discussion is given for the implementation of a boundary integral equation based onpotential representations.

Recall the block Gauss–Seidel iteration (4.28) for the extended obstacle scatterers with more than one component, it is onlyrequired to solve a boundary integral equation for a single component at each step. To present the Nyström method, we mayassume that the boundary curve C possesses a regular analytic and 2p-periodic parametric representation of theform

rðtÞ ¼ ðx1ðtÞ; x2ðtÞÞ; 0 6 t 6 2p;

in counterclockwise orientation satisfying jr0ðtÞj ¼ ðjx01ðtÞj2 þ jx02ðtÞj

2Þ1=2> 0 for all t. Hence we assume that the obstacle has

only one component. By straightforward calculations using Hð1Þ0

0 ¼ �Hð1Þ1 , we transform (3.8) into the parametric form

uðtÞ þZ 2p

0Mðt; sÞ � igNðt; sÞ½ �uðsÞds ¼ f ðtÞ; ð6:1Þ

where uðtÞ :¼ @n/ðrðtÞÞ; f ðtÞ :¼ 2ð@n � igÞ/incðrðtÞÞ, and the integral kernels are given by

Mðt; sÞ ¼ ij2

Hð1Þ1 ðjq1ðt; sÞÞq2ðt; sÞq1ðt; sÞ

jr0ðsÞjjr0ðtÞj ;

Nðt; sÞ ¼ i2

Hð1Þ0 ðjq1ðt; sÞÞ jx01ðsÞj2 þ jx02ðsÞj

2h i1=2

;

for t – s. Here

q1ðt; sÞ ¼ jx1ðtÞ � x1ðsÞj2 þ jx2ðtÞ � x2ðsÞj2� �1=2

;

q2ðt; sÞ ¼ x02ðtÞðx1ðsÞ � x1ðtÞÞ � x01ðtÞðx2ðsÞ � x2ðtÞÞ:

It follows from the expansion of the Hankel function that the kernels M and N have logarithmic singularities at t ¼ s.Hence, following [2], we split the kernels into

Mðt; sÞ ¼ M1ðt; sÞ ln 4 sin2 t � s2

� �þM2ðt; sÞ;

Nðt; sÞ ¼ N1ðt; sÞ ln 4 sin2 t � s2

� �þ N2ðt; sÞ;

where

M1ðt; sÞ ¼ �j

2pJ1ðjq1ðt; sÞÞ

q2ðt; sÞq1ðt; sÞ

jr0ðsÞjjr0ðtÞj ;

M2ðt; sÞ ¼ Mðt; sÞ �M1ðt; sÞ ln 4 sin2 t � s2

� �;

N1ðt; sÞ ¼ �1

2pJ0ðjq1ðt; sÞÞ jx01ðsÞj

2 þ jx02ðsÞj2

h i1=2;

N2ðt; sÞ ¼ Nðt; sÞ � N1ðt; sÞ ln 4 sin2 t � s2

� �:


The kernels M1;M2;N1, and N2 turn out to be analytic. In particular, using the expansions of the Bessel functions, we candeduce the diagonal terms

M1ðt; tÞ ¼ 0; M2ðt; tÞ ¼1

2px02ðtÞx001ðtÞ � x01ðtÞx002ðtÞjx01ðtÞj

2 þ jx02ðtÞj2 ;

and

N2ðt; tÞ ¼i2� E

p� 1

2pln

j2

4jr0ðtÞj2

� � jr0ðtÞj;

where E is Euler’s constant. We note that despite the continuity of the kernel M, for numerical accuracy it is advantageous toseparate the logarithmic part of M since the derivatives of M fail to be continuous at t ¼ s.

