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Comparison of asynchronous particleswarm optimization and dynamicdifferential evolution for partiallyimmersed conductorChien-Ching Chiu a & Wei-Chun Hsiao aa Electrical Engineering Department , Tamkang University Tamsui ,Taiwan, R.O.C.Published online: 20 Jun 2011.

To cite this article: Chien-Ching Chiu & Wei-Chun Hsiao (2011) Comparison of asynchronous particleswarm optimization and dynamic differential evolution for partially immersed conductor, Waves inRandom and Complex Media, 21:3, 485-500, DOI: 10.1080/17455030.2011.588271

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Waves in Random and Complex MediaVol. 21, No. 3, August 2011, 485–500

Comparison of asynchronous particle swarm optimization and

dynamic differential evolution for partially immersed conductor

Chien-Ching Chiu* and Wei-Chun Hsiao

Electrical Engineering Department, Tamkang University Tamsui, Taiwan, R.O.C.

(Received 25 January 2011; final version received 6 May 2011)

The application of two techniques for the of shape reconstruction of aperfectly two-dimensional conducting cylinder from mimic measurementdata is studied in the present paper. After an integral formulation, themicrowave imaging is recast as a nonlinear optimization problem; a costfunction is defined by the norm of a difference between the measuredscattered electric fields and the calculated scattered fields for an estimatedshape of a conductor. Thus, the shape of conductor can be obtained byminimizing the cost function. In order to solve this inverse scatteringproblem, transverse electric (TE) waves are incident upon the objects andtwo techniques are employed to solve these problems. The first is based onan asynchronous particle swarm optimization (APSO) and the second is adynamic differential evolution (DDE). Both techniques have been tested inthe case of simulated mimic measurement data contaminated by additivewhite Gaussian noise. Numerical results indicate that the DDE algorithmand the APSO have almost the same reconstructed accuracy.

1. Introduction

The detection and reconstruction of buried and inaccessible scatterers by invertingmicrowave electromagnetic data is a research field of considerable interest because ofnumerous applications in geophysical prospecting, civil engineering, and non-destructive testing and medical imaging [1]. Numerical inverse scattering studiesfound in the literature are based on either frequency or time domain approaches.However, it is well known that one major difficulty of inverse scattering is its ill-posedness in nature [2].

Another inverse scattering problem is the nonlinearity because it involves theproduct of two unknowns: the electrical property of object, and the electric fieldwithin the object. In general, the nonlinearity of the problem is coped with by thetraditional deterministic methods which are founded on a functional minimizationby way of some gradient-type scheme. Furthermore, for a gradient-type method, it iswell known that the convergence of the iteration depends highly on the initial guess.

*Corresponding author. Email: chiu@ee.tku.edu.tw

ISSN 1745–5030 print/ISSN 1745–5049 online

� 2011 Taylor & Francis

DOI: 10.1080/17455030.2011.588271

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If a good initial guess is given, the speed of the convergence can be very fast. On theother hand, if a good initial guess is used, a good solution can be obtained very

quickly [3]. In general, they tend to become trapped in local minima when the initialtrial solution is far away from the exact one applying iterative optimizationtechniques [4–6]. Such promising technique yields to efficient direct imaging

approaches [7–12]. These algorithms based on stochastic strategies, offer advantagesrelative to local inversion algorithms including strong search ability simplicity,robustness, and insensitivity to ill-posedness. In contrast to traditional computationsystems, evolutionary computation [13–16] provides a more robust and efficient

approach for solving inverse scattering problems. It has been demonstrated thaparticle swarm optimization (PSO) is a useful method of optimization for difficultand discontinuous multidimensional engineering problems [17,18]. PSO is very

efficient at exploring the entire search space. A new method of optimization,dynamic differential evolution (DDE) [19] is able to accomplish the same goal asgenetic algorithm (GA) optimization in a new and faster way. Since APSO and DDE

both work with a population of solutions, combining the searching abilities of bothmethods seems to be a good approach.

