Research Article Inverse Transient Radiative Analysis in Two …downloads.hindawi.com/journals/mpe/2015/680823.pdf · 2019. 7. 31. · laser machining, and medical diagnostics, to

Research ArticleInverse Transient Radiative Analysis in Two-DimensionalTurbid Media by Particle Swarm Optimizations

Yatao Ren Hong Qi Qin Chen and Liming Ruan

School of Energy Science and Engineering Harbin Institute of Technology Harbin 150001 China

Correspondence should be addressed to Hong Qi qihonghiteducn and Liming Ruan ruanlmhiteducn

Received 30 September 2014 Revised 4 December 2014 Accepted 9 December 2014

Academic Editor Subhash Chandra Mishra

Copyright copy 2015 Yatao Ren et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Three intelligent optimization algorithms namely the standard Particle SwarmOptimization (PSO) the Stochastic Particle SwarmOptimization (SPSO) and the hybrid Differential Evolution-Particle Swarm Optimization (DE-PSO) were applied to solve theinverse transient radiation problem in two-dimensional (2D) turbid media irradiated by the short pulse laser The time-resolvedradiative intensity signals simulated by finite volume method (FVM) were served as input for the inverse analysis The sensitivitiesof the time-resolved radiation signals to the geometric parameters of the circular inclusions were also investigated To illustratethe performance of these PSO algorithms the optical properties the size and location of the circular inclusion were retrievedrespectively The results showed that all these radiative parameters could be estimated accurately even with noisy data Comparedwith the PSO algorithm with inertia weights the SPSO and DE-PSO algorithm were demonstrated to be more effective and robustwhich had the potential to be implemented in 2D transient radiative transfer inverse problems

1 Introduction

The problem of transient radiative transfer in participatingmedia has attracted increasing interest over the last twodecades due to the emergence of ultrashort pulse laser and itsapplication to the picosecond level or higher accuracy levelof time-resolved techniques [1] Lasers with pulse durationsof pico- to femtoseconds have been widely utilized in theareas of laser light-scattering flame diagnose the predictionof the temperature distribution in combustion chambersatmospheric science remote sensing nondestructive testinglaser machining and medical diagnostics to name a few [2ndash4] In recent years the medical applications of short pulselaser such as the laser tissue welding [5 6] the laser tissueablation [7 8] the optical tomography in medical imaging[9] and the photodynamic therapy [10] have drawn moreand more attentions all over the world In particular theshort pulse laser of near-infrared spectrum is extensivelyused in noninvasive techniques to retrieve the character-istics of the semitransparent media such as the biologicaltissues and turbid media [11ndash14] Theoretically speaking theresearch emphases of noninvasive reconstruction technology

are focused on using the model-based iterative imagingreconstruction techniques which employ a precise and effi-cient numerical forward model based on solving completetransient radiative transfer equation (TRTE) and then estab-lish a proper inverse algorithm to retrieve the optical propertyparameters or internal geometry of the medium by using themeasured time-resolved transmittance and reflectance sig-nals Compared with the traditional measurement methodsits biggest advantages are noninvasive in situ measurementcontaining a lot of time-resolved information about internalcomposition and characteristic of the participating media

For solving the inverse transient radiation problems inthe noninvasive reconstruction the first step is to solve thedirect TRTE model accurately and efficiently Nowadaysseveral numerical strategies have been developed to solvethe forward model including the discrete ordinate method(DOM) [15ndash18] the finite volume method (FVM) [18ndash21]the integral equation (IE) method [22] the finite-elementmethod FEM [23 24] and the Monte Carlo method [25]Each of these methods has its own relative advantages anddisadvantages and none of them is superior to others inall aspects However many numerical complications may

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 680823 15 pageshttpdxdoiorg1011552015680823

2 Mathematical Problems in Engineering

arise from incompatibilities between the radiative transportmodel and the discrete formulation employed for the fluiddynamics and other heat transfer modes The FVM for RTEavoids many of these complications and has the advantageof greater compatibility with existing finite volume basedheat transfer software Besides the FVM also could simplifythe problems of complex geometries using the unstructuredmeshes Through decades of development FVM has becomea sophisticated technology in mechanics and thermal anal-ysis There have been several investigations focused on theinteraction of ultrafast radiation with participating mediausing FVM [19 20] In these studies both the homogeneousand inhomogeneous media have been considered Howevermost of the studies of inverse radiation problems concentrateon one-dimensional (1D) cases and there are very fewresearches about multidimensional problems especially theinverse transient radiative problems

To solve the inverse problems a wide variety of inversetechniques have been successfully employed in the inverseradiation analyses which can be grouped into two categoriesthat is the traditional algorithm based on gradient and theintelligent optimizations [26] To date many gradient-basedtechniques have been employed in the inverse radiationanalysis such as the constrained least-squares method [27]the conjugate gradient (CG) method [28] and Levenberg-Marquardt method [29] However all these traditional meth-ods depend on the initial value or the derivatives and gradi-ents which are difficult to be solved accurately by numericalsimulation in some cases Furthermore if the initial value isnot chosen properly the solution may be infeasible in theoriginal domain or in a worse case not be convergent Inaddition these conventional methods using derivatives aregenerally restricted to the conditions of continuity sensitivityconvexity monotonicity and nonlinearity of both objectivefunctions and constraints [30] Compared with traditionalgradient-based methods the intelligent optimization algo-rithms have a couple of outstanding characteristics thatis the ill-posed inverse problem could also be solved thederivative of the objective function is not necessary and the apriori information is not needed A typical characteristic ofthe intelligent optimization methods is that they can solvethe global optimal problem reliably and obtain high qualityglobal solutions with enough computational time especiallyfor higher problem dimensions [31] This is primarily due tothe fact that the random search technique is used to modifylocally examined regions which gives the algorithm a globaloptimization capability The Particle Swarm Optimization(PSO) as an intelligent swarm optimization first introducedin 1995 by Eberhart and Kennedy [32] has been studiedextensively by many researchers in recent years It is apotential heuristic bionic evolutionary algorithm originallyinspired by the locking behavior of birds Generally speak-ing PSO is characterized to be simple in concept easy toimplement and computationally efficient Early studies havefound the implementation of PSO to be effective and robust insolving problems featuring nonlinearly nondifferentiabilityand high dimensionality Many modifications have beenmade to improve the convergence rate of the original PSOalgorithmMore recently our research grouphas done several

studies on the PSO algorithm to solve the one-dimensionalinverse radiation problems [33ndash37] However to the best ofour knowledge few researches are presented to solve themultidimensional inverse transient radiation problems by theintelligent optimization methods

The objective of present work is to apply the PSO-basedalgorithms to solve the inverse transient radiation problemin two-dimensional (2D) participating medium The opticalproperties the size and location of the circular inclusionwere retrieved respectively The remainder of the paper isorganized as follows The detailed mathematical formulationand computational steps of the direct model are describedin Section 2 The theoretical models of the standard PSOthe Stochastic PSO and the hybrid Differential Evolutionand Particle Swarm Optimization (DE-PSO) algorithms aredescribed in Section 3 The solving model for direct problemand the inverse transient radiation analysis by the PSO-basedalgorithms are examined in Section 4 In Section 5 the mainconclusions and perspectives are provided

2 The Direct Model

In the present work the thermal effect caused by the incidentlaser was ignored and only the transient radiative transferwas considered in the numerical model Thus the transientradiative transfer equation can be expressed as [38]

1

119888

120597119868 (119904Ω 119905)

120597119905+120597119868 (119904Ω 119905)

120597119904

= minus120573119868 (119904Ω 119905) +120590119904

4120587int

4120587

119868 (119904Ω1015840

119905)Φ (Ω1015840

Ω) dΩ1015840

(1)

where 119904 is the length along the directionΩ radiative intensity119868 is the function of position 119904 direction Ω and time 119905 120573 120581

119886

and 120590119904denotes extinction absorption and scattering coeffi-

cients respectively and 120573 = 120581119886+ 120590

119904 Φ(Ω1015840Ω) represents the

scattering phase function between incoming directionΩ1015840 andscattering direction Ω Ω1015840 is the solid angle in the directionΩ1015840 In the 2D transient radiative transfer problem caused

by collimated irradiation the radiative intensity within themedium is separated into two parts [35] (a) 119868

119888 the remnant

of the collimated beam after partial extinction by absorptionand scattering along its path (b) 119868

119889 the diffuse part which

is the result of emission from the boundaries emission fromthe medium inside and the radiation scattered away fromthe collimated irradiationThus the radiative intensity in themedia can be written as