Hence, we have to numerically solve an integral of the form

uðtÞ þZ 2p

0Kðt; sÞuðsÞds ¼ f ðtÞ; 0 6 t 6 2p; ð6:2Þ

where the kernel can be written in the form

Kðt; sÞ ¼ K1ðt; sÞ ln 4 sin2 t � s2

� �þ K2ðt; sÞ:

Here

K1ðt; sÞ ¼ M1ðt; sÞ � igN1ðt; sÞ;K2ðt; sÞ ¼ M2ðt; sÞ � igN2ðt; sÞ:

The Nyström method consists in the straightforward approximation of the integrals by quadrature formulas. In our case, forthe 2p-periodic integrands, we choose an equidistant set of knots tj :¼ jp=n; j ¼ 0; . . . ;2n� 1, and use the quadraturerule
Z 2p
0ln 4 sin2 t � s

2

� �gðsÞds �

X2n�1

j¼0

RðnÞj ðtÞgðtjÞ; 0 6 t 6 2p; ð6:3Þ

with the quadrature weights given by

RðnÞj ðtÞ :¼ �2pn

Xn�1

m¼1

1m

cos mðt � tjÞ �pn2 cos nðt � tjÞ; j ¼ 0; . . . ;2n� 1;

and the trapezoidal rule
Z 2p
0gðtÞdt � p

n

X2n�1

j¼1

gðtjÞ: ð6:4Þ

In the Nyström method, the integral Eq. (6.2) is replaced by the approximation equation after applying the (6.3) and (6.4)

uðnÞðtÞ þX2n�1

j¼0

RðnÞj ðtÞK1ðt; tjÞ þpn

K2ðt; tjÞh i

uðnÞðtjÞ ¼ f ðtÞ ð6:5Þ

for 0 6 t 6 2p. In particular, for any solution of (6.5) the value uðnÞi ¼ uðnÞðtiÞ; i ¼ 0; . . . ;2n� 1, at the quadrature points triv-ially satisfy the linear system

uðnÞi þX2n�1

j¼0

RðnÞji�jjK1ðti; tjÞ þpn

K2ðti; tjÞh i

uðnÞj ¼ f ðtiÞ ð6:6Þ

for i ¼ 0; . . . ;2n� 1, where

Rð0Þj :¼ RðnÞj ð0Þ ¼ �2pn

Xn�1

m¼1

1m

cosjmp

n� ð�1Þjp

n2 ; j ¼ 0; . . . ;2n� 1:

To demonstrate the performance of the above method for solving the boundary integral Eq. (3.8), we take the benchmarkexample in [2] and consider the scattering of a plane wave by a kite-shaped obstacle, which is described by the parametricrepresentation

rðtÞ ¼ ðcos t þ 0:65 cos 2t � 0:65;1:5 sin tÞ; 0 6 t 6 2p;

−4 −2 0 2 4−4

−2

0

2

4

x

y

−4 −2 0 2 4−4

−2

0

2

4

x

y

−4 −2 0 2 4−4

−2

0

2

4

x

y

Fig. 2. Uniformly distributed point scatterers and a kite-shaped obstacle scatterer in Example 1. (Left) a set of 20 equally spaced point scatterers on a circlewith radius 3; (middle) a kite-shaped extended obstacle scatterer; (right) mixed scatterers of a kite-shaped obstacle surrounded by a set of 20 equallyspaced point scatterers.

Table 1Numerical results of the Nyström’s method to solve the boundary integral equation for a kite-shaped obstacle.

n Rew1;Eðd;dÞ Imw1;Eðd;dÞ Rew1;Eð�d;dÞ Imw1;Eð�d;dÞ

j ¼ 1 8 �1.62399616 0.60308809 1.39015276 0.0942514316 �1.62746853 0.60222418 1.39696605 0.0949946732 �1.62745750 0.60222603 1.39694483 0.0949965064 �1.62745750 0.60222603 1.39694483 0.09499650

j ¼ 58 �2.47821459 1.72919957 �0.39350495 0.1189706716 �2.46812551 1.68955122 �0.19892628 0.0613890732 �2.47554472 1.68747934 �0.19945835 0.0601572864 �2.47554477 1.68747932 �0.19945836 0.06015726


as seen in the middle of Fig. 2. The far-field pattern of the scattered wave for the obstacle scattering problem is given by (5.4),which can be evaluated again by the trapezoidal rule after solving the integral equation for @n/. Table 1 gives numericalapproximations for the far-field pattern w1;Eðr̂;dÞ in the forward direction r̂ ¼ d and the backward direction r̂ ¼ �d. Thedirection d of the incident wave is d ¼ ð1;0Þ and the coupling parameter g ¼ j. Note that the exponential convergence isclearly exhibited.