Frequency domain inverse scattering by population-based stochastic algorithmsare published in the last 10 years. Concerning the shape reconstruction ofconducting scatterers, the PSO has been investigated whereas the GA has

been utilized in the reconstruction of periodic conductor scatterers [20,21]. In thiscase, the reported results indicate that the PSO is reliable tools for inversescattering applications. Moreover, it has been shown that both DE and PSOoutperform real-coded GA in terms of convergence speed. In recent decade years,

some papers have compared different algorithm in inverse scattering [13,14].However, to our knowledge, a comparative study about the performances ofasynchronous particle swarm optimization and dynamic differential evolution when

applied to inverse scattering problems has not yet been investigated under frequencydomain.

There are several papers for the shape reconstruction of two-dimension objectsby using transverse magnetic (TM) waves [22–24]. However, from a mathematical

point of view, the 2D TE problems where both electric field components in the

transverse plane need to be taken into account in the formulation which results in a

more complex formulation compared with the TM case. Therefore, the computa-

tional load for exploiting such positive features is unavoidably increased because of

the need to solve a vectorial problem characterized by a stronger nonlinearity. From

a physical perspective, the TE-polarized case includes polarization charges at

dielectric discontinuities, which are difficult to model numerically. Besides, TE-

polarized data may contain more useful information about the object of interest as it

is based on two different components of the electric field as opposed to one in the

TM-polarized case. Note that these two polarizations are physically uncoupled:

they provide independent information about the object being imaged [25]. Recently,

there are a few reports on subject of 2D object about shape reconstruction problems

by TE experimental data, such as genetic algorithms (GAs) [26] and level-set

algorithm [27].

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In this paper, the inverse scattering problem of the partially immersed perfectly

conducting cylinder by TE wave illumination is investigated on the application ofboth APSO and DDE. The objective function of both APSO and DDE is tominimize the discrepancy between measured and estimated scattered field data.

Numerical results show that the APSO and DDE have almost the same recon-structed accuracy when the same number of iterations is applied. In Section 2, the

solution of the forward scattering problem is presented. In Sections 3 and 4, inverseproblem and the numerical results of the proposed inverse problem are given,respectively. Section 5 gives the conclusions.

2. Forward problem

Let us consider a perfectly conducting cylinder which is partially immersed in a lossy

homogeneous half-space, as shown in Figure 1 Media in regions 1 and 2 arecharacterized by permittivities and conductivities ð"1, �1Þ and ð"2, �2Þ, respectively.A perfectly conducting cylinder is illuminated by a TE plane wave. The cylinder is of

an infinite extent in the z direction, and its cross-section is described in polarcoordinates in the x, y plane by the equation � ¼ Fð�Þ. We assume that the time

dependence of the field is harmonic with the factor ej!t. Let ~Hinc denote the incidentfield form region 1 with incident angle �1 as follows:

~Hinc ¼ e�jk1ð y cos�1þx sin�1Þz: ð1Þ

Owing to the interface between regions 1 and 2, the incident plane wave generatestwo waves that would exist in the absence of the conducting object. Thus, the

unperturbed field is given by:

H* i

¼½e�jk1ð y cos�1þx sin�1Þ þH1e

�jk1ð�y cos�1þx sin �1Þ�z, y � �aH2e

�jk2ð y cos�2þx sin�2Þz, y4 � a

�ð2Þ

Hinc

1φregion 1(y<-a)

region 2(y>-a)

(ε1,σ1)

(ε2,σ2)

y a= −

θ X

Y

F(θ)ρ =

Figure 1. Geometry of the problem in (x, y) plane.

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where

H1 ¼Z1 � Z2

Z1 þ Z2

� �e2jk1a cos�1

H2 ¼2Z1e

jk1a cos�1e�jk2a cos�2

Z1 þ Z2

�2 ¼ sin�1k1k2

sin�1

Z1 ¼ �1 cos�1, Z2 ¼ �2 cos�2, �1 ¼

ffiffiffiffiffiffi�1

"1

r, �2 ¼

ffiffiffiffiffiffi�2

"2

r:

Since the cylinder is partially immersed, the equivalent current exists both in theupper half space and the lower half space.