119868 = 119868119888+ 119868

119889 (2)

For the square pulse laser the intensity within theparticipating media can be expressed as [35]

119868119888= 119902in exp [minusint

1199040

0

(120581119886+ 120590

119904) 119889119904]

sdot [119867 (119888119905 minus 1199040) minus 119867 (119888119905 minus 119888119905

119901minus 119904

0)] 120575 (Ω minusΩ

0)

(3)

Mathematical Problems in Engineering 3

where119867 is Heaviside function [35] 120575 denotes Dirac function[35] 119902in is the heat flux of incident laser at boundary 119909 = 0Ω0is the incident direction and 119904

0represents the geometric

distance along the directionΩ0

For the 2D participating media the discrete equationcould be obtained using the FVMmodel

1

119888

120597119868119889(119904Ω 119905)

120597119905+120597119868

119889(119904Ω 119905)

120597119904

= minus120573119868119889(119904Ω 119905) + 119878

119888(119904Ω 119905) + 119878

119889(119904Ω 119905)

= minus120573119868119889(119904Ω 119905) + 119878

119905(119904Ω 119905)

(4)

where

119878119888=120590119904

4120587int

4120587

119868119888(119904Ω 119905) Φ (Ω

1015840

Ω) dΩ1015840

119878119889=120590119904

4120587int

4120587

119868119889(119904Ω 119905) Φ (Ω

1015840

Ω) dΩ1015840

(5)

In (4) 119878119905= 119878

119888+ 119878

119889denotes the total source term The

source terms 119878119888and 119878

119889result from the collimated radiation

119868119888and the diffused radiation 119868

119889 respectively For the diffuse

reflection boundary the boundary condition is given as [35]

119868 (119904119908Ω 119905) = 120576

119908119868119887(119904119908 119905)

+1 minus 120576

119908

120587int

nsdotΩ1015840gt0119868 (119904

119908Ω

1015840

119905)10038161003816100381610038161003816n sdotΩ101584010038161003816100381610038161003816 dΩ

1015840

(n sdotΩ) lt 0

(6)

where 119868(119904119908Ω 119905) and 119868(119904

119908Ω

1015840

119905) are the outgoing and theincoming intensities at the boundary respectively n is theoutward unit normal vector at the boundary 120576

119908is the emis-

sivity of the boundary Substituting the length-dimensionaltime term 119905

lowast

= 119888119905 into (4) we can get

120597119868119889(119904Ω 119905

lowast

)

120597119905lowast

+120597119868

119889(119904Ω 119905

lowast

)

120597119904=minus120573119868

119889(119904Ω 119905

lowast

) + 119878119905(119904Ω 119905

lowast

)

(7)

The total source term 119878119905can be depicted as

119878119905(119904Ω 119905

lowast

) =120590119904

4120587int

4120587

119868119888(119904Ω 119905

lowast

)Φ (Ω0Ω) dΩ1015840

+120590119904

4120587int

4120587

119868119889(119904Ω 119905

lowast

)Φ (Ω1015840

Ω) dΩ1015840

(8)

where Ω0denotes the direction of incident radiation Using

the fully implicit scheme the diffusion intensity 119868119889for the

control volume 119875 in the direction s119898 at time 119905lowast can be writtenas

119868119898

119889119875(119905lowast

) =

1003816100381610038161003816119863119898

119909

1003816100381610038161003816 119868119898

119889119909119906(119905lowast

) 119891119910+10038161003816100381610038161003816119863119898

119910

10038161003816100381610038161003816119868119898

119889119910119906(119905lowast

) 119891119909+ 119878

119898

119875

1003816100381610038161003816119863119898

119909

1003816100381610038161003816 +10038161003816100381610038161003816119863119898

119910

10038161003816100381610038161003816+ (120573Δ119909ΔΩ

119898119861) 119891

119909119891119910

119878119898

119875= Δ119881ΔΩ

119898

119878119898

119905119901119891119909119891119910+119862Δ119881ΔΩ

119898

119861119868119898

119889119875(119905lowast

minus Δ119905lowast

) 119891119909119891119910

119878119898

119905119875= 119878

119898

119888+ 119878

119898

119889=120590119904

4120587119868119888Φ(Ω

0Ω) +

120590119904

4120587119868119898

119889119875Φ(Ω

119898

Ω) ΔΩ119898

119863119898

119909= int

ΔΩ119898

(n sdot s119898) dΩ(9)

where 119888 represents the velocity of light Δ119905lowast is the time step119863119898 represents the directionweight119861 and119862 are set asΔ119905lowast(1+

120573Δ119905lowast

) and 1(1+120573Δ119905lowast) respectively 119868119898119889119909119906

and 119868119898119889119910119906

denote the119898-direction radiative intensities of node 119875 at the upstreamboundaries of control volume which are along axis 119909 andaxis 119910 respectivelyThe time-domain thermal signals that isthe transmittance signal 120588

119879and reflectance signal 120588

119877 can be

expressed as

120588119879(119897119909 119910 119905

lowast

)

=1

119902in[2120587int

120583gt0

119868119889(119897119909 119910 120583 119905

lowast

) d120583 + 119902119888(119897119909 119910Ω

0 119905lowast

)]

120588119877(0 119910 119905

lowast

) = minus2120587

119902inint

120583lt0

119868119889(0 119910 120583 119905

lowast

) d120583

(10)

where 120583 is the direction cosine and 119902119888denotes the direct

transmission flux of the collimated light The FVM waschosen to solve the equation of TRT For the sake of simplicitythe details of FVM are available in [20] and are not repeatedhere

3 The Inverse Model

Commonly there are three ways to obtain the internal infor-mation of media that is the continuous wave method thetime-domain method and the frequency-domain method[39] All these methods have the similar procedure in theexperimental studies and the numerical simulations Takingthe time-domain method as an example the reconstructionscheme consists of three major parts [40] (1) a forwardmodel that predicts the detector signals based on the solutionof the transient radiative transfer equation (2) an objectivefunction that provides criterion of the differences betweenthe detected and the predicted data (3) the reconstructiontechnique which can minimize the objective function toget new guesses of the estimated parameters such as theoptical properties or the geometrical parameters Based onthe new guesses of the optical properties a new forward


calculation is performed to get the corresponding detectorpredictions The reconstruction process is completed whenthe value of the objective function is less than a preset valueor the number of iterations exceeds the maximum numberof iterations Foremost the forward model must be solvedprecisely enough so that the measurement data obtainedby detectors could be simulated correctly Consequentlythe forward model based on the complete transport-theoryTRTE should be utilized [40] The details of the inverseoptimization algorithms are shown as follows

31 Theoretical Model of Basic PSO Algorithm The PSOalgorithmwas first introduced by Eberhart and Kennedy [32]inspired by social behavior simulations of flocking birds Theflocking population is called a swarm and the individuals arecalled the particles The basic idea of PSO can be describedsimply as follows Each particle in a swarm represents apotential solution which changes its position and updates itsvelocity based on its own and its neighborsrsquo experience Thevelocity and position of the 119894th particle at generation 119905 + 1could be expressed as [32]

V119894(119905 + 1) = 119908V

119894(119905) + 119888

1sdot 119903

1sdot [P

119894(119905) minus X

119894(119905)]

+ 1198882sdot 119903

2sdot [P

119892(119905) minus X

119894(119905)]

(11)

X119894(119905 + 1) = X

119894(119905) + V

119894(119905 + 1) (12)

where 1198881and 119888

2represent two positive constants called

acceleration coefficients 1199031and 119903

2are uniformly distributed

random numbers in the interval [0 1] 119908 is the inertiaweight factor which is used to control the impact of theprevious velocities on the current velocity In the presentwork 119908 is defined as 119908 = 10 minus 06 times 119905119873 where119873 represents the maximum generation number X

119894(119905) =

[1198831198941(119905) 119883

1198942(119905) sdot sdot sdot 119883in(119905)]

T denotes the present location ofparticle 119894 which represents a potential solution V

119894(119905) =

[1198811198941(119905) 119881

1198942(119905) sdot sdot sdot 119881in(119905)]

T is the present velocity of particle 119894which is based on its own and its neighborsrsquo flying experienceP119894(119905) = [119875

1198941(119905) 119875

1198942(119905) 119875in(119905)]

T is the ldquolocal bestrdquo in the119905th generation of the swarm The global best position of theswarm is expressed as P

