numerical integration of multi-dimensional highly oscillatory, gentle oscillatory and...

Engineering Analysis with Boundary Elements 36 (2012) 1284–1295

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Engineering Analysis with Boundary Elements

0955-79

doi:10.1

n Corr

E-m

siraj.isla

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

Numerical integration of multi-dimensional highly oscillatory, gentleoscillatory and non-oscillatory integrands based on wavelets andradial basis functions

Siraj-ul-Islam b,n, Imran Aziz a,b, Wajid Khan b

a Department of Mathematics, University of Peshawar, Pakistanb Department of Basic Sciences, University of Engineering and Technology, Peshawar, Pakistan

a r t i c l e i n f o

Article history:

Received 29 September 2011

Accepted 26 January 2012Available online 15 March 2012

Keywords:

Haar wavelets

Hybrid functions

Meshless method

RBFs

Multi-dimensional highly oscillatory

kernels

Non-oscillatory kernels

Quadrature rule

Numerical method

Double integrals

Triple integrals

97/$ - see front matter & 2012 Elsevier Ltd. A

016/j.enganabound.2012.01.008

esponding author.

ail addresses: [email protected].

[email protected] (Siraj-ul-Islam).

a b s t r a c t

In this paper Haar wavelets (HWs), hybrid functions (HFs) and radial basis functions (RBFs) are used for

the numerical solution of multi-dimensional mild, highly oscillatory and non-oscillatory integrals. Part

of this study is extension of our earlier work [9,47] to multi-dimensional oscillatory and non-oscillatory

integrals. Second part of the study is focused on coupling Levin’s approach [30] with meshless methods.

In first part of the paper, application of the numerical algorithms based on HWs and HFs [9] is extended

to integrals having a varying oscillatory and non-oscillatory integrands defined on circular and

rectangular domains. In second part of the paper, we propose a meshless method based on multi-

quadric (MQ) RBF for highly oscillatory multi-dimensional integrals. The first approach is directly

related to numerical quadrature with wavelets basis. Like classical numerical quadrature, this approach

does not need any intermediate numerical technique. The second approach based on meshless method

of Levin’s type converts numerical integration problem to a partial differential equation (PDE) and

subsequently finding numerical solution of the PDE by a meshless method. The computational

algorithms thus derived are tested on a number of benchmark kernel functions having varying

oscillatory character or integrands with critical points at the origin. The novel methods are compared

with the existing methods as well. Accuracy of the methods is measured in terms of absolute and

relative errors. The new methods are simple, more efficient and numerically stable. Theoretical and

numerical convergence analysis of the HWs and HFs is also given.

& 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Numerical investigation of integrals with gentle oscillatorycharacter is important due to its frequent use in other numericalmethods like finite element method (FEM) or meshless localPetrov–Galerkin (MLPG) method in evaluating the so-called stiff-ness matrices [35,36]. Stiffness matrices are encountered when agiven PDE is discretized by these methods. Numerical integrationof the resulting discrete set of algebraic equations is a complexprocedure and requires strategies using a large number of integra-tion points, and may, therefore, be time-consuming [35,43] to getmore accurate results. The existing methods which are in use andare very popular in the MLPG literature include product Gaussnumerical quadrature rules [35,36], convectional Gaussian quad-rature and mid-point rules [14,39]. Numerical integration in thecontext of FEM or MLPG is a crucial issue and hence requires

ll rights reserved.

special attention. Numerical integration can affect accuracy of theoverall numerical method if lower order algorithm is used. Numer-ical quadrature which has been used for evaluation of theseintegrals bears some drawbacks which are listed below:

(i)
Integrands in most of the cases are transcendental functionsand conventional schemes, such as the Gaussian quadraturerule, require considerable number of points to attain accep-table level of accuracy [5].
(ii)
Accuracy of the Gaussian quadrature rule is sometimesadversely affected by the fact that the Gaussian quadraturerule is based on an infinitely differentiable interpolation func-tion, whereas the functions involved in the MLPG integrals areoften not of this type [3,35].
(iii)
The use of large number of equally spaced nodes in the caseof Newton–Cotes quadrature rule may cause erratic behaviorwith high degree polynomial interpolation.
Various techniques like volume or area segmentation of theintegration domain has been proposed by Atluri et al. [4,5] toimprove accuracy of quadrature rule. Another approach [38] is

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Siraj-ul-Islam et al. / Engineering Analysis with Boundary Elements 36 (2012) 1284–1295 1285

based on few integration points in MLPG with imbedded localsub-domain. Different types of polynomial based quadraturemethods have been discussed in [1,29,34,37,40–42,44] and thereferences therein.

In order to overcome some of the difficulties listed above, wepropose new methods based on HWs and HFs to find numericalsolutions of multi-dimensional integrals. This paper should beconsidered as a logical continuation of our previous papers [9,47].Most recent developments about HWs and HFs have beendiscussed in [9,47] and the references therein.