7. Numerical experiments

In this section, we present some numerical examples for the computation of the far-field patterns for the incident anglea 2 ½0;2p� and the observation angle b 2 ½0;2p�. As comparisons, we will show the different far-field patterns: the far-fieldpattern via the Born approximation w1;B for a group of point scatterers, the far-field pattern via the original Foldy–Lax for-mulation w1;FL for a group of point scatterers, the far-field pattern via the boundary integral equation for extended obstaclescatterers w1;E, and the far-field pattern via the generalized Foldy–Lax formulation for both point and extended obstaclescatterers. In Example 1 and Example 2, the wavenumber j is taken to be 4p, i.e., the wavelength is taken to bek ¼ 2p=j ¼ 0:5. In Example 3, the wavenumber j is taken to be 2p which accounts for the wavelength k ¼ 2p=j ¼ 1:0.

Example 1. Consider a group of twenty equally distributed point scatterers on a circle with radius 3 and a kite-shapedextended obstacle with the parametric representation given by

rðtÞ ¼ ðcos t þ 0:65 cos 2t � 0:65;1:5 sin tÞ; 0 6 t 6 2p;

as seen in Fig. 2. The scattering coefficient rj is the same for all the twenty point scatterers, which is rj ¼ r ¼ 1:0 for Figs. 3–7. Figs. 3 and 4 show the far-field patterns for the point scatterers, as seen in the left of Fig. 2. Fig. 3 shows the real and ima-ginary parts of the far-field pattern for the scattered field computed from the Born approximation (2.10), where the multiplescattering is neglected among the point scatterers. As a comparison, Fig. 4 shows the real and imaginary parts of the far-fieldpattern for the scattered field computed from the Foldy–Lax formulation (2.5), where the multiple scattering is taken intoaccount among the point scatterers. As we can see from Fig. 3 and Fig. 4, the difference is clear between the far-field patternw1;B and the far-field pattern w1;FL. Fig. 5 shows the far-field pattern for the single kite-shaped extended obstacle scatterer, asseen in the middle of Fig. 2. It shows the real and imaginary parts of the far-field pattern for the scattered field (3.7) com-

incident angle

obse

rvat

ion

angl

e

0 2 4 60

2

4

6

−0.5

0

0.5

1

incident angle

obse

rvat

ion

angl

e

0 2 4 60

2

4

6

−0.5

0

0.5

1

Fig. 3. The far-field pattern w1;B of the scattered field from the uniformly distributed point scatterers in Example 1 via the Born approximation. (Left) realpart; (right) imaginary part.

incident angle

obse

rvat

ion

angl

e

0 2 4 60

2

4

6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

incident angle

obse

rvat

ion

angl

e

0 2 4 60

2

4

6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

Fig. 4. The far-field pattern w1;FL of the scattered field from the uniformly distributed point scatterers in Example 1 via the Foldy–Lax formulation. (Left)real part; (right) imaginary part.

incident angle

obse

rvat

ion

angl

e

0 2 4 60

2

4

6

−3

−2

−1

0

1

incident angle

obse

rvat

ion

angl

e

0 2 4 60

2

4

6

−1

0

1

2

3

Fig. 5. The far-field pattern w1;E of the scattered field from the kite-shaped extended obstacle scatterer in Example 1 via the boundary integral equation.(Left) real part; (right) imaginary part.


puted from the boundary integral Eq. (3.8). Fig. 6 shows the far-field pattern for the heterogeneous medium with both pointand extended obstacle scatterers, as seen in the right of Fig. 2. It shows the real and imaginary parts of the far-field patternfor the scattered field (4.8) computed from the generalized Foldy–Lax formulation (4.10) and (4.11). Comparing with the far-field pattern in Fig. 5, the far-field pattern in Fig. 6displays a lot of small oscillation, which comes from the presence of thepoint scatterers. In order to show whether the generalized Foldy–Lax formulation captures the multiple scattering betweenthe point scatterers and the extended kite-shaped obstacle scatterer, Fig. 7 shows only the portion of the far-field pattern dueto the multiple scattering by subtracting w1;E and w1;FL from w1;GFL, i.e., the far-field pattern w1;GFL � w1;E � w1;FL.