119892

32 Theoretical Model of DE-PSO Algorithm In order toavoid premature convergence of the standard PSO and ensurethe overall convergence a considerable number of modifiedPSO algorithms were proposed in recent years like theStochastic PSO (SPSO) the Multiphase PSO (MPPSO) theQuantum-Behaved PSO (QPSO) the hybrid Ant ColonyOptimization and PSO (ACO-PSO) and so forth [33 36 41]All these improved PSO-based algorithms are proved to beadaptive and robust parameter-searching techniques In thepresent work the SPSO and the hybrid Differential Evolutionand PSO (DE-PSO) are utilized in the inverse procedure [42]The DE-PSO is the combination of two stochastic algorithmsand is expected to accelerate the convergence process byintroducing the DEA which has strong global searchingability to the PSOThe detailed model description of SPSO is

The object function

Newly generated vector

x2

x1

lowast

lowast

Vectors from generation i

Xr3

Xr2 Tk

d[Xr2(t) minus Xr3

(t)]

Xr1(t)

Figure 1 The schematic diagram of vector-generation process forDEA

available in [41] and will not be repeated hereThe theoreticalmodel of DE-PSO is described as follows

The differential evolution algorithm (DEA) is a novelparallel direct and stochastic searching method which wasproposed to solve the continuous domain problems in 1995[43] The basic idea of DEA is to use the differential of twoindividuals in the current generation namely vectors 119903

2and

1199033 to combinewith a third vector 119903

1to generate a newparticle

which is called the trial vector Then the objective functionwas calculated to decide whether the trial vector is to replacevector 119896 or not The trial vector T is generated according to[43]

T119896(119905) = X

1199031(119905) + 119889 [X

1199032(119905) minus X

1199033(119905)] (13)

where 119889 is a real and constant factor which controls theamplification of the differential variation [X

1199032(119905)minusX

1199033(119905)] and

in this paper it was set as 05 The integers 1199031 119903

2 and 119903

3are

chosen randomly from the interval [1 119899] and are differentfrom running index 119894 Other parameters are defined as sameas those of the PSO Figure 1 illustrates the vector-generationprocess defined by (13)

In the process of DE-PSO the differential evolutionoperator is introduced to the position updating equationwhich can be expressed as [42]

X119894(119905 + 1) = X

119894(119905) + V

119894(119905 + 1) + 120573 (120576DE minus 119865min) (14)

where 120573 = 119889[X1199032(119905) minus X

1199033(119905)] which is named as differential

evolution operator 119865min represents the minimum value ofobjective function in the present generation 120576DE denotes apreset constant value which satisfies 120576DE le 119865min

The main function of the third term on the right side of(14) is to avoid local convergence However the differential


evolution operation may lead to instability of the algorithmso only when the objective function of the new particle issmaller than that of the last generation (14) is applied as theposition update equation Otherwise the position updatesaccording to (12) The implementation of the approach forsolving the inverse transient radiation problem by DE-PSOalgorithm can be carried out according to the followingroutine

Step 1 Initialize the positionX119894(119905) and the speedV

119894(119905) of each

particle and calculate the objective function Afterwards theglobal best position P

119892and every local best position P

119894(119905) can

be determined

Step 2 Check if the value of the objective function is smallerthan the preset parameter 120576 or the maximum number ofiterations is reached If so go to Step 6 otherwise go to thenext step

Step 3 Update the velocity and position of each particleaccording to (11) and (14) respectively Each velocity andposition component is bounded by the maximum and mini-mumvalues towhich each particle in the space is constrainedThe range of the particle position is decided by the physicalsituation for example the single scattering albedo can only beset in the range of [0 1]Then calculate the objective functionof each particle and update the global beat position and localbest positions

Step 4 Calculate the objective function of each newly gener-ated particle If the value of the objective function is smallerthan that of the corresponding particle in the last generationthen the new particle can be maintained Otherwise regen-erate the particle according to (12)

Step 5 Update the generation from 119905 to 119905+1 and go to Step 2

Step 6 Output the global best position and its correspondingvalue of objective function

To illustrate the difference between PSO SPSO and DE-PSO algorithms more clearly the flowcharts of the threealgorithms are shown in Figure 2

33 Inverse Analysis Procedure The transient radiativeinverse problems for estimating the internal radiative proper-ties of the 2D participating media are solved by minimizingthe objective function which can be defined as

119865 (a) = 1

2

4

sum

119895=1

int

1199051199042

1199051199041

[120588119895est (a) minus 120588119895mea (a)]

2

d119905 (15)

where 119895 represents the different wall of the 2D media120588119895mea(a) denotes the measured time-resolved transmittanceor reflectance signals which will be simulated by the forwardmodel using the FVM and 120588

119895est(a) is the estimated time-resolved transmittance and reflectance for an estimatedvector a = (119886

0 119886

1 119886

119873)T Moreover 119905

1199041and 119905

1199042are start

and end point of the sampling span respectively

To demonstrate the effects of measurement errors on thepredicted terms the random errors were considered In thepresent paper the simulated measured signals with randomerrors were obtained by adding normally distributed errors toexact reflectance and transmittance as follows

120588mea = 120588exa + rand ( ) sdot 120590 (16)

where 120588mea and 120588exa represent the measured reflectance ortransmittance signals with and without noisy data respec-tively and rand( ) denotes a normal distributed randomvariable with zero mean and unit standard deviation Thestandard deviation of the measured values is denoted by 120590which is defined as

120590 =120588exa sdot 1205742576

(17)

where 120574 is themeasured error and 2576 arises from the factthat 99 of a normally distributed population is containedwithin plusmn2567 standard deviation of the mean value Theabsorption and scattering coefficients or the inclusion loca-tions were estimated by minimizing the objective functionusing SPSO and DE-PSO algorithms The inverse algorithmswere adopted for minimization of the objective functionwhich is also called the fitness function The stop criterionof the inverse algorithms is that the number of iterationsexceeds themaximumnumber of iterations or the best fitnessfunction is less than a specified small value

4 Results and Discussion

41HomogeneousMedia Consider a 2Dhomogeneous semi-transparent medium with length 10m times 10m whose leftside exposes to a collimated square pulse laser beam in the119909-direction with pulse width 119888119905

119901= 01m The absorption

coefficient 120581119886and the scattering coefficient 120590

119904are set as

002mminus1 and 998mminus1 respectively The emissions of themedium and its boundaries are neglected The media isassumed to be anisotropic scattering with black boundariesand the phase function is defined as

Φ(Ω1015840

Ω) = 1 + 1198860(120583120583

1015840

+ 1205781205781015840

+ 1205851205851015840

) (18)

where 1198860= 1 119886

0= 0 and 119886

0= minus1 represent the back-

ward scattering isotropic scattering and forward scatteringrespectively 120583 120578 and 120585 and 1205831015840 1205781015840 and 1205851015840 denote the directioncosines of the incoming direction Ω and scattering directionΩ1015840 respectively The control volumes were set as119873

119909times 119873

119910=

500times500 and solid angles were divided into119873120579times119873

120601= 20times10

in the FVM The total observed time span 119888119905 was 10m Asshown in Figure 3 the results of FVM model are comparedwith the solutions of Discrete Ordinate Method (DOM) [16]and Least Square Finite Element Method (LSFEM) [44] Itcan be seen that the results calculated by FVMapproximationin the present study are in good agreement with those in[16 44] which means the FVMmodel in this paper is provedto be valid


Initialization

Output results

Yes

No

Calculate the object function

Yes

Calculate velocity and position by (11) and (12)

No

t = t + 1

t gt Nt

Update Pi and Pg

Fobj (Pg) lt 120576

(a)

Initialization

Output results

Yes

No


Yes

Calculate velocity and position by

No

Select the random particle j

No

i = j

(11) and (12) with w = 0

Pg = Pj

Pg = P998400g

or t gt Nt

t = t + 1

Fobj (Pg) lt 120576

P998400g = arg min Fobj (Pi)Fobj(P998400

g) Pg = arg min Fobj (Pg)

(b)

Initialization

Update velocity and position by (11) and (14)

Output results

Yes

No


Yes

No

No

Yes

Update position by (12)

t = t + 1

t gt Nt

Update Pi and Pg

Fobj (Pg) lt 120576Calculate Fobj [X(t + 1)]

Ft+1obj le Ft

obj

(c)