First part of this paper extends application of the new methodsbased on the simple HWs and HFs to integrals with mildlyand non-oscillatory behavior. This approach has the followingadvantages:

(i)
It provides an accurate solution with less collocation pointsfor the integral with less or no oscillatory character,
(ii)
Simple and direct applicability of new methods omit therequirement of using any other intermediate numerical tech-nique and hence the methods are not susceptible to theproblem of ill-conditioning.
Numerical evaluation of oscillatory integrals accurately andefficiently is one of the challenging problems in scientific comput-ing. Most of the existing algorithms and commercial softwares(Mathematica, Maple, Matlab) cannot handle numerical solutionof highly oscillatory integrals. These integrals occur in a varietyof applications including electromagnetics, optics, quantummechanics, etc. [12,31,45], and hence numerical investigation ofthese integrals is worth attention. The main stream methodswhich are used for the evaluation of these type of integrals includemainly the Filon and Levin methods [16,30]. More work on thenumerical solutions of multi-dimensional oscillatory integrals canbe found in [8,20,21]. Methods based on Filon’s approach areasymptotic, whereas Levin’s approach is based on finding numer-ical solution of the PDE (which is obtained after convertingintegration to PDE) by collocation method with monomial basis.Despite their popularity, Levin’s method is prone to the problemof ill-conditioning due to the use of monomial basis functions[30,31]. In [31] the authors have used Chebychev spectral methodby separating the inner and outer integrals along with TSVDtechnique to address the problem of ill-conditioning. Keeping inview geometric flexibility and higher accuracy of multiquadric(MQ) RBF and the successful use of meshless method for thenumerical solution of PDEs [2,6,7,11,15,17–19,22,23,25–28,32,33,46,48,50–56], it is desirable to exploit potentials of meshlessmethod for numerical solution of highly oscillatory integrals aswell. In the second part of this paper we propose a meshlessmethod based on MQ RBFs for the numerical solution of suchintegrals.

The organization of this paper is as follows. In Section 2 thenumerical methods based on HWs and HFs for multi-dimensionalintegrals are described. In Section 3 convergence analysis of HWsand HFs is given. In Section 4 meshless method of Levin’s type isderived. Numerical results are reported in Section 5 and someconclusions are drawn in Section 6.

2. Numerical integration using Haar wavelets andhybrid functions

In the subsequent sections we adapt the same notations usedin [9,47]. Interpolation condition of the HWs, HFs and basicdefinitions of Haar wavelets can be found in [9,47]. A new formula(10) based on HFs for three-dimensional integrals is also derived.The rest of the formulae is also summarized in the following

sections. The main reason for including this section is to extendapplicability of the methods [9,47] for the integrals with mildlyoscillatory kernels and integrals with critical points. Stableperformance of HWs and HFs based algorithms is reported inthe case when integrand is either non-oscillatory or mildlyoscillatory. It has also been experimented that the methods basedon HWs and HFs are not suitable for the integrals having highlyoscillatory kernel functions.

2.1. Numerical formula for single integrals using Haar wavelets

Formula for numerical integration of single integrals withconstant limits is given byZ b

af ðxÞ dx�

ðb�aÞ

2M

X2M

i ¼ 1

f ðxkÞ ¼ðb�aÞ

2M

X2M

i ¼ 1

f aþðb�aÞði�0:5Þ

2M

� �: ð1Þ

Similarly, formulae for two- and three-dimensional integrals aregiven byZ b

a

Z dðyÞ

cðyÞFðx,yÞ dx dy�

ðb�aÞ

2N

X2N

i ¼ 1

G aþðb�aÞði�0:5Þ

2N

� �, ð2Þ

where

GðyÞ ¼fdðyÞ�cðyÞg

2M

X2M

i ¼ 1

F cðyÞþfdðyÞ�cðyÞgði�0:5Þ

2M,y

� �: ð3Þ

Z b

a

Z dðzÞ

cðzÞ

Z f ðy,zÞ

eðy,zÞFðx,y,zÞ dx dy dz�

ðb�aÞ

2P

X2P

i ¼ 1

H aþðb�aÞði�0:5Þ

2P

� �,

ð4Þ

where

HðzÞ ¼fdðzÞ�cðzÞg

2N

X2N

i ¼ 1

G cðzÞþfdðzÞ�cðzÞgði�0:5Þ

2N,z

� �ð5Þ

and

Gðy,zÞ ¼ff ðy,zÞ�eðy,zÞg

2M

X2M

i ¼ 1

F eðy,zÞþff ðy,zÞ�eðy,zÞgði�0:5Þ

2M,y,z

� �:

ð6Þ

In the above formulae, M¼ 2J1 , N¼ 2J2 and P¼ 2J3 , where J1, J2

and J3 are the levels of resolutions of the HWs in the x-, y- andz-directions, respectively.