incident angle

obse

rvat

ion

angl

e

0 2 4 60

2

4

6

−3

−2

−1

0

1

incident angle

obse

rvat

ion

angl

e

0 2 4 60

2

4

6

−1

0

1

2

3

Fig. 6. The far-field pattern w1;GFL of the scattered field from the uniformly distributed point scatterers and the kite-shaped extended obstacle scatterer inExample 1 via the generalized Foldy–Lax formulation. (Left) real part; (right) imaginary part.

incident angle

obse

rvat

ion

angl

e

0 2 4 60

1

2

3

4

5

6

−0.6

−0.4

−0.2

0

0.2

0.4

incident angle

obse

rvat

ion

angl

e

0 2 4 60

1

2

3

4

5

6

−0.6

−0.4

−0.2

0

0.2

0.4

Fig. 7. The difference of the far-field patterns w1;GFL—w1;E—w1;FL from the uniformly distributed point scatterers and the kite-shaped extended obstaclescatterer in Example 1. (Left) real part; (right) imaginary part.

0 5 10 15 20 2510−8

10−6

10−4

10−2

100

102

number of iterations

erro

r e

GFL

σ=1σ=2σ=4σ=8

0 10 20 30 40 50 60 7010−8

10−6

10−4

10−2

100

102


erro

r e

GFL

σ=1σ=2σ=4σ=8

Fig. 8. The error of two consecutive approximations is given against the number of iterations with different scattering coefficients for Example 1. (Left)block Gauss–Seidel iteration; (right) block Jacobi iteration.


To show how the convergence of the Block Gauss–Seidel iteration and the block Jacobi iteration depend on the point scat-terers, such as the scattering coefficients and the number of point scatterers, we define the error between two consecutiveapproximations

eFL ¼ k/ðmÞ � /ðm�1Þk2 and eE ¼ kuðmÞ �uðm�1ÞkL2ðCÞ;

where m is the number of iteration. Define

0 5 10 15 20 2510−8

10−6

10−4

10−2

100

102


erro

r e

GFL

m=20m=100m=200m=300

0 5 10 15 20 25 30 35 40 4510−8

10−6

10−4

10−2

100

102

104


erro

r e

GFL

m=20m=100m=200m=300

Fig. 9. The error of two consecutive approximations is given against the number of iterations with different number of point scatterers for Example 1. (Left)block Gauss–Seidel iteration; (right) block Jacobi iteration.

−−4

−2

0

2

4

y

Fig. 10scatteremixed s


eGFL ¼maxfeFL; eEg:

Given the 20 equally distributed point scatterers with equal scattering coefficient on the circle with radius 3, Fig. 8 shows theerror eGFL of two consecutive approximations using rj ¼ r ¼ 1;2;4;8. As can be seen from Fig. 8, the larger of the scatteringcoefficient is, the stronger the multiple scattering among the point scatterers, and thus more iterations are needed to reachthe same level of accuracy. Given the scattering coefficients rj ¼ r ¼ 1:0, Fig. 9 investigates the error eGFL of two consecutiveapproximations using different number of equally spaced point scatterers m ¼ 20;100;200;300 on the circle with radius 3.Similarly, it can be observed that more iterations are needed to reach the same level of accuracy for more number of pointscatterers. As expected, it can be seen from Figs. 8 and 9 that the block Gauss–Seidel iteration displays a faster convergencethan the block Jacobi iteration.

Example 2. Consider a group of one hundred randomly distributed point scatterers in annulus with radii bounded blow by 3and above by 4 and two extended obstacle scatterers: one is a circular extended scatterer with the parametric representationgiven by

r1ðtÞ ¼ ðcos t � 1:5; sin tÞ; 0 6 t 6 2p;

and the other one is a kite-shaped extended scatterer with the parametric representation given by

r2ðtÞ ¼ ðcos t þ 0:65 cos 2t þ 0:85;1:5 sin tÞ; 0 6 t 6 2p;

as seen in Fig. 10. The scattering coefficient rj is taken as a random number from the interval ð0;1Þ for j ¼ 1; . . . ;100 in Ex-ample 2. Figs. 11 and 12 concern with the far-field patterns for the randomly distributed point scatterers, as seen in the left ofFig. 10. Fig. 11 shows the real and imaginary parts of the far-field pattern for the scattered field computed from the Bornapproximation (2.10). As a comparison, Fig. 12 shows the real and imaginary parts of the far-field pattern for the scatteredfield computed from the Foldy–Lax formulation (2.5), where the multiple scattering is taken into account among the pointscatterers. Due to the randomly distributed point scatterers, Figs. 11 and 12 show no symmetric patterns, which are different