Figure 2 The flowcharts of (a) PSO (b) SPSO and (c) DE-PSO algorithms


0 20 40 60 80 100

Tran

smitt

ance

DOM 2002 Sakami et al FEM 2007 An et al

FVM

120573ct

a0 = 10

a0 = 0

a0 = minus101E minus 3

1E minus 4

1E minus 5

(a)

0 20 40 60 80 100

001

01

1

Refle

ctan

ce

DOM 2002 Sakami et al LS-FEM 2007 An et al

FVM

120573ct

a0 = 10a0 = 0

a0 = minus10

1E minus 3

1E minus 4

1E minus 5

(b)

Figure 3 Validation of the FVM solution for transient radiative transfer of a 2D homogeneous rectangle medium (a) Transmittance and (b)reflectance

Tumor

Liver

z2

z1

(120581a0 120590s0)

(120581a1 120590s1)

Figure 4 The simplification of actual models

42 Nonhomogeneous Media with Circular Inclusions Theestimation in this paper is expected to be applied in thenondestructive detection of humanor small animals Figure 4shows the model of a rat liver with tumor which is simplifiedto a rectangle medium with a circular inclusion To illustratethe influence of the inclusions to the radiative signals fourcases are investigated in this section that is two circular inclu-sions with different radiuses location and optical propertiesin a rectangular medium The media size is 10m times 10mwhich exposed to a pulse laser with pulse width 119888119905

119901= 01m

The absorption coefficient and scattering coefficient of the

2D medium are set as 1205811198860

= 20mminus1 and 1205901199040

= 30mminus1The geometric locations of the two inclusions are identifiedby three parameters namely 119911

1 119911

2 and 119903 among which

1199111and 119911

2represent the coordinate values of the inclusion

center and 119903 denotes the radius of the circular inclusionsTable 1 shows the control parameters of the four cases Itis worthy noticing that the four cases are designed on theprinciple that compared to Case 1 the other three casesonly change one set of parameters that is the size locationor optical properties for both inclusions For the case ofclarity the diagrams of the four cases are shown in Figure 5


Table 1 The control parameters of the four cases

Parameters Case 1 Case 2 Case 3 Case 4

Size [m] 1199031

01 02 01 01

1199032

02 01 02 02

Location [m] (11991111 119911

21) (03 045) (03 045) (07 065) (03 045)

(11991112 119911

22) (075 065) (075 065) (035 025) (075 065)

Optical properties [mminus1] (1205811198861 120590

1199041) (10 75) (10 75) (10 75) (40 55)

(1205811198862 120590

1199042) (02 98) (02 98) (02 98) (15 90)

120581a2 = 02mminus1

120590s2 = 98mminus1

120581a0 = 20mminus1

120590s0 = 30mminus1

120581a1 = 10mminus1

120590s1 = 75mminus1

z22

z21

z11 z12

(a)

120581a0 = 20mminus1

120590s0 = 30mminus1 120581a2 = 02mminus1

120590s2 = 98mminus1

120581a1 = 10mminus1

120590s1 = 75mminus1

z22

z21

z11 z12

(b)

120581a0 = 20mminus1

120590s0 = 30mminus1

120581a2 = 02mminus1

120590s2 = 98mminus1

120581a1 = 10mminus1

120590s1 = 75mminus1

z22

z21

z11 z12

(c)

120581a2 = 15mminus1

120590s2 = 90mminus1

120581a0 = 20mminus1

120590s0 = 30mminus1

120581a1 = 40mminus1

120590s1 = 55mminus1

z22

z21

z11 z12

(d)

Figure 5 The physical model contains two inclusions

The control volumes are set as 119873119909times 119873

119910= 100 times 100

and the time step is taken as Δ(119888119905) = 001m The time-resolved reflectance and transmittance signals are shown inFigures 6(a) and 6(b) It is apparent that the size locationand properties of the inclusions have a significant influenceon the radiative signals which can be concluded that theseparameters can be retrieved theoretically All the cases were

implemented using FORTRAN code and the developedprogram was executed on an Intel Xeon E5-2670 PC

43 Reconstruction of the Inclusion Parameters431 Estimation of the Optical Parameters To test the recon-struction ability of the three swarm intelligence optimizationalgorithms two cases for estimating the optical parameters


0 1 2 3 4 5

Refle

ctan

ce

ct (m)

1E minus 4

1E minus 3

1E minus 5

1E minus 6

1E minus 7

01

001

Case 1

Case 2

Case 3

Case 4

(a)

0 1 2 3 4 5

001

Tran

smitt

ance

ct (m)

1E minus 4

1E minus 3

1E minus 5

1E minus 6

1E minus 7

Case 1

Case 2

Case 3

Case 4

(b)

Figure 6 The (a) reflectance and (b) transmittance signals in the function of ct

Point-irradiated

Face-irradiated

120590s1)(120581a1

(120581a0 120590s0)

(120581a0 120590s0)

Figure 7 The incident types of pulse laser

were considered For the laser incidence two different con-ditions were included (see Figure 7) (1) point-irradiated (PI)condition which means one single laser beam irradiates onespot of the media wall (2) face-irradiated (FI) conditionwhichmeans the laser beams incident on thewhole boundarywall In this section the two cases were proceeded under bothtwo laser incident conditions In Case 1 the inclusion was inthe center of the medium which means 119911

1= 119911

2= 05m and

119903 = 025m The responses of detectors D1 D2 D3 and D4which are shown in Figure 8(a) were simulated by the for-ward model The properties of the inclusion were estimatedand the inverse analysis was taken by using the PSO SPSOand DE-PSO algorithms Meanwhile the optical propertiesof the 2D medium were taken as 120581

1198860= 020mminus1 and

1205901199040= 030mminus1 respectively The true values of the inclusion

properties were 1205811198861= 10mminus1 and 120590

1199041= 90mminus1 respectively

The searching intervals were set as [0 10mminus1] The estimatedresults are listed in Table 2 In Case 2 two inclusions insidethe medium were considered as shown in Figure 8(b) Theoptical properties of both inclusions were estimated by PSOSPSO and DE-PSO In this case the optical properties ofthe medium were taken as 120581

1198860= 020mminus1 and 120590

1199040=

Table 2 The retrieving results of the optical properties for Case 1using PSO SPSO and DE-PSO without measurement error

Algorithm ConditionTrue values or inversion

results [mminus1]Relative errors

[]1205811198861= 10 120590

1199041= 90 120581

11988611205901199041

PSO PI 10185 91372 1846 1524

FI 09964 89671 0356 0366

SPSO PI 09880 88960 1200 1155

FI 09979 90151 0206 0167

DE-PSO PI 10098 89138 0983 0958

FI 10008 89956 0077 0049

030mminus1 respectively The true value of the inclusions whichneeded to be estimated was (120581

1198861 120590

1199041) = (10mminus1

90mminus1

)

and (1205811198862 120590

1199042) = (45mminus1

55mminus1

) respectively The sizeand location of the two inclusions were (119911

11 119911

21 1199031) =

(03 03 02)m and (11991112 119911

22 1199032) = (07 06 015)m The

retrieving results of optical parameters are shown in Tables3 and 4

It can be seen from Table 2 that 1205811198861

and 1205901199041

can beretrieved accurately under both PI and FI conditions byPSO SPSO and DE-PSO when there is only one inclusionFurthermore it also demonstrates that the retrieving resultsunder FI condition are more accurate than those under PIcondition the reason of which is that under FI conditionthe reflectance and transmittance signals are stronger andare carrying more information available Meanwhile theDE-PSO algorithm shows its priority in reconstructing theinclusion properties Under both conditions the relativeerrors of retrieval results are smaller than those of the SPSOand PSO Table 3 demonstrates the retrieving results of theoptical properties for Case 2 of different generations Asshown in Table 4 the results of SPSO and DE-PSO are


D4

D1

D3

D2

120581a0 = 02mminus1

120590s0 = 03mminus1

120581a1 = 10mminus1

120590s1 = 90mminus1

(a)

D4

D1D2

D3

120581a0 = 02mminus1

120590s0 = 03mminus1

120581a2 = 45mminus1

120590s2 = 55mminus1

120581a1 = 10mminus1

120590s1 = 90mminus1

A

B

(b)

Figure 8 The schematic of pulse laser irradiation and the measurement position of time-domain signals of the two test cases

Table 3 The retrieving results of the optical properties for Case 2 using PSO SPSO and DE-PSO of different generations

Generations True value [mminus1] Algorithm Retrieving results [mminus1] Relative errors []

10 (1205811198861 120590

1199041) = (10 90)

(1205811198862 120590

1199042) = (45 55)