For a perfectly conducting scatterer, the total tangential electric field at thesurface of the scatterer is equal to zero.

n^�

1

j!"r � H

* tot� �¼ 0 ð3Þ

with H* tot

¼ H* i

þ H* s

where n is the outward unit vector normal to thesurface of the scatterer and H

* s

is the scattered fielded.For the direct scattering problem, the scattered field Hs is calculated by assuming

that the shape is known. For the inverse problem, assume the approximate center ofscatterer, which in fact can be any point inside the scatterer, is known. Then the

shape function Fð�Þ can be expanded as:

Fð�Þ ¼XN=2n¼0

Bn cosðn�Þ þXN=2n¼1

Cn sinðn�Þ ð4Þ

where Bn and Cn are real coefficients to be determined, and Nþ 1 is the number ofunknowns for the shape function. In the inversion procedure, the APSO and DDE

are used to minimize the following cost function [28]:

CF ¼1

Mt

XMt

m¼1

Hsexpð r

*

mÞ �Hscalð r

*

mÞ

��� ���2. Hsexpð r

*

mÞ

��� ���2( )1=2

ð5Þ

where Mt is the total number of mimic measurement data points. Hsexpð r

*

mÞ andHs

calð r*

mÞ are the measured and calculated scattered fields, respectively.

3. Inverse problem

3.1. Particle swarm optimization

Particle swarm optimization is a class of derivative-free, population-based and self-adaptive search optimization technique which introduced by Kennedy and Eberhart

[30]. Particles (potential solutions) are distributed throughout the searching space

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and their positions and velocities are modified based on social behavior. The social

behavior in PSO is a population of particles moving towards the most promising

region of the search space. Clerc [31] proposed the constriction factor to adjust the

velocity of the particle for obtaining the better convergence; the algorithm was

named as constriction factor method. PSO starts with an initial population of

potential solutions that is randomly generated. After the initialization step, each

particle of population has assigned a randomized velocity and position. Thus, each

particle has a position and velocity vector, and can move through the problem space.

In each generation, the particle changes its velocity according to its best experience,

called xpbest, and the best particle in the swarm, called xgbest.Assume there are Np particles in the swarm that is in a search space in D

dimensions, the position and velocity could be determine according to the following

equations (constriction factor method):

vkid ¼ � � vk�1id þ c1 � ’1 � xpbest,id � xk�1id

� �þ c2 � ’2 � xgbest,id � xk�1id

� �� �ð6Þ

xkid ¼ xk�1id þ vkid ð7Þ

where � ¼ 2

2���ffiffiffiffiffiffiffiffiffiffi�2�4�p�� ��, � ¼ c1 þ c2 � 4. c1 and c2 are learning coefficients, used to

control the impact of the local and global component. vkid and xkid are the velocity and

position of the ith particle in the dth dimension at kth generation; ’1 and ’2 are boththe random number between 0 and 1. It should be mentioned that the fastest speed

(Vmax) method is also applied to control the particle’s searching velocity and to

confine the particle within the search space [32]. The value of Vmax is set to be half of

Xmax, where Xmax is the upper limits of the search space. Note that the Vmax and Xmax

are maximum velocity and maximum distance, respectively. As an extreme case, if

the maximum velocity Vmax is set to Xmax, the exploration to the inverse scattering

problem space is not limited. Occasionally, the particles may move out of their search

space. This problem could be remedied by applying the boundary condition to draw

the foul particles back to the normal space. In many practical optimization problems,

the dimensionality and the location of the global optimum is usually difficult to

know a priori. It is therefore desirable to have a single boundary condition that can

offer a robust and consistent performance for the PSO technique regardless of the

problem dimensionality and the location of the global optimum. When the particle

exceed a certain dimension of the solution search space, the position of particle will

be re-located in the search boundary and reverse its velocity component in this

dimension and multiplied by a random number. Its rate of return to the search space

is based on a random number. So in this paper, we apply the ‘damping boundary

condition,’ as Figure 2, is proposed by Huang and Mohan to ensure the particles

move within the legal search space [33]. The key distinction between asynchronous

PSO and a typical PSO is on the population updating mechanism. In the

synchronous PSO, the algorithm updates all the particles velocities and positions

using Equations (6) and (7) at end of the generation. And then update the best

positions, xpbest andxgbest. Alternatively, the updating process of asynchronous PSO

is that if the latest and the best position is better than the current position, we set that

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position to replace the current position after each particle position have updated.

This latest and the best position will be used in following particles swarm

immediately. The swarm reacts more quickly to speed up the convergence because

the updating occurs immediately after objective function evaluation for each particle.The flowchart of the modified asynchronous PSO (APSO) is shown in Figure 3.