4 −2 0 2 4x

−4 −2 0 2 4−4

−2

0

2

4

x

y

−4 −2 0 2 4−4

−2

0

2

4

x

y

. Randomly distributed point scatterers and two obstacle scatterers in Example 2. (Left) a group of one hundred of randomly distributed pointrs in the annulus with radii bounded blow by 3 and above by 4; (middle) a kite-shaped and a circle-shaped extended obstacle scatterers; (right)catterers of a kite-shaped and a circle-shaped obstacle scatterers surrounded by a set of one hundred randomly distributed point scatterers.

incident angle

obse

rvat

ion

angl

e

0 2 4 60

2

4

6

−1

−0.5

0

0.5

1

1.5

2

incident angle

obse

rvat

ion

angl

e

0 2 4 60

2

4

6

−1

−0.5

0

0.5

1

1.5

2

Fig. 11. The far-field pattern w1;B of the scattered field from the uniformly distributed point scatterers in Example 2 via the Born approximation. (Left) realpart; (right) imaginary part.

incident angle

obse

rvat

ion

angl

e

0 2 4 60

2

4

6

−1

−0.5

0

0.5

1

1.5

incident angle

obse

rvat

ion

angl

e

0 2 4 60

2

4

6

−1

−0.5

0

0.5

1

1.5

2

2.5


incident angle

obse

rvat

ion

angl

e

0 2 4 60

2

4

6

−6

−4

−2

0

2

incident angle

obse

rvat

ion

angl

e

0 2 4 60

2

4

6

−2

−1

0

1

2

3

4

5

Fig. 13. The far-field pattern w1;E of the scattered field from the kite-shaped extended obstacle scatterer in Example 2 via the boundary integral equation.(Left) real part; (right) imaginary part.


from Fig. 3 and 4 for the equally distributed point scatterers. Fig. 13 shows the far-field pattern for the two extended obstaclescatterers, as seen in the middle of Fig. 10. It shows the real and imaginary parts of the far-field pattern for the scattered field(3.7) computed from the boundary integral Eq. (3.8). Fig. 14 shows the far-field pattern for the heterogeneous medium withboth point and extended obstacle scatterers, as seen on the right of Fig. 10. It shows the real and imaginary parts of the far-field pattern for the scattered field (4.8) computed from the generalized Foldy–Lax formulation (4.10) and (4.11). Again, inorder to show whether the generalized Foldy–Lax formulation captures the multiple scattering between the point scatterersand the extended kite-shaped obstacle scatterer, Fig. 15 shows only the portion of the far-field pattern due to the multiplescattering by subtracting w1;E andw1;FL from w1;GFL, i.e., the far-field pattern w1;GFL � w1;E � w1;FL.

incident angle

obse

rvat

ion

angl

e

0 2 4 60

2

4

6

−5

−4

−3

−2

−1

0

1

2

incident angle

obse

rvat

ion

angl

e

0 2 4 60

2

4

6

−2

0

2

4

6

Fig. 14. The far-field pattern w1;GFL of the scattered field from the uniformly distributed point scatterers and the kite-shaped extended obstacle scatterer inExample 2 via the generalized Foldy–Lax formulation. (Left) real part; (right) imaginary part.

incident angle

obse

rvat

ion

angl

e

0 2 4 60

2

4

6

−0.5

0

0.5

incident angle

obse

rvat

ion

angl

e

0 2 4 60

2

4

6

−1

−0.5

0

0.5

1

Fig. 15. The difference of the far-field patterns w1;GFL—w1;E—w1;FL from the uniformly distributed point scatterers and the kite-shaped extended obstaclescatterer in Example 2. (Left) real part; (right) imaginary part.

0 200 400 600 800 100010−8

10−6

10−4

10−2

100

102


erro

r e

GFL

ρ E=0.25λ

ρ E=0.5λ

ρ E=1.0λ

ρ E=2.0λ

0 300 600 900 1200 1,50010−8

10−6

10−4

10−2

100

102


erro

r e

GFL

ρ E=0.25λ

ρ E=0.5λ

ρ E=1.0λ

ρ E=2.0λ

Fig. 16. Error of two consecutive approximations using different separation distance for Example 2. (Left) block Gauss–Seidel iteration; (right) block Jacobiiteration.