2.2. Numerical formulas for single integrals using hybrid functions

For m¼1, the formula based on hybrid function is given byZ b

af ðxÞ dx�

ðb�aÞ

n

Xn

i ¼ 1

f aþðb�aÞð2i�1Þ

2n

� �: ð7Þ

Similarly, for two-dimensional integral the formula based on HFsfor m¼9 is given byZ b

a

Z ðdðyÞcðyÞ

Fðx,yÞ dx dy

�b�a

5 734 400n

Xn

i ¼ 1

832 221G aþðb�aÞð18i�17Þ

18n

� ��

�260 808G aþðb�aÞð18i�15Þ

18n

� �

þ2 903 148G aþðb�aÞð18i�13Þ

18n

� �

�3 227 256G aþðb�aÞð18i�11Þ

18n

� �


18n

� �

Siraj-ul-Islam et al. / Engineering Analysis with Boundary Elements 36 (2012) 1284–12951286

�3 227 256G aþðb�aÞð18i�7Þ

18n

� �


18n

� �

�260 808G aþðb�aÞð18i�3Þ

18n

� �

þ832 221G aþðb�aÞð18i�1Þ

18n

� ��, ð8Þ

where

GðyÞ ¼fdðyÞ�cðyÞg

5 734 400n

Xn

i ¼ 1

832 221f cðyÞþfdðyÞ�cðyÞgð18i�17Þ

18n

� ��

�260 808f cðyÞþfdðyÞ�cðyÞgð18i�15Þ

18n

� �

þ2 903 148f cðyÞþfdðyÞ�cðyÞgð18i�13Þ

18n

� �

�3 227 256f cðyÞþfdðyÞ�cðyÞgð18i�11Þ

18n

� �


18n

� �

�3 227 256f cðyÞþfdðyÞ�cðyÞgð18i�7Þ

18n

� �


18n

� �

�260 808f cðyÞþfdðyÞ�cðyÞgð18i�3Þ

18n

� �

þ832 221f cðyÞþfdðyÞ�cðyÞgð18i�1Þ

18n

� ��:

2.3. Numerical formula for triple integrals

The method used in the last section is extended for numericalintegration to triple or even integrals in higher dimensions withvariable limits of integration. Formula for triple integrals usingHFs with m¼9 is derived in the following manner. Consider thetriple integralZ b

a

Z dðzÞ

cðzÞ

Z f ðy,zÞ

eðy,zÞFðx,y,zÞ dx dy dz: ð9Þ

Applying formula (7) three times, we obtainZ b

a

Z dðzÞ

cðzÞ

Z f ðy,zÞ

eðy,zÞFðx,y,zÞ dx dy dz

�b�a

5 734 400n

Xn

i ¼ 1

832 221H aþðb�aÞð18i�17Þ

18n

� ��

�260 808H aþðb�aÞð18i�15Þ

18n

� �

þ2 903 148H aþðb�aÞð18i�13Þ

18n

� �:

�3 227 256H aþðb�aÞð18i�11Þ

18n

� �


18n

� �

�3 227 256H aþðb�aÞð18i�7Þ

18n

� �


18n

� �

�260 808H aþðb�aÞð18i�3Þ

18n

� �

þ832 221H aþðb�aÞð18i�1Þ

18n

� ��, ð10Þ

where

HðzÞ ¼fdðzÞ�cðzÞg

5 734 400n

Xn

i ¼ 1

832 221G cðzÞþfdðzÞ�cðzÞgð18i�17Þ

18n

� ��

�260 808G cðzÞþfdðzÞ�cðzÞgð18i�15Þ

18n

� �

þ2 903 148G cðzÞþfdðzÞ�cðzÞgð18i�13Þ

18n

� �

�3 227 256G cðzÞþfdðzÞ�cðzÞgð18i�11Þ

18n

� �


18n

� �

�3 227 256G cðzÞþfdðzÞ�cðzÞgð18i�7Þ

18n

� �


18n

� �

�260 808G cðzÞþfdðzÞ�cðzÞgð18i�3Þ

18n

� �

þ832 221G cðzÞþfdðzÞ�cðzÞgð18i�1Þ

18n

� ��, ð11Þ

Gðy,zÞ ¼ff ðy,zÞ�eðy,zÞg

5 734 400n

�Xn

i ¼ 1

832 221F eðy,zÞþff ðy,zÞ�eðy,zÞgð18i�17Þ

18n

� ��

�260 808F eðy,zÞþff ðy,zÞ�eðy,zÞgð18i�15Þ

18n

� �

þ2 903 148F eðy,zÞþff ðy,zÞ�eðy,zÞgð18i�13Þ

18n

� �

�3 227 256F eðy,zÞþff ðy,zÞ�eðy,zÞgð18i�11Þ

18n

� �


18n

� �

�3 227 256F eðy,zÞþff ðy,zÞ�eðy,zÞgð18i�7Þ

18n

� �


18n

� �

�260 808F eðy,zÞþff ðy,zÞ�eðy,zÞgð18i�3Þ

18n

� �

þ832 221F eðy,zÞþff ðy,zÞ�eðy,zÞgð18i�1Þ

18n

� ��: ð12Þ

Formulas for other values of m can be derived in a similar fashionas well.

3. Convergence

3.1. Haar wavelets

Lemma 1. Assume that f ðxÞAL2ðRÞ with the bounded first derivative

on (0,1), then the error norm at the level of resolution J satisfies the

following inequality:

JeJðxÞJr

ffiffiffiffiK

7

rC2�ð3Þ2

J�1

, ð13Þ

where K, C are some real constants.