PSO (09750 93216) (24962 35738)(74025 59205) (645009 76456)

SPSO (09803 89818) (19718 02021)(54861 03347) (219131 939149)

DE-PSO (09844 88901) (15635 12207)(33674 37704) (251700 314477)

50 (1205811198861 120590

1199041) = (10 90)

(1205811198862 120590

1199042) = (45 55)

PSO (09790 90331) (21025 03679)(64496 75222) (433244 367678)

SPSO (09995 89996) (00478 00040)(35513 46093) (210814 161946)

DE-PSO (10021 90036) (02071 00397)(33049 40275) (265584 267718)

100 (1205811198861 120590

1199041) = (10 90)

(1205811198862 120590

1199042) = (45 55)

PSO (09790 90331) (21025 03679)(6449 75222) (433244 367678)

SPSO (09998 89995) (00234 00054)(35119 48832) (219588 112147)

DE-PSO (10007 90042) (00696 00472)(36868 48677) (180717 114957)

Table 4 The retrieving results of the optical properties for Case 2 using PSO SPSO and DE-PSO without measurement error

Algorithm Time [s] True values [mminus1] Retrieving results [mminus1] Relative errors []

PSO 25476 (1205811198861 120590

1199041) = (10 90) (09985 90019) (01486 00207)

(1205811198862 120590

1199042) = (45 55) (69803 75019) (55118 36398)

SPSO 32775 (1205811198861 120590

1199041) = (10 90) (10000 89996) (00034 00046)

(1205811198862 120590

1199042) = (45 55) (44444 54556) (12349 08077)

DE-PSO 49554 (1205811198861 120590

1199041) = (10 90) (10005 90036) (00544 00397)

(1205811198862 120590

1199042) = (45 55) (45248 54454) (05508 09925)


0 20 40 60 80 100 120

Obj

ect f

unct

ion

Generation

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

PSO SPSO

DE-PSO

Figure 9 The objective function of PSO DE-PSO and SPSO inCase 2

more accurate than those of the PSO Furthermore underFI condition for Case 2 the optical properties of bothinclusions can be estimated accurately without measurementerrors and with the fact that the estimated relative errors ofresults for inclusion B (see Figure 8) are larger than those ofinclusion A However the calculation time required of theDE-PSO is more than that of the SPSO and PSO Thus itcan be concluded that SPSO and DE-PSO can retrieved theoptical properties in a 2Dmedium accurately but with longercalculation time The value of the objective function of PSODE-PSO and SPSO with two inclusions is shown in Figure 9It can be seen that the PSO fall into local convergence quicklyand stop searching In the earlier stage of the inverse processthe objective function of SPSO declines faster than that ofDE-PSO However in the later period of the inverse processthe SPSO almost stops searching and the objective functionof DE-PSO continues declining which leads to the result thatthe retrieving results of DE-PSO aremore accurate than thoseof the SPSO

432 Estimation of Geometrical Parameters In the inverseprocedure the long sampling span will reduce the computa-tional efficiency Hence it is of great significance to select theproper sampling spanThus the sensitivity of radiative signalto the inverse parameters needs to be analyzedThe sensitivitycoefficient is one of themost important characteristic param-eters in the sensitivity analysis which is the first derivativeof the radiative signals to a certain inverse parameter Thesensitivity coefficient is defined as

119878119898119894(120588

119895) =

120597120588119895

120597119898119894

100381610038161003816100381610038161003816100381610038161003816119898119894=1198980

=

120588119895(119898

0+ 119898

0Δ) minus 120588

119895(119898

0minus 119898

0Δ)

21198980Δ

(19)

where 119898119894denotes the independent variable which stands for

1199111 119911

2 and 119903 Δ represents a tiny change 120588

119895is the radiative

signals of each boundaryThe sensitivities of the inclusion location (119911

1 119911

2) and size

119903 for the signals D1ndashD4 are shown in Figure 10 The param-eters are set the same as Case 1 described in Section 431 Itdemonstrates that the sensitivity coefficient is relatively largerin the span of 119888119905 isin [0 3m] So this span can be chosen asthe sampling span for inverse analysis The same results areexpected in other cases

In many conditions the shapes and optical properties ofinclusions in the media are known in the actual applicationcases of reconstructing the mediumrsquos inside information Forexample a tumor or cyst in the biological tissue is spherical orspherical-like and the defects in the specimen such as the airpore are mainly the shape of a cube sphere or ellipsoid Inaddition the optical properties of human and animals tissuesin vitro or in vivo are studied thoroughly [45] Consequentlythe geometrical feature and optical properties of inclusionscould be used as a priori information in these cases whichwill reduce the complexity of the reconstruction techniqueand improve the quality and efficiency of the reconstructionresults Therefore on the condition that the inclusion shapeand the optical properties are known the inverse problemturns into the retrieval of the size and location of theinclusions

In this section the inclusion location (1199111 119911

2) and size

119903 of cases (1199111= 119911

2= 025m 119903 = 025m) (119911

1= 119911

2=

05m 119903 = 025m) and (1199111= 119911

2= 075m 119903 = 025m)

were estimated simultaneously The size of the 2D mediumwas set as 10m times 10m The absorption coefficient 120581

1198860and

scattering coefficient 1205901199040

of the medium were 02mminus1 and03mminus1 respectively There was one circular inclusion in the2D medium with absorption coefficient 120581

1198861of 10mminus1 and

scattering coefficient 1205901199041of 90mminus1 but the inclusion size and

location were unknown which needed to be retrievedThe FIincidence type and the data of the detectors 1 2 3 and 4 wereused for inverse analysis by using the PSO SPSO and DE-PSO algorithms withmeasurement errorsThe search span of1199111 119911

2 and 119903was set as [0 1]mThe results are shown inTables

5 and 6 Figure 11 shows the comparison of the exact andreconstructed reflectance signal when 119911

1= 119911

2= 119903 = 025m

with 10 measurement errorIt can be seen that the size and location can be esti-

mated accurately using PSO SPSO and DE-PSO withoutmeasurement errors under FI condition Furthermore theresults of DE-PSO are more accurate than those of theSPSO and PSO The relative errors are reasonable even with10 measurement error However it is worth noting thatthe relative errors of DE-PSO with 10 measurement errorare bigger than those of the SPSO which means the SPSOalgorithm is more robust than the DE-PSO

5 ConclusionsIn the present study the standard PSO SPSO and DE-PSOalgorithms were applied to estimate the size location andoptical parameters of the circular inclusions in a 2D rectangu-lar medium The conclusion was obtained that the retrieving


Table 5 The inverse results of size and location of circular inclusion using PSO SPSO and DE-PSO without measurement error under FIcondition

Algorithm True values or inverse results [m] Relative errors []1199111

1199112

119903 1199111

1199112

119903

True value 02500 02500 02500 mdash mdash mdashPSO 02446 02457 02479 2163 1712 0837

SPSO 02499 02500 02500 0027 0007 0017

DE-PSO 02500 02500 02500 0015 0005 0007


SPSO 0500 0500 02499 0042 0002 0033

DE-PSO 04999 0500 02499 0019 0006 0021


SPSO 07497 07500 02500 0041 0005 0105

DE-PSO 07498 07500 02498 0026 0005 0067

0 1 2 3 4 5

000

002

ct (m)

minus002

minus004

minus006

minus008

minus010

r = 025m

D1D2

D3D4

z1 = 05m

z2 = 05m

Δz1 = 0005

S z1

(120588)

(a)

0 1 2 3 4 5

0000

0003

minus0003

minus0006

D1D2

D3D4

ct (m)

r = 025m

z1 = 05m

z2 = 05m

Δz2 = 0005

S z2

(120588)

(b)

0 1 2 3 4 5

000

001

002

003

004

005

006

007

ct (m)

minus001

D1D2

D3D4

r = 025m

Δr = 0005

z1 = 05m

z2 = 05m

S r(120588)

(c)

Figure 10 The sensitivity of the radiation signals for the geometric parameters of the inclusion


Table 6 The inverse results of size and location of circular inclusion using PSO SPSO and DE-PSO with 10 measurement error under FIcondition