Given the one hundred randomly distributed point scatterers with scattering coefficient chosen as a random number fromthe interval ð0;1Þ, Fig. 16 plots the error eGFL of two consecutive approximations with different separation distance of the twoextended scatterers, qE ¼ 0:25k;0:5k;1:0k;2:0k, in order to show how the convergence of the Block Gauss–Seidel iterationand the block Jacobi iteration depend on the separation distance between the extended scatterers qE. Theorem 4.2 gives asufficient condition for the convergence of the physically motivated Gauss–Seidel iteration, which requires that the separa-tion distance among scatterers to be sufficiently large. In practice, the iteration still converges when the separation distanceqE of the two extended scatterers is as close as 0:25k as long as the discretization of the boundary can resolve the separation

−2−1

01

2

−2−1

01

2−1

0

1

z

xy −2−1

01

2

−2−1

01

2−1

0

1

xy

z

−2−1

01

2

−2−1

01

2−1

0

1

xy

z

Fig. 17. Uniformly distributed point scatterers and two spherical obstacle scatterer in Example 3. (Left) a set of 20 equally spaced point scatterers on a circlewith radius 2 on the xy-plane; (middle) two spherical extended obstacle scatterers; (right) mixed scatterers of two spherical obstacle surrounded by a set of20 equally spaced point scatterers.

−10

1

−1

0

1−1

0

1

xy

z

−0.5

0

0.5

1

−10

1

−1

0

1−1

0

1

xy

z

−0.15

−0.1

−0.05

0

0.05


−10

1

−1

0

1−1

0

1

xy

z

−0.3

−0.2

−0.1

0

0.1

0.2

−10

1

−1

0

1−1

0

1

xy

z

−0.05

0

0.05

Fig. 19. The far-field pattern w1;E of the scattered field from the two spherical obstacle scatterers in Example 3 via the boundary integral equation. (Left)real part; (right) imaginary part.


distance. Surprisingly, the number of iterations is not necessary to increase as the separation distance between the two ex-tended scatterer qE decreases for both the block Gauss–Seidel iteration and the block Jacobi iteration. The number of itera-tions increases as the separation distance qE decreases to 2k; and then it decreases as the separation distance furtherdecreases to 0:25k, as seen in Fig. 16. As usual, the block Gauss–Seidel iteration exhibits a faster convergence than the blockJacobi iteration.

Example 3. We show an example to illustrate that the developed generalized Foldy–Lax formulation and the proposed blockGauss–Seidel method can be extended to the three-dimensional Helmholtz equation, with the power of the separationdistance changed to �1 and the constant in the far-field pattern changed to 1=4p. We choose the Galerkin boundary elementmethod for solving the three-dimensional boundary integral equations, and refer to [5] for detailed descriptions of themethod. In this example, we consider a group of 20 equally distributed point scatterers on a circle with radius 2 on the xy-plane, as seen in left of Fig. 17. The scattering coefficients rj is chosen to be all equal, i.e., rj ¼ r ¼ 1:0. The centers of two

−10

1

−1

0

1−1

0

1

xy

z

−0.5

0

0.5

1

−10

1

−1

0

1−1

0

1

xy

z

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

Fig. 20. The far-field pattern w1;GFL of the scattered field from the uniformly distributed point scatterers and the two spherical obstacle scatterers inExample 3 via the generalized Foldy–Lax formulation. (Left) real part; (right) imaginary part.

−10

1

−1

0

1−1

0

1

xy

z

−0.02

−0.01

0

0.01

0.02

−10

1

−1

0

1−1

0

1

xy

z

−0.02

−0.01

0

0.01

0.02

Fig. 21. The difference of the far-field patterns w1;GFL—w1;E—w1;FL from the uniformly distributed point scatterers and the two spherical obstacle scatterersin Example 3. (Left) real part; (right) imaginary part.