APSO goes through seven procedures as follows:

(1) Initialize a starting population: A swarm of particles is randomly generated.(2) Calculate H fields by method of moment [34].(3) Evaluate cost function: The asynchronous PSO algorithm evaluates the cost

function (5) for each individual in the population.(4) Find xpbest and xgbest.(5) Mutation scheme: the particle swarm optimization (PSO) algorithm has

been shown to converge rapidly during the initial stages of a global search,

but when around global optimum, the search can become very slow. For the

reason, mutation scheme is introduced in this algorithm to speed up the

convergence when particles are around global optimum. The mutation

scheme can also avoid premature convergences in searching procedure and

help the xgbest escape from the local optimal position. As shown in Figure 3,

there is an additional competition between the xgbest and xmugbest. The current

xgbest will be replaced by the xmugbest if the x

mugbest is better than the current xgbest.

The xmugbest is generated by following way:

Xmugbest ¼

Xgbest � ’3 � c3 � ðc3 � c4Þ �k

kmax

� xmax � xminð Þ, if ’mu 5 0:5

Xgbest þ ’3 � c3 � ðc3 � c4Þ �k

kmax

� xmax � xminð Þ, if ’mu � 0:5

8>>><>>>:

ð8Þ

where c3 and c4 are the scaling parameter. ’3 and ’mu are both the random

number between 0 and 1. k is the current iteration number. kmax is the

1kidx−k

idv

kidx

y

x

'kidx

ˆ ˆ' () kkkid xid yidv rand v x v y= − ⋅ +

Figure 2. (Color online) The damping map.

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maximum iteration number. xmax and xmin are the upper limit and lower limitof the search space, respectively.

(6) Update the velocity and position.(7) Stop the process and print the best individual if the termination criterion is

satisfied, else go to step 2.

3.2. Dynamic differential evolution (DDE)

Each individual in DDE algorithm is a D-dimensional vector consisting of Doptimization parameters. After initialization, DDE algorithm performs the geneticevolution until the termination criterion is met. DDE algorithm, like other EAs, alsorelies on the genetic operations (mutation, crossover and selection) to evolve

Generate initial population x0 randomly and uniformly in the search range, k=0

k>kmaxStop: giving , optimal solution.

id=1

Evaluate cost function for the id-th particle’s positionin the population,

id=id+1No

k=k+1Yes

No

Yes

Find

Find

Update the velocity and position of the id-th particle in the

population

id=population size

mutationYes

No

=

Calculate H field by method of moment

pbestx

gbestx

= gbestxgbestx gbestx

) )mugbest gbestCF x CF x<( (

mugbestx

gbestx

oNseY

1

1{ ( ) ( ) / ( ) }CF H r H r H r

M= −

Figure 3. The flowchart of the modified asynchronous PSO (APSO).

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generation by generation. The flowchart of the DDE algorithm is shown in Figure 4.DDE algorithm goes through six procedures as follows:

(1) Initialize a starting population: DDE algorithm is initialized with apopulation that is composed by a group of randomly generated candidateindividuals. Individuals in DDE algorithm represent a set of D-dimensionalvectors in the parameter space for the problem, fxi : i ¼ 1, 2, . . . ,Npg,where D is the number of parameters to be optimized and Np is thepopulation size.

(2) Evaluate the population using cost function: After initialization, DDEalgorithm evaluates the cost function (5) for each individual in thepopulation.

(3) Perform mutation operation to generate trial vectors: the mutation operationof DDE algorithm is performed by arithmetical combination of individual.

Generate initial population x0 randomlyand uniformly in the search range, g=0

g>gmax stop

j=0

1gjv +

))1 gj

gj ff ( x( u <+

Yesgj

gj xx =+1

11 ++ = gj

gj ux

No

j=populationsize

j=j+1Nog=g+1 Yes

No

Yes)) best

gj ff ( x( x <

No

Yes

1gj

1gbest xx ++ =

Mutate to have trial vector

ex:

Crossover with to deliver a crossover vector1+gjv g

jx 1+gju

≥<

=+

+

Crx

Crv

igj

igjg

j ζζ

,)(

,)( 11u

Evaluates the cost function for each individual in the population

11 2( ) ( ) [( ) ( ) ] [( ) ( ) ]g g g g g g

i i i i i ij j best j r rv x F x x x xλ+ = + ⋅ − + ⋅ −

Figure 4. The flowchart of the dynamic differential evolution (DDE).