1199112

119903 1199111

1199112

119903


SPSO 02509 02506 02492 0356 0243 0339

DE-PSO 02340 02412 02452 6395 3514 1934


SPSO 05006 05267 02453 0120 5331 1894

DE-PSO 05150 05302 02386 2996 6045 4549


SPSO 07356 07681 02265 1926 2419 9418

DE-PSO 07237 07786 02324 3508 3821 7041

0 1 2 3 4 5

001

Results of PSOResults of SPSOResults of DE-PSO

Refle

ctan

ce

ct (m)

z1 = 025 z2 = 025 r = 025

1E minus 3

1E minus 4

1E minus 5

1E minus 6

Figure 11 The comparison of the exact and reconstructedreflectance signals

results of absorption coefficient and scattering coefficientunder FI condition are more accurate than those under PIcondition the reason of which is that under FI conditionthe reflectance and transmittance signals are stronger andare carrying more information available In addition theDE-PSO algorithm shows its priority in reconstructing theinclusion properties When only one inclusion is consideredunder both laser incident conditions the resultsrsquo relativeerrors of DE-PSO are smaller than those of the SPSOHowever when estimating the size and location of inclusionmedia the SPSO is more robust than DE-PSO Furthermorecompared with the performance of standard PSO both theSPSO and DE-PSO are proved to be more accurate androbust which have the potential to be applied in the field of2D inverse transient radiative problems

Nomenclature

a The vector of estimated properties119888 The speed of light ms1198881 1198882 The acceleration coefficients of PSO

119889 The control variable of DEA119865 The objective function119868 The radiation intensity W(m sdot sr)119899 The number of the population119873119909 119873

119910 The number of grids

119873120579 119873

120601 The number of polar angles and azimuthal angles

P119892(119905) The global best position discovered by all particles

at generation 119905P119894(119905) The local best position of particle 119894 discovered at

generation 119905 or earlier119903 The radius of the inclusions1199031 1199032 The uniformly distributed random numbers

119878 The sensitivity coefficient119905 Time or generation in PSO algorithm119905119901 The incident pulse width s

1199051199041 1199051199042 The start and end point of sampling span

Δ119905 The time step s119879119894(119905) The trial vector of DEA

V119894(119905) The velocity array of the 119894th particle at the 119905th

generation in PSO119908 The inertia weight coefficientX119894(119905) The position array of the 119894th particle at the 119905th

generation in PSO1199111 119911

2 The coordinate values of the inclusionsrsquo center

Greeks Symbols

120573 The extinction coefficient mminus1

120576 The tolerance for minimizing the objective function120576rel The relative error Φ The scattering phase function120574 The measured error120581119886 Absorption coefficient mminus1

120583 The direction cosine


120588mea The measured time-resolved reflectance signalswith noisy data

120588exa The measured time-resolved reflectance signalswithout noisy data

120588est The estimated time-resolved reflectance signals120590119904 The scattering coefficient mminus1

Ω The directionΩ1015840 The solid angle in the directionΩ1015840

Subscripts

119886 The average relative error of the reflectance signals119892 The global best position in PSO119894 The searching index119901 The pulse widthrel The relative error

Superscript

119879 The inverse of the matrixlowast Dimensionless

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The support of this work by the National Natural ScienceFoundation of China (no 51476043) the Major NationalScientific Instruments and Equipment Development SpecialFoundation of China (no 51327803) and the TechnologicalInnovation Talent Research Special Foundation of Harbin(no 2014RFQXJ047) is gratefully acknowledged A veryspecial acknowledgement is made to the editors and refereeswho make important comments to improve this paper

References

[1] H Tan L Ruan and T W Tong ldquoTemperature response inabsorbing isotropic scattering medium caused by laser pulserdquoInternational Journal of Heat and Mass Transfer vol 43 no 2pp 311ndash320 2000

[2] Z Guo and K Kim ldquoUltrafast-laser-radiation transfer in het-erogeneous tissues with the discrete-ordinatesmethodrdquoAppliedOptics vol 42 no 16 pp 2897ndash2905 2003

[3] W An L M Ruan and H Qi ldquoInverse radiation problem inone-dimensional slab by time-resolved reflected and transmit-ted signalsrdquo Journal of Quantitative Spectroscopy and RadiativeTransfer vol 107 no 1 pp 47ndash60 2007

[4] Y Yuan H L Yi Y Shuai B Liu and H P Tan ldquoInverseproblem for aerosol particle size distribution using SPSO asso-ciated with multi-lognormal distribution modelrdquo AtmosphericEnvironment vol 45 no 28 pp 4892ndash4897 2011

[5] K Kim and Z Guo ldquoUltrafast radiation heat transfer in lasertissue welding and solderingrdquo Numerical Heat Transfer Part AApplications vol 46 no 1 pp 23ndash40 2004

[6] P Rath S C Mishra P Mahanta U K Saha and K MitraldquoDiscrete transfermethod applied to transient radiative transfer

problems in participating mediumrdquo Numerical Heat TransferPart A Applications vol 44 no 2 pp 183ndash197 2003

[7] F H Loesel F P Fisher H Suhan and J F Bille ldquoNon-thermal ablation of neutral tissue with femto-second laserpulsesrdquo Applied Physics B vol 66 no 1 pp 12ndash18 1998

[8] J Jiao and Z Guo ldquoModeling of ultrashort pulsed laser ablationin water and biological tissues in cylindrical coordinatesrdquoApplied Physics B Lasers and Optics vol 103 no 1 pp 195ndash2052011

[9] B A Brooksby H Dehghani B W Poque and K D PaulsenldquoNear-infrared (NIR) tomography breast image reconstructionwith a priori structural information from MRI algorithmdevelopment for reconstructing heterogeneitiesrdquo IEEE Journalon Selected Topics in Quantum Electronics vol 9 no 2 pp 199ndash209 2003

[10] D E J G J Dolmans D Fukumura and R K Jain ldquoPhotody-namic therapy for cancerrdquo Nature Reviews Cancer vol 3 no 5pp 380ndash387 2003

[11] A D Klose ldquoThe forward and inverse problem in tissue opticsbased on the radiative transfer equation a brief reviewrdquo Journalof Quantitative Spectroscopy and Radiative Transfer vol 111 no11 pp 1852ndash1853 2010

[12] A Charette J Boulanger and H K Kim ldquoAn overview onrecent radiation transport algorithm development for opticaltomography imagingrdquo Journal of Quantitative Spectroscopy andRadiative Transfer vol 109 no 17-18 pp 2743ndash2766 2008

[13] J Boulanger and A Charette ldquoNumerical developments forshort-pulsed near infra-red laser spectroscopy Part II inversetreatmentrdquo Journal of Quantitative Spectroscopy and RadiativeTransfer vol 91 no 3 pp 297ndash318 2005

[14] M Akamatsu and Z Guo ldquoTransient prediction of radiationresponse in a 3-d scattering-absorbing medium subjected to acollimated short square pulse trainrdquo Numerical Heat TransferPart A Applications vol 63 no 5 pp 327ndash346 2013

[15] Z Guo and S Kumar ldquoThree-dimensional discrete ordinatesmethod in transient radiative transferrdquo Journal ofThermophysicsand Heat Transfer vol 16 no 3 pp 289ndash296 2002

[16] M Sakami K Mitra and P-F Hsu ldquoAnalysis of light pulsetransport through two-dimensional scattering and absorbingmediardquo Journal of Quantitative Spectroscopy and RadiativeTransfer vol 73 no 2-5 pp 169ndash179 2002

[17] L H Liu L M Ruan and H P Tan ldquoOn the discrete ordinatesmethod for radiative heat transfer in anisotropically scatteringmediardquo International Journal of Heat andMass Transfer vol 45no 15 pp 3259ndash3262 2002

[18] B Hunter and Z Guo ldquoComparison of the discrete-ordinatesmethod and the finite-volume method for steady-state andultrafast radiative transfer analysis in cylindrical coordinatesrdquoNumerical Heat Transfer Part B Fundamentals vol 59 no 5pp 339ndash359 2011

[19] J C Chai ldquoOne-dimensional transient radiation heat trans-fer modeling using a finite-volume methodrdquo Numerical HeatTransfer Part B Fundamentals vol 44 no 2 pp 187ndash208 2003

[20] S C Mishra P Chugh P Kumar and K Mitra ldquoDevelopmentand comparison of the DTM the DOM and the FVM formula-tions for the short-pulse laser transport through a participatingmediumrdquo International Journal of Heat and Mass Transfer vol49 no 11-12 pp 1820ndash1832 2006

[21] J C Chai P F Hsu and Y C Lam ldquoThree-dimensionaltransient radiative transfer modeling using the finite-volumemethodrdquo Journal of Quantitative Spectroscopy and RadiativeTransfer vol 86 no 3 pp 299ndash313 2004