Proof. For proof see [49]. &

3.2. Legendre wavelets

Lemma 2. Suppose that the function f(x) is piecewise constant or may

be approximated as piecewise constant, then we can approximates


f(x) as

f ðxÞ �X2k�1

i ¼ 1

XM�1

m ¼ 0

ai,mcm,iðxÞ ¼ f MðxÞ

then fM(x) approximate f(x) with the following error norm:

Jf ðxÞ�f MðxÞJ2r1

M!

1

2MksupxA ½0;1�9f

ðMÞðxÞ9:

Proof.

Jf ðxÞ�f MðxÞJ22 ¼

Z 1

0½f ðxÞ�f MðxÞ�

2 dx

¼X2k�1

i ¼ 1

Z 2i=2k

ð2i�2Þ=2k½f ðxÞ�f MðxÞ�

2 dx

rX2k�1

i ¼ 1

Z 2i=2k

ð2i�2Þ=2k½f ðxÞ�f �MðxÞ�

2 dx

rX2k�1

i ¼ 1

Z 2i=2k

ð2i�2Þ=2k

1

M!

1

2Mksup

xA ½0;1�9f ðMÞðxÞ9

!2

dx

¼

Z 1

0

1

M!

1

2Mksup


!2

dx

¼1

M!

1

2Mksup


!2

:

Taking the square root we have the required result. Here f nMðxÞ

denotes the interpolating polynomial of f(x) of order M. We haveapplied the maximum error bound for interpolation by using theresults given [10,13]. &

3.3. HFs interpolation

Consider the following one-dimensional function:

f ðxÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffix4þ1

pcosðo

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2þ4xþ5

pÞ, xA ða,bÞ, ð14Þ

where o is the frequency controlling parameter. As shown in Fig. 1(left top), for smaller value of or20 the function given in Eq. (14) isinterpolated exactly by HFs on the interval [0, 1]. As the value of oincreases the function approximation gets worse. It is clear fromFig. 1, re-construction of such functions through interpolation failsbadly for large values of o irrespective of the type of basis functionsused in the interpolation procedure. For many Fourier-type integralswhose evaluation is expensive in the high frequency range, themethods presented in the previous sections fail to attain reasonableaccuracy. These integrals are generally of the type:Z 1

�1

Z 1

�1Pðx,ZÞeiozðx,ZÞ dx dZ: ð15Þ

The reason for this failure is that like other quadrature rules, methodsusing wavelets basis cannot interpolate oscillatory kernels accurately.This phenomenon has been shown in Fig. 1 where the functiongiven in Eq. (14) is reconstructed for different values of o by HFs.Subsequently, the quadrature rules based on HFs and HWs willproduce inaccurate results. Inaccurate evaluation of highly oscillatorykernels by quadrature rules is the compelling reason to explore othermethods for accurate and stable computations of these integrals.

4. Meshless method of Levin’s type

Keeping in view discussion of the previous sections, weintroduce a new meshless method based on MQ RBFs fornumerical solution of integrals with highly oscillatory kernels.

We modify Levin’s method [30] by replacing monomials basisfunctions with MQ RBFs to obtain a meshless numerical solutionof the PDE. This modification improves accuracy and the condi-tion number of coefficient matrix of the resultant linear systemis reduced by the virtue of TSVD technique. In this approach,the formulation of the problem starts with the representation ofpðx,yÞ with MQ RBFs on the entire domain. The derivatives arethen calculated by differentiation of the MQ RBF. The RBFapproximation for pðx,yÞ is given in the following form:

pðx,yÞ ¼XN

k ¼ 1

fðr,cÞak, xAO, ð16Þ

where ak, k¼ 1;2, . . . ,N, are the real RBFs coefficients andr2 ¼ ðx�xkÞ

2þðy�ykÞ

2.MQ RBFs are defined as

fðr,cÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffir2þc2

p, ð17Þ

where c is the shape parameter. RBFs approximation for thederivatives of pðx,yÞ can be represented by

Lpðx,yÞ ¼XN

k ¼ 1

Lfðr,cÞak: ð18Þ

The coefficients ak,k¼ 1;2, . . . ,N, can be found using the colloca-tion points in the following form:

pðx1,y1Þ

pðx2,y2Þ

^

pðxN ,yNÞ

266664

377775¼

f11 f12 . . . f1N

f21 f22 . . . f2N

^ ^ ^ ^

fN1 fN2 . . . fNN

266664

377775

a1

a2

^

aN

266664

377775: ð19Þ

This can be written in the matrix notation as

p¼Ua, ð20Þ

where

p¼ ½pðx1,y1Þ,pðx2,y2Þ, . . . ,pðxN ,yNÞ�t ,

a¼ ½a1,a2, . . . ,aN�t

and

Fsk ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðxs�xkÞ

2þðys�ykÞ

2þc2

qis the sk-th matrix element of the N�N matrix U.

Levin [30] studied the following two-dimensional oscillatoryintegrals on the regular domain ½a,b� � ½c,d�:

I¼

Z b

a

Z d

cf ðx,yÞeigðx,yÞ dy dx: ð21Þ

In order to use MQ RBF instead of monomials in Levin’s method,we first give a brief introduction of Levin’s method and extends itto three-dimensional integrals as well.