0 2 4 6 810−6

10−4

10−2

100


erro

r e

GFL

ρ E=0.25λ

ρ E=0.5λ

ρ E=1.0λ

0 2 4 6 8

10−6

10−4

10−2

100


erro

r e

GFL

ρ E=0.25λ

ρ E=0.5λ

ρ E=1.0λ

Fig. 22. Error of two consecutive approximations using different separation distance for Example 3. (Left) block Gauss–Seidel iteration; (right) block Jacobiiteration.


spherical obstacle scatterers with the same radius of 0:5 are located at ð�1:0;0:0;0:0Þ and ð1:0;0:0;0:0Þ, respectively, as seenin the middle of Fig. 17. The far-field patterns are plotted on the unit sphere as a function of the variables for the latitudinalangle and the longitudinal angle varying from 0 to p and from 0 to 2p, respectively. All the figures are shown from theincident direction of d ¼ ð0;0;1Þ. Fig. 18 shows the real and imaginary parts of the far-field pattern, w1;FL, for the scatteredfield computed from the original Foldy–Lax formulation. Fig. 19 shows the far-field pattern, w1;E, for the two sphericalscatterers by using the boundary integral equation, and Fig. 20 plots the far-field pattern, w1;GFL, for the mixed scatterers byusing the generalized Foldy–Lax formulation. Finally, Fig. 21 shows the portion of the far-field pattern arising from themultiple scattering among all scatterers by subtracting w1;E and w1;FL from w1;GFL, i.e., the far-field patternw1;GFL � w1;E � w1;FL. Fig. 22 plots the error eGFL of two consecutive approximations with different separation distance of


the two spheres, qE ¼ 0:25k;0:5k;1:0kto show how the convergence of the Block Gauss–Seidel iteration and the block Jacobiiteration depend on the separation distance between the extended scatterers for the three-dimensional Helmholtz equation.Remarkably, the number of iterations and convergence is almost independent of the separation distance in 3D for bothmethods of iteration; while the block Gauss–Seidel iteration shows a faster convergence than the block Jacobi iteration.

8. Conclusions

We presented an efficient algorithm for the generalized Foldy–Lax formulation to capture multiple scattering among agroup of isotropic point scatterers and extended obstacle scatterers. The improved formulation is based on a coupled systemwhich combines the original Foldy–Lax formulation and a uniquely solvable boundary integral equation for exterior bound-ary value problem. The uniqueness and existence of the solution for the improved formulation were given in terms of thephysical parameters such as the scattering coefficient and the separation distances. Computationally, an efficient blockGauss–Seidel iterative method was proposed to solve the coupled system. At each step of iteration, only a linear systemof algebraic equations or a boundary integral equation for a single obstacle scatterer needs to be solved. Sufficient conditionfor the convergence of the iteration is provided in terms of physical parameters. Convergence is investigated for the effects ofthe number of point scatterers, the scattering coefficients, and the separation distances through numerical experiments. Weare intended to extend the proposed method to the three-dimensional Maxwell equations and will report the progresselsewhere.

References

[1] W. Chew, Waves and Fields in Inhomogeneous Media, Van Nostrand Reinhold, New York, 1990.[2] D. Colton, R. Kress, Integral Equation Methods in Scattering Theory, John Wiley & Sons, 1983.[3] D. Colton, R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Springer-Verlag, Berlin, 1998.[4] L. Foldy, The multiple scattering of waves. I. General theory of isotropic scattering by randomly distributed scatterers, Phys. Rev. 67 (1945) 107–119.[5] K. Huang, P. Li, A two-scale multiple scattering problem, Multiscale Model. Simul. 8 (2010) 1511–11534.[6] K. Huang, K. Solna, H. Zhao, Generalized Foldy–Lax formulation, J. Comput. Phys. 229 (2010) 4544–4553.[7] R. Kress, Minimizing the condition number of boundary integral operators in acoustic and electromagnetic scattering, Q. J. Mech. Appl. Math. 38 (1985)

323–341.[8] M. Lax, Multiple scattering of waves, Rev. Mod. Phys. 23 (1951) 287–310.[9] P. Martin, Multiple Scattering Interaction of Time-Harmonic Wave with N Obstacles, Cambridge University Press, 2006.

[10] J. Sun, S. Carney, J. Schotland, Strong tip effects in near-field optical tomography, J. Appl. Phys. 102 (2007) 103103.

an efficient algorithm for the generalized foldy–lax formulation

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