[22] C-Y Wu ldquoPropagation of scattered radiation in a participatingplanar medium with pulse irradiationrdquo Journal of QuantitativeSpectroscopy and Radiative Transfer vol 64 no 5 pp 537ndash5482000

[23] L M Ruan W An H P Tan and H Qi ldquoLeast-squares finite-element method of multidimensional radiative heat transfer inabsorbing and scattering mediardquo Numerical Heat Transfer PartA Applications vol 51 no 7 pp 657ndash677 2007

[24] L-M Ruan W An and H-P Tan ldquoTransient radiative trans-fer of ultra-short pulse in two-dimensional inhomogeneousmediardquo Journal of EngineeringThermophysics vol 28 no 6 pp998ndash1000 2007

[25] P-F Hsu ldquoEffects ofmultiple scattering and reflective boundaryon the transient radiative transfer processrdquo International Jour-nal of Thermal Sciences vol 40 no 6 pp 539ndash549 2001

[26] B Zhang H Qi Y T Ren S C Sun and L M RuanldquoApplication of homogenous continuous Ant ColonyOptimiza-tion algorithm to inverse problem of one-dimensional coupledradiation and conduction heat transferrdquo International Journal ofHeat and Mass Transfer vol 66 pp 507ndash516 2013

[27] K Kamiuto ldquoA constrained least-squares method for limitedinverse scattering problemsrdquo Journal of Quantitative Spec-troscopy and Radiative Transfer vol 40 no 1 pp 47ndash50 1988

[28] H Y Li ldquoInverse radiation problem in two-dimensional rectan-gular mediardquo Journal of Thermophysics and Heat Transfer vol11 no 4 pp 556ndash561 1997

[29] M P Menguc and S Manickavasagam ldquoInverse radiationproblem in axisymmetric cylindrical scattering mediardquo Journalof Thermophysics and Heat Transfer vol 7 no 3 pp 479ndash4861993

[30] C Elegbede ldquoStructural reliability assessment based on parti-cles swarm optimizationrdquo Structural Safety vol 27 no 2 pp171ndash186 2005

[31] S Hajimirza G El Hitti A Heltzel and J Howell ldquoUsinginverse analysis to find optimum nano-scale radiative surfacepatterns to enhance solar cell performancerdquo International Jour-nal of Thermal Sciences vol 62 pp 93ndash102 2012

[32] R C Eberhart and J Kennedy ldquoA new optimizer using particleswarm theoryrdquo in Proceedings of the 6th International Sympo-sium on Micro Machine and Human Science vol 1 pp 39ndash431995

[33] H Qi L M Ruan M Shi W An and H P Tan ldquoApplication ofmulti-phase particle swarm optimization technique to inverseradiation problemrdquo Journal of Quantitative Spectroscopy andRadiative Transfer vol 109 no 3 pp 476ndash493 2008

[34] H Qi L Ruan S Wang M Shi and H Zhao ldquoApplication ofmulti-phase particle swarm optimization technique to retrievethe particle size distributionrdquo Chinese Optics Letters vol 6 no5 pp 346ndash349 2008

[35] H Qi D L Wang S G Wang and L M Ruan ldquoInverse tran-sient radiation analysis in one-dimensional non-homogeneousparticipating slabs using particle swarm optimization algo-rithmsrdquo Journal of Quantitative Spectroscopy and RadiativeTransfer vol 112 no 15 pp 2507ndash2519 2011

[36] B Zhang H Qi S C Sun L M Ruan and H P Tan ldquoA novelhybrid ant colony optimization and particle swarm optimiza-tion algorithm for inverse problems of coupled radiative andconductive heat transferrdquoThermal Science pp 23ndash23 2014

[37] D L Wang H Qi and L M Ruan ldquoRetrieve propertiesof participating media by different spans of radiative signalsusing the SPSO algorithmrdquo Inverse Problems in Science andEngineering vol 21 no 5 pp 888ndash915 2013

[38] M F Modest Radiative Heat Transfer Academic PressWaltham Mass USA 2003

[39] J C Hebden S R Arridge andD T Delpy ldquoOptical imaging inmedicine I Experimental techniquesrdquo Physics in Medicine andBiology vol 42 no 5 pp 825ndash840 1997

[40] A H Hielscher A D Klose and K M Hanson ldquoGradient-based iterative image reconstruction scheme for time-resolvedoptical tomographyrdquo IEEE Transactions on Medical Imagingvol 18 no 3 pp 262ndash271 1999

[41] H Qi L M Ruan H C Zhang Y M Wang and H P TanldquoInverse radiation analysis of a one-dimensional participatingslab by stochastic particle swarm optimizer algorithmrdquo Inter-national Journal of Thermal Sciences vol 46 no 7 pp 649ndash6612007

[42] J Liao and Q T Shen ldquoParticle swarm optimization based ondifferential evolutionrdquo Science Technology and Engineering vol7 no 8 pp 1628ndash1630 2007

[43] R Storn and K Price Differential Evolution-A Simple andEfficient Adaptive Scheme for Global Optimization over Continu-ous Spaces International Computer Science Institute BerkeleyCalif USA 1995

[44] W An L M Ruan H P Tan and H Qi ldquoLeast-squares finiteelement analysis for transient radiative transfer in absorbingand scattering mediardquo Journal of Heat Transfer vol 128 no 5pp 499ndash503 2006

[45] W F Cheong S A Prahl and A J Welch ldquoA review ofthe optical properties of biological tissuesrdquo IEEE Journal ofQuantum Electronics vol 26 no 12 pp 2166ndash2185 1990

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


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Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

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Applied MathematicsJournal of


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Algebra

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Stochastic AnalysisInternational Journal of


arise from incompatibilities between the radiative transportmodel and the discrete formulation employed for the fluiddynamics and other heat transfer modes The FVM for RTEavoids many of these complications and has the advantageof greater compatibility with existing finite volume basedheat transfer software Besides the FVM also could simplifythe problems of complex geometries using the unstructuredmeshes Through decades of development FVM has becomea sophisticated technology in mechanics and thermal anal-ysis There have been several investigations focused on theinteraction of ultrafast radiation with participating mediausing FVM [19 20] In these studies both the homogeneousand inhomogeneous media have been considered Howevermost of the studies of inverse radiation problems concentrateon one-dimensional (1D) cases and there are very fewresearches about multidimensional problems especially theinverse transient radiative problems

To solve the inverse problems a wide variety of inversetechniques have been successfully employed in the inverseradiation analyses which can be grouped into two categoriesthat is the traditional algorithm based on gradient and theintelligent optimizations [26] To date many gradient-basedtechniques have been employed in the inverse radiationanalysis such as the constrained least-squares method [27]the conjugate gradient (CG) method [28] and Levenberg-Marquardt method [29] However all these traditional meth-ods depend on the initial value or the derivatives and gradi-ents which are difficult to be solved accurately by numericalsimulation in some cases Furthermore if the initial value isnot chosen properly the solution may be infeasible in theoriginal domain or in a worse case not be convergent Inaddition these conventional methods using derivatives aregenerally restricted to the conditions of continuity sensitivityconvexity monotonicity and nonlinearity of both objectivefunctions and constraints [30] Compared with traditionalgradient-based methods the intelligent optimization algo-rithms have a couple of outstanding characteristics thatis the ill-posed inverse problem could also be solved thederivative of the objective function is not necessary and the apriori information is not needed A typical characteristic ofthe intelligent optimization methods is that they can solvethe global optimal problem reliably and obtain high qualityglobal solutions with enough computational time especiallyfor higher problem dimensions [31] This is primarily due tothe fact that the random search technique is used to modifylocally examined regions which gives the algorithm a globaloptimization capability The Particle Swarm Optimization(PSO) as an intelligent swarm optimization first introducedin 1995 by Eberhart and Kennedy [32] has been studiedextensively by many researchers in recent years It is apotential heuristic bionic evolutionary algorithm originallyinspired by the locking behavior of birds Generally speak-ing PSO is characterized to be simple in concept easy toimplement and computationally efficient Early studies havefound the implementation of PSO to be effective and robust insolving problems featuring nonlinearly nondifferentiabilityand high dimensionality Many modifications have beenmade to improve the convergence rate of the original PSOalgorithmMore recently our research grouphas done several

studies on the PSO algorithm to solve the one-dimensionalinverse radiation problems [33ndash37] However to the best ofour knowledge few researches are presented to solve themultidimensional inverse transient radiation problems by theintelligent optimization methods