According to Levin’s method, the integral in Eq. (21) istransformed into the following PDE:

@2ðpeigÞ

@x@y¼ ½pxyþ igypxþ igxpyþðigxy�gxgyÞp�e

ig ¼ feig , ð22Þ

where pðx,yÞ is a proper function to be determined.Making use of MQ RBF we can write Eq. (22) in the following

form:

XN

k ¼ 1

@2fjk

@x@yakþ igyðxjÞ

XN

k ¼ 1

@fjk

@xakþ igxðxjÞ

XN

k ¼ 1

@fjk

@yak

þðigxyðxjÞ�gxðxjÞgyðxjÞÞXN

k ¼ 1

fjkak ¼ f ðxjÞeigðxjÞ, ð23Þ

where j¼ 1;2,3, . . . ,N.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1.5

−1

−0.5

0

0.5

1

1.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−2

−1.5

−1

−0.5

0

0.5

1

1.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1.5

−1

−0.5

0

0.5

1

1.5

Fig. 1. Comparison of interpolation through HFs for M¼8, k¼3, a¼ 0,b¼ 1 with exact value of the function for different values of o.


The above linear system of equations can be written in matrixnotation as

Wa¼ b, ð24Þ

where W is the matrix in N � N system of linear equations(Eq. (23)), and the unknown coefficients on the left-hand side ofEq. (23) is denoted by a¼ ½a1,a2, . . . ,aN�

t , and b¼ ½b1,b2, . . . ,bN�t is

the right-hand side of Eq. (24). The matrices W and b are defined as:

Csk ¼@2fsk

@x@yþ igyðxsÞ

@fsk

@xþ igxðxsÞ

@fsk

@yþðigxyðxsÞ�gxðxsÞgyðxsÞÞfsk,

bs ¼ f ðxsÞeigðxsÞ, s,k¼ 1;2, . . . ,N:

We determine the coefficients a by inverting W

a¼W�1b, ð25Þ

which implies that

as ¼XN

k ¼ 1

C�1sk bk, s¼ 1;2, . . . ,N, ð26Þ

where C�1sk denotes the matrix element of the matrix W�1: The

solution p can now be readily found, which leads to the evaluationof the following integral.


By putting (22) into (21), the integral thus obtained is given by

I¼

Z b

a

Z d

cf ðx,yÞeigðx,yÞ dy dx¼

Z b

a

Z d

c

@2ðpeigÞ

@x@ydy dx

¼ pðb,dÞeigðb,dÞ�pða,dÞeigða,dÞ�pðb,cÞeigðb,cÞ þpða,cÞeigða,cÞ: ð27Þ

The above equations show that it is equivalent to find numericalsolution of the PDE (22) and then compute integral (21) with thehelp of Eq. (27).

In the similar way if we consider the following three-dimen-sional oscillatory integral:

I¼

Z b

a

Z e

d

Z t

sf ðx,y,zÞeigðx,y,zÞ dz dy dx ð28Þ

and then extend Levin’s method to convert Eq. (28) into thefollowing PDE:

@3ðpeigÞ

@x@y@z¼ ½pxyzþ iðgxpyzþgypxzþgzpxyÞþ iðgyzpxþgxzpyþgxypzÞ

�ðgxgypzþgygzpyþgxgzpxÞ

þðigxyz�gxgyz�gygxz�gzgxy�gxgygzÞp� ¼ feig : ð29Þ

Using the same meshless procedure shown above we can get thevalue pðx,y,zÞ. Subsequently, by substituting Eq. (29) into Eq. (28)the three-dimensional integral becomes

I¼ pðb,e,tÞeigðb,e,tÞ�pða,e,tÞeigða,e,tÞ�pðb,d,tÞeigðb,d,tÞ þpða,d,tÞeigða,d,tÞ

�pðb,e,sÞeigðb,e,sÞ þpða,e,sÞeigða,e,sÞ þpðb,d,sÞeigðb,d,sÞ�pða,d,sÞeigða,d,sÞ:

ð30Þ

Hence, the main task in meshless method of Levin’s type is to findpðx,yÞ from (22) for two-dimensional oscillatory integrals andpðx,y,zÞ from (29) for three-dimensional oscillatory integrals.

5. Numerical experiments and discussions

In this section we test the efficiency of the new methods basedon HWs, HFs and RBF on several benchmark problems andcompare the numerical results with existing methods in theliterature. These include Gaussian Method [35] for one- and two-dimensional integrals in polar co-ordinates, delaminating quad-rature method for multi-dimensional gentle and highly oscillatoryintegrals [31] and Levin’s method [30]. Six test problems areconsidered to show good approximating properties of the newmethods in terms of absolute and relative errors. In Test Problem 1,the integrand is axi-symmetric mildly oscillatory. In Test Problem 2,the integrand is non-axi-symmetric non-oscillatory. Test Problem 3is related to non-oscillatory integral and Test Problem 4 is to mildlyoscillatory integrals with stationary points. Test Problems 5 and 6are related to two- and three-dimensional highly oscillatory inte-grals. The absolute error is defined as