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For each parameter vector xgj of the parent generation, a trial vector vgþ1j isgenerated according to following equation:

ðvgþ1j Þi ¼ ðxgj Þi þ F � ½ðxgbestÞi � ðx

gj Þi� þ � ½ðx

gr1Þi � ðx

gr2Þi�, i 6¼ r1 6¼ r2 ð9Þ

where F4 0 and 4 0 both are scaling factors that control the amplificationof the differential variationðxgbest � xgj Þ and ðx

gr1Þ � ðx

gr2Þ. The indices i, r1 and

r2 of individuals are randomly chosen. The superscipt g stands for thegeneration index of the parent generation. The subscript best refers to theoptimal individual in the parent population.

(4) Perform crossover operation with probability of crossover Cr to delivercrossover vectors: The crossover operation in DDE algorithm is performed toincrease the diversity of the parameter vectors. This operation is similar tothe crossover process in GAs. However, the crossover operation in DDEalgorithm just allows delivering the crossover vector ugþ1j by mixingcomponent of the current vector xgj and the trial vector vgþ1j . It can beexpressed as:

ugþ1j ¼ðvgþ1j Þi, 5Cr

ðxgj Þi, � Cr

(ð10Þ

where Cr is the probability of crossover, is the random number generateduniformly between 0 and 1.

(5) Perform selection operation to produce offspring: selection operation isconducted by comparing the current vector xgj with the crossover vector ugj .The vector with smaller cost function value is selected as a member of thenext generation. Explicitly, the selection operation for the minimizationproblem is given by:

xgþ1i ¼ugi , if CFðugi Þ5CFðxgi Þxgi , otherwise

�ð11Þ

DDE algorithm is carried out in a dynamic way: each parent individualwould be replaced by its offspring if the offspring has obtained a better costfunction value than its parent. Thus, DDE algorithm can respond theprogress of population status immediately and yield faster convergence speedthan the typical DE. Based on the convergent characteristic of DDEalgorithm, we are able to reduce the numbers of cost function evaluation andfind the solution efficiently.

(6) Stop the process and obtain the best individual if the termination criterion issatisfied, else go to step 2.

The key distinction between a DDE algorithm and a typical DE [35] is on thepopulation updating mechanism. In a typical DE, all the update actions of thepopulation are performed at the end of the generation of which the implementationis referred to as static updating mechanism. Alternatively, the updating mechanismof DDE algorithm is carried out in a dynamic way: each parent individual would bereplaced by its offspring if the offspring has obtained a better cost function value

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than its parent. Thus, DDE algorithm can respond the progress of population statusimmediately and yield faster convergence speed than the typical DE. Based on theconvergent characteristic of DDE algorithm, we are able to reduce the numbers ofcost function evaluation and reconstruct the microwave image efficiently.

4. Numerical results

We illustrate the performance of the proposed inversion algorithm and its sensitivityto random noise in the scattered field. Let us consider a perfectly conducting cylinderburied in a lossless half-space (�1 ¼ �2 ¼ 0). The permittivity in each region ischaracterized by "1 ¼ "0 and "2 ¼ 2:56 "0, respectively. The frequency of the incidentwave is chosen to be 1GHz with incident angles �1 equal to �45

�

, 0�

and 45�

,respectively. The wavelength 0 is 0.3m. For each incident wave, 8 measurements aremade at the points equally separated on a semi-circle with the radius of 3m inregion 1. There are 24 measurement points in each simulation. We set the generationto be 500, c1 and c2 to be 1.3 and 2.8, respectively. Number of unknowns is set to be 7(i.e. Nþ 1¼ 7). The search range for the unknown coefficient of the shape function ischosen to be from 0.07 to 0.15, B0 and C0 is chosen to be 0.0 to 0.05. Our purpose isto reconstruct the shape of the object by using the scattered field at different incidentangles.