The objective of present work is to apply the PSO-basedalgorithms to solve the inverse transient radiation problemin two-dimensional (2D) participating medium The opticalproperties the size and location of the circular inclusionwere retrieved respectively The remainder of the paper isorganized as follows The detailed mathematical formulationand computational steps of the direct model are describedin Section 2 The theoretical models of the standard PSOthe Stochastic PSO and the hybrid Differential Evolutionand Particle Swarm Optimization (DE-PSO) algorithms aredescribed in Section 3 The solving model for direct problemand the inverse transient radiation analysis by the PSO-basedalgorithms are examined in Section 4 In Section 5 the mainconclusions and perspectives are provided

2 The Direct Model

In the present work the thermal effect caused by the incidentlaser was ignored and only the transient radiative transferwas considered in the numerical model Thus the transientradiative transfer equation can be expressed as [38]

1

119888

120597119868 (119904Ω 119905)

120597119905+120597119868 (119904Ω 119905)

120597119904

= minus120573119868 (119904Ω 119905) +120590119904

4120587int

4120587

119868 (119904Ω1015840

119905)Φ (Ω1015840

Ω) dΩ1015840

(1)

where 119904 is the length along the directionΩ radiative intensity119868 is the function of position 119904 direction Ω and time 119905 120573 120581

119886

and 120590119904denotes extinction absorption and scattering coeffi-

cients respectively and 120573 = 120581119886+ 120590

119904 Φ(Ω1015840Ω) represents the

scattering phase function between incoming directionΩ1015840 andscattering direction Ω Ω1015840 is the solid angle in the directionΩ1015840 In the 2D transient radiative transfer problem caused

by collimated irradiation the radiative intensity within themedium is separated into two parts [35] (a) 119868

119888 the remnant

of the collimated beam after partial extinction by absorptionand scattering along its path (b) 119868

119889 the diffuse part which

is the result of emission from the boundaries emission fromthe medium inside and the radiation scattered away fromthe collimated irradiationThus the radiative intensity in themedia can be written as

119868 = 119868119888+ 119868

119889 (2)

For the square pulse laser the intensity within theparticipating media can be expressed as [35]

119868119888= 119902in exp [minusint

1199040

0

(120581119886+ 120590

119904) 119889119904]

sdot [119867 (119888119905 minus 1199040) minus 119867 (119888119905 minus 119888119905

119901minus 119904

0)] 120575 (Ω minusΩ

0)

(3)






1

119888

120597119868119889(119904Ω 119905)

120597119905+120597119868

119889(119904Ω 119905)

120597119904

= minus120573119868119889(119904Ω 119905) + 119878

119888(119904Ω 119905) + 119878

119889(119904Ω 119905)

= minus120573119868119889(119904Ω 119905) + 119878

119905(119904Ω 119905)

(4)

where

119878119888=120590119904

4120587int

4120587

119868119888(119904Ω 119905) Φ (Ω

1015840

Ω) dΩ1015840

119878119889=120590119904

4120587int

4120587

119868119889(119904Ω 119905) Φ (Ω

1015840

Ω) dΩ1015840

(5)

In (4) 119878119905= 119878

119888+ 119878







119868 (119904119908Ω 119905) = 120576

119908119868119887(119904119908 119905)

+1 minus 120576

119908

120587int

nsdotΩ1015840gt0119868 (119904

119908Ω

1015840

119905)10038161003816100381610038161003816n sdotΩ101584010038161003816100381610038161003816 dΩ

1015840

(n sdotΩ) lt 0

(6)

where 119868(119904119908Ω 119905) and 119868(119904

119908Ω

1015840


119908is the emis-


lowast

= 119888119905 into (4) we can get

120597119868119889(119904Ω 119905

lowast

)

120597119905lowast

+120597119868

119889(119904Ω 119905

lowast

)

120597119904=minus120573119868

119889(119904Ω 119905

lowast

) + 119878119905(119904Ω 119905

lowast

)

(7)


119878119905(119904Ω 119905

lowast

) =120590119904

4120587int

4120587

119868119888(119904Ω 119905

lowast

)Φ (Ω0Ω) dΩ1015840

+120590119904

4120587int

4120587

119868119889(119904Ω 119905

lowast

)Φ (Ω1015840

Ω) dΩ1015840

(8)




119868119898

119889119875(119905lowast

) =

1003816100381610038161003816119863119898

119909

1003816100381610038161003816 119868119898

119889119909119906(119905lowast

) 119891119910+10038161003816100381610038161003816119863119898

119910

10038161003816100381610038161003816119868119898

119889119910119906(119905lowast

) 119891119909+ 119878

119898

119875

1003816100381610038161003816119863119898

119909

1003816100381610038161003816 +10038161003816100381610038161003816119863119898

119910

10038161003816100381610038161003816+ (120573Δ119909ΔΩ

119898119861) 119891

119909119891119910

119878119898

119875= Δ119881ΔΩ

119898

119878119898

119905119901119891119909119891119910+119862Δ119881ΔΩ

119898

119861119868119898

119889119875(119905lowast


) 119891119909119891119910

119878119898

119905119875= 119878

119898

119888+ 119878

119898

119889=120590119904

4120587119868119888Φ(Ω

0Ω) +

120590119904

4120587119868119898

119889119875Φ(Ω

119898

Ω) ΔΩ119898

119863119898

119909= int

ΔΩ119898

(n sdot s119898) dΩ(9)


120573Δ119905lowast


and 119868119898119889119910119906



119877 can be

expressed as

120588119879(119897119909 119910 119905

lowast

)

=1

119902in[2120587int

120583gt0

119868119889(119897119909 119910 120583 119905

lowast

) d120583 + 119902119888(119897119909 119910Ω

0 119905lowast

)]

120588119877(0 119910 119905

lowast

) = minus2120587

119902inint

120583lt0

119868119889(0 119910 120583 119905

lowast

) d120583

(10)



3 The Inverse Model





V119894(119905 + 1) = 119908V

119894(119905) + 119888

1sdot 119903

1sdot [P

119894(119905) minus X

119894(119905)]

+ 1198882sdot 119903

2sdot [P

119892(119905) minus X

119894(119905)]

(11)

X119894(119905 + 1) = X

119894(119905) + V

119894(119905 + 1) (12)

where 1198881and 119888





119894(119905) =

[1198831198941(119905) 119883

1198942(119905) sdot sdot sdot 119883in(119905)]


119894(119905) =

[1198811198941(119905) 119881

1198942(119905) sdot sdot sdot 119881in(119905)]


1198941(119905) 119875

1198942(119905) 119875in(119905)]


119892


The object function


x2

x1

lowast

lowast


Xr3

Xr2 Tk

d[Xr2(t) minus Xr3

(t)]

Xr1(t)




2and




T119896(119905) = X

1199031(119905) + 119889 [X

1199032(119905) minus X

1199033(119905)] (13)


1199032(119905)minusX

1199033(119905)] and


2 and 119903

3are



X119894(119905 + 1) = X

119894(119905) + V

119894(119905 + 1) + 120573 (120576DE minus 119865min) (14)

where 120573 = 119889[X1199032(119905) minus X







119894(119905) of each



119894(119905) can

be determined








119865 (a) = 1

2

4

sum

119895=1

int

1199051199042

1199051199041

[120588119895est (a) minus 120588119895mea (a)]

2

d119905 (15)



0 119886

1 119886

119873)T Moreover 119905

1199041and 119905

1199042are start



120588mea = 120588exa + rand ( ) sdot 120590 (16)


120590 =120588exa sdot 1205742576

(17)






119904are set as


Φ(Ω1015840

Ω) = 1 + 1198860(120583120583

1015840

+ 1205781205781015840

+ 1205851205851015840

) (18)

where 1198860= 1 119886

0= 0 and 119886



119909times 119873

119910=


120601= 20times10



Initialization

Output results

Yes

No


Yes


No

t = t + 1

t gt Nt

Update Pi and Pg

Fobj (Pg) lt 120576

(a)

Initialization

Output results

Yes

No


Yes


No


No

i = j

(11) and (12) with w = 0

Pg = Pj

Pg = P998400g

or t gt Nt

t = t + 1

Fobj (Pg) lt 120576



(b)

Initialization


Output results

Yes

No


Yes

No

No

Yes


t = t + 1

t gt Nt

Update Pi and Pg


Ft+1obj le Ft

obj

(c)



0 20 40 60 80 100

Tran

smitt

ance


FVM

120573ct

a0 = 10

a0 = 0