Labs ¼ 9u�u9,

Lrel ¼Labs

u,

where u and u are the actual and numerical solutions, respectively.The notation Ei is used for the error at i-th mesh arrangement

with step size hi and the experimental rate of convergence Rc(N) isdefined as

RcðNÞ ¼log Ei

Eiþ 1

� �log hi

hiþ 1

� � : ð31Þ

Based on our previous experience we have chosen the shapeparameter c¼ 1=

ffiffiffiffiffiffiffiffiffiffiffiN�1p

(where N is the number of collocationpoints) in the MQ RBF. The notation N¼Nx � Ny (in two-dimen-sional case) and N¼Nx � Ny � Nz (in three-dimensional case) isused for the collocation points, where Nx, Ny and Nz are the numberof collocation points along the x-, y- and z-axis, respectively.

5.1. Axi-symmetric mildly oscillatory integrals

Test Problem 1.Z y2

y1

Z r0

0cosðr2Þr dr dy¼

sinðr20Þ

2ðy2�y1Þ

The problem 1 has been considered in [35] to select a bestsubsidiary method in the context of preserving the MLPG solutionaccuracy of the underlying numerical quadrature rules. Our mainmotivation is to suggest further improvement in the numericalintegration in terms of accuracy with minimum number of colloca-tion points, so that the intermediate numerical techniques canevaluate the given integrals with optimal accuracy in an efficientmanner. To pursue this objective, we compare performance of thenew methods with existing methods in terms of absolute errors.We assume a circular integration domain or a circular sector withcenter (0, 0) as considered in [35]. The numerical results corre-sponding to Test Problem 1 are shown in Fig. 2 where the newmethods are compared with Rules I, II and III given in [35]. Theintegrand in this case is an axi-symmetric non-polynomial mildlyoscillatory function. Accuracy wise performance of these rules isinvestigated on a circular disk or a circular sector. As shown in Fig. 2(corresponding to Test Problem 1), performance of the new methodbased on HFs is better than Rules I, II, III. Also the new methodbased on HFs produces more accurate results than HWs. Accurateresults of our methods based on HWs and HFs against Rules I, II, III[35] for the circle of different radii (small radius r0 ¼ 1 (left) andlarge radius r0 ¼ 4 (right)) are shown in Fig. 2 with improvedperformance of the new method based on HFs. As reported in [35],Rules I and III cannot establish even a single digit accuracy forN¼16 collocation points and hence are unable to capture oscilla-tory behavior of the integrand for small number of collocationpoints. Even Rule II which performs better than Rules I and IIIrequires at least N¼64 points to approximate solution with an errorOð10�4

Þ. Accuracy of the HFs based method is up to Oð10�3Þ for grid

size N¼16 and Oð10�5Þ for N¼49. From overall comparison it is

clear that HFs based method shows best accuracy and stableperformance as compared with the existing method as well as thenew method based on HWs for reasonably small number ofcollocation points. In Fig. 3 we have performed CPU time (inseconds) comparison of our method based on HFs and Rule I. TheCPU time of Rule I is slightly less than that of HFs. This differencehas a very little impact when it comes to implementation. InTable 1 numerical order of convergence is shown for TestProblem 1. It is clear from this table that HWs based method issecond order convergent, whereas the method based on HFs iseight-order convergent. This evidence clearly indicates that themethod based on HFs is superior in terms of better accuracy andfaster convergence.

5.2. Non-axi-symmetric non-oscillatory integrals

Test Problem 2.Z y2

y1

Z r0

0r3 cos2ðyÞ dr dy¼

r40

8y2�y1þ

sinð2y2Þ�sinð2y1Þ

2

� �:

0 10 20 30 40 50 60 70 80 90 10010

10

10

10

Fig. 3. Comparison of CPU time of Rules I given [35], the new method based on

HFs for the Test Problem 1, r0 ¼ 1.

Table 1Numerical rate of convergence for Test Problem 1 (r0 ¼ 1).

Nodes Haar rate of

convergence

Hybrid rate of

convergence

3�3 1.9876 5.1909

4�4 2.0075 7.3077

5�5 1.9743 7.6544

6�6 1.9844 7.7857

7�7 1.9219 7.8554

8�8 2.0688 7.8986

9�9 2.1890 7.8985

10�10 1.8428 7.9411

0 10 20 30 40 50 60 70 80 90 10010

10

10

10

10

10

10

10

10

Rule I

Rule II

Rule III

HWs

HFs

0 10 20 30 40 50 60 70 80 90 10010

10

10

10

10

10

10

10

10

10

Rule I

Rule II

Rule III

HWs

HFs

Fig. 2. Comparison of Labs of Rules I, II, III given [35], the new methods based on HWs and HFs for the Test Problem 1, for two different values of r0.