Three examples are investigated for the inverse scattering of the proposedstructure by using the APSO and DDE. In the first example, the shape function ischosen to be Fð�Þ ¼ ð0:1þ 0:04 cos 2�Þm. The reconstructed shape function of APSOand DDE with the best population member is plotted in Figure 5 and the value ofcost function versus the number of function calls for example 1 is shown in Figure 6.The reconstructed shape error by APSO and DDE are 4.93% and 5.8%, respectively.It is clear that the reconstructed result is good. The algorithm is executed on an

–0.2 –0.1 0 0.1 0.2–0.2

–0.1

0

0.1

0.2

X-coordinate, m

Y-c

oord

inat

e, m

ExactAPSO=500thDDE=500th

Figure 5. The reconstructed shape of the cylinder for example 1.

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Intel Core 2 CPU at 2.66GHz with 2 GB of RAM. The CPU time for reconstructingthe shape is about half hour for PSO and it takes about 40min for DDE.

In the second example, the shape function is chosen to be Fð�Þ ¼ ð0:1þ0:02 sin 3�Þ m. This example shows that the proposed scheme can reconstruct morecomplicated scatterer whose shape function has three concavities. The reconstructedshape function with the best population member is plotted in Figure 7 and thevalue of cost function versus the number of function calls for example 2 is shown inFigure 8. The reconstructed shape error by APSO and DDE are 9.25% and 9.12%,respectively. There is a little discrepancy in the bottom of the shape since TE wavesare incident from the top of the shape.

100 101 102 103

100

101

Number of function calls

Val

ue o

f co

st fu

nctio

n

TE-PSOTE-DDE

Figure 6. The value of cost function versus the number of function calls for example 1.

–0.2 –0.1 0 0.1 0.2–0.2

–0.1

0

0.1

0.2

X-coordinate, m

Y-c

oord

inat

e, m

ExactAPSO=500thDDE=500th

Figure 7. The reconstructed shape of the cylinder for example 2.

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In the third example, the shape function is chosen to be Fð�Þ ¼ ð0:1þ0:03 cos 2� þ 0:025 sin 2� þ 0:015 sin 3�Þm. The reconstructed shape function withthe best population member is plotted in Figure 9 and the value of cost functionversus the number of function calls for example 3 is shown in Figure 10.The reconstructed shape error by APSO and DDE are 5.59% and 5.35%,respectively. It is seen that the error comes from the bottom of the shape, but westill can obtain good results by APSO and DDE, respectively.

To investigate the effects of noise, we add to each complex scattered field aquantity bþ cj, where b and c are independent random numbers having auniform distribution over 0 to the noise level times the R.M.S value of the

100 101 102 103

100

101

Number of function calls

Val

ue o

f co

st fu

nctio

n

TE-PSOTE-DDE

Figure 8. The value of cost function versus the number of function calls for example 2.

–0.2 –0.1 0 0.1 0.2–0.2

–0.1

0

0.1

0.2

X-coordinate, m

Y-c

oord

inat

e, m

ExactAPSO=500thDDE=500th

Figure 9. The reconstructed shape of the cylinder for example 3.

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scattered field. Normalized standard deviations of 10�4, 10�3, 10�2 and 10�1,respectively, are used in the simulations. Figure 11 shows the reconstructed resultsunder the condition that the scattered H fields to mimic measurement datacontaminated by the noise. The discrepancy of shape function of the reconstructedshape is shown in Figure 11. It could be observed that good reconstruction has beenobtained for shape of the perfectly conducting cylinder when the relative noiselevel is below 10�2.

5. Conclusion

The problem of shape reconstruction of perfectly conducting cylinder has beeninvestigated by applying the APSO and DDE techniques. The inverse problem isreformulated into an optimization one. Numerical results show that the APSO and

100 101 102 103

100

101

Number of function calls

Val

ue o

f cos

t fu

nctio

n

TM-PSOTM-DDE

Figure 10. The value of cost function versus the number of function calls for example 3.

10–4 10–3 10–2 10–10

10

20

30

40

50

Noise level

Rel

ativ

e er

ror

(%)

DDEAPSO

Figure 11. Shape error as functions by APSO and DDE, respectively.

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DDE have almost the same reconstructed accuracy when the same number of

iterations is applied. However, both techniques are reliable and give accurate shape

reconstruction even when the initial guess is far from the exact one.

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