The numerical results corresponding to Ex. 2 are shown in Fig. 4corresponding to r0 ¼ 1, y1 ¼ 0, y2 ¼ 2p (left) and corresponding tor0 ¼ 1, y1 ¼ 0, y2 ¼ 2:5 (right). The integrand in this case is a non-axi-symmetric non-oscillatory function. Like the pervious problem,accuracy of the new method based on HFs is better than Rules I, II, IIIand HWs. For N¼4 (Fig. 4 left) the accuracy of HFs is little lowerthan Rule II. Performance of the Haar based algorithm is steady butlower in accuracy in comparison with the method based on HFs.Similarly, from Fig. 4 (right) it is clear that the new method based onHFs defined on circular sector gives best results in comparison withthe methods given in [35] (Rules I, II, III) and our method based on

HWs especially for small number of collocation points. From thesenumerical experiments we observed that the new methods based onHWs and HFs produce uniformly accurate results for differentgeometries like circular domains with different radii and circularsectors. Method based on HFs has been found more accurate androbust for the integrals having non-oscillatory and mildly oscillatoryintegrands and for the cases where the integrand requires moreintegration points close to center of the circular sector as well.

The numerical solutions of Test Problems 3 and 4 are reported in[31] by using delaminating quadrature method which is modifiedform of Levin’s method. The general form of these integrals isZ

Vf ðx,y,z, . . .Þeigðx,y,z,...Þ dV , ð32Þ

where f ðx,y,z, . . .Þ and gðx,y,z . . .Þ are often called amplitude andphase functions, respectively. Modified Levin’s method has beenproposed for these integrals in [31]. These integrals are very difficultto calculate by standard classical quadrature methods (such as theSimpson rule [24,31]), so efforts should be made to construct anaccurate and efficient method for this class of numerical integration.The new method based on HFs is fundamentally different fromLevin’s method [30], the modified Levin’s method [31], our meshlessmethod (discussed in the previous section) and is recommended fornumerical solution of the non-oscillatory integrals of type given in

0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 60 70 80 90 100

Fig. 4. Comparison of Labs of Rules I, II, III given [35], the methods based on HWs and HFs for the Test Problem 2, r0 ¼ 1, y1 ¼ 0.

0 50 100 150 200 250 300 0 50 100 150 200 250 300

Fig. 5. Comparison of the method based on HFs and the method based on RBF for Test Problem 3 (Left) Labs (Right) Lrel.


Eq. (32) or mildly oscillatory integrals of type given in Test Problems1, 2 and 4, for the following reasons:

(i)
The present approach is computationally less intensive thanthe methods given [30,31] as it uses simple quadratureprinciple and it does not need an intermediate PDE to besolved numerically.
(ii)
This approach does not encounter solution of system of linearequations and hence the problem of ill-conditioning doesnot arise.
5.3. Non-oscillatory integrals

Test Problem 3.Z 1

0

Z 1

0ex2þy cosðxþy2Þei

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2þð104

�yÞ2p

dx dy

¼�1:180502347994539þ i� 0:1589906984095899:

This type of integrals is given in [31]. For the oscillator g, thecoefficient matrix is ill-conditioned if the existing methods


[31,24,30] are used without some regularization technique.Oscillators of this type are very familiar in the computationalelectromagnetic [31]. In Fig. 5 (left absolute error and rightrelative error), we compared results of both meshless method ofLevin’s type (developed in the earlier section) and method basedon HFs. Superior performance of the method based on HFs is clearfrom the figure. It has been observed from the comparison thatbetter accuracy of Oð10�9

Þ is obtained through HFs with only N¼4collocation points. Maximum accuracy reported in Fig. 5 of [31] isof order 10�10 for N¼196 collocation points by using TSVD basedalgorithm, whereas the results of LU based methods are reportedunstable corresponding to Test Problem 3. In our case as shown in

0 50 100 150 200 250 30010

10

10

10

10

Fig. 6. Comparison of CPU time (in seconds) of HFs and RBF based methods for

Test Problem 3.

0 500 1000 1500 2000 2500 3000 3500 400010

10

10

10

10

10

10

10

10

10

10

Fig. 7. Real and imaginary parts Labs and Lrel of the

Fig. 5 we have achieved the same accuracy corresponding to N¼4nodal points. The new method extends accuracy up to 10�13 forreal and imaginary parts. Unlike [24,30,31] and our meshlessmethod, the method based on HFs is straightforward and wellconditioned and hence it does not need any regularizationtechnique like TSVD. This is very important contribution fromapplication point of view as TSVD becomes computationallyintensive as N gets larger and larger. Computational efficiencyof the method based on HFs is also shown in Fig. 6. Thus, ourmethod based on HFs has bypassed the difficulty of ill-condi-tioned matrices which plagues other methods ([24,30,31] includ-ing meshless method).

5.4. Gentle oscillators with stationary points

Test Problem 4.

Z 1

�1

Z 1

�1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðx3þy4þ5Þ

px2þyþ3

ei100ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðx2þy2þ2Þp

dx dy

¼�0:01260046717�i� 0:05903285625:

Numerical results corresponding to the real part of the oscillatorgiven in Test Problem 4 are shown in Fig. 7. The oscillator hascritical point at (0,0) (for critical points rg ¼ 0) and specialnumerical treatment (like the methods named Cand I and CandII) has been proposed by Ji et al. [31] to obtain numerical solutionof the problem. Although the methods [31] suggest an accuratenumerical solution for the given class of integral, however, thesemethods have the following fundamental difficulties which haveto be taken care of during implementation. These are

(i)

10

1

1

1

1

1

1

1

1

1

new

For Cand I [31], the authors have used Chebyshev–Lobattonodes that generates sparse grid in the center and dense atthe end points. This is problematic when stationary points liein the center. In this case a fine mesh is required to havereasonable accuracy at the stationary points, which in turnsincrease computational cost of the algorithm.

500 1000 1500 2000 2500 3000 3500 40000

0

0

0

0

0

0

0

0

0

method based HFs for the Test Problem 4.


(ii)

z

Subdivision of the domain of the given integral is needed tomove stationary points from interior of the domain to thecorner of the sub-domain.

The added advantage of our method HFs is that it does not requirespecial nodal arrangement and subdivision of the domain tocope with the oscillators having stationary or critical points isnot needed. Accuracy of the new method based on HFs is ofthe order 10�9 in terms of absolute error and 10�7 in termsof relative error for N¼900 number of points. Accuracy of theHFs ascends to 10�12 for n¼3600 number of collocation points.The results reported in Fig. 6 of the paper [31] show accuracy10�7 for N¼8100 corresponding to Cand. 1 and 10�7 for N¼1600corresponding to Cand. 2. It has been reported in [31] thatLevin’s method is unable to solve integral given in Ex. 4.Satisfactory results of method based on HFs are obtained in thiscase as well.

0 50 100 150 200 250 30010

10

10

10

Fig. 9. Comparison of Labs of HFs and RBF based metho

00.2

0.40.6

0.81

00.2

0.40.6

0.81

−1

−0.5

0

0.5

1

xy

Fig. 8. Plot of the real part of the integrand in Test Problem 5.

5.5. Highly oscillatory integrals

Test Problem 5.Z 1

0

Z 1

0cosðoðxþyÞÞei104

ðxþyþx2þy2Þ dy dx:

The above Test Problem 5 which is highly oscillatory is solved byboth the quadrature method based on HFs and meshless methodof Levin’s type. The numerical results are shown in Fig. 9 (whereHfre, Hfim, RBFre, RBFim denote Hybrid real, Hybrid imaginary,RBF real and RBF imaginary parts respectively). Real part of theintegrand given in the Test Problem 5 is shown in Fig. 8 (foro¼ 15) which shows highly oscillatory character of the oscillator.The meshless and HFs based methods are tested on the problemfor different values of frequency (o¼ 1;15). From the Fig. 9 it isclear that unlike previous cases, performance of the meshlessmethod is stable and superior than its counterpart numericalmethod based on HFs. Accuracy wise performance of the meshlessapproach is comparable with graphical results shown in [31]. Theadded advantage of our approach is its meshless character, simplederivation and easy extension to higher dimensions.

5.6. Three-dimensional oscillatory integrals

Test Problem 6.Z 1

0

Z 1

0

Z 1

0cosð10ðxþyþzÞÞei104

ðxþyþ zÞ dz dy dx:

The exact solution is (0.6675648262455628þ i�

3.083181356966526)�10�13. Test problem 6 is a three-dimen-sional highly oscillatory problem. Numerical results are shown inFig. 10. Like the previous oscillatory problems, performance of themeshless approach is superior than the quadrature method basedon HFs. Performance of HFs is unstable in this case due to highlyoscillatory integrand. From the two numerical experiments it

0 50 100 150 200 250 30010

10

10

10

10

10

10

10

ds for Test Problem 5 using different values of o.

0 100 200 300 400 500 600 700 800 900 100010

10

10

10

10

10

10

10

10

10

Fig. 10. Comparison of Labs of HFs and RBF Test Problem 6.


becomes clear that the meshless approach of Levin’s type is moresuitable for the numerical solution of highly oscillatory integralsin higher dimensions.

6. Conclusion

New quadrature methods based on HWs and HFs and MQ RBFsare being proposed for the numerical solution of multi-dimen-sional low, highly oscillatory and non-oscillatory integrals. It isshown that the methods based on HWs and HFs can be useful forthe numerical solution of the integrals with no or gentle oscilla-tory behavior. It is also observed that the methods based on HWsand HFs have been found suitable and accurate for the integralswith stationary points and integrals having phase function with asmaller norm. Like other quadrature methods and commercialsoftwares these methods cannot handle oscillatory integrals aswell. Due to sparse availability of numerical methods for highlyoscillatory integrals, we proposed a meshless method of Levin’stype for such type of integrals. Best performance of the meshlessmethod of Levin’s type is observed for highly oscillatory integrals.In the case of mildly oscillatory integrals, meshless method is oflower accuracy than the methods based on HWs and HFs. Thenew meshless method is not susceptible to ill-conditioned beha-vior of the system of linear equations by virtue of regularizationtechnique TSVD. Benchmark problems testifies improved accu-racy and stable behavior of the new methods.

Acknowledgments

The authors are thankful to the reviewers for the valuablesuggestions which improved the first draft of the paper. Thesecond author is thankful to the Higher Education Commission(HEC) Islamabad Pakistan for the financial support during his PhDstudies.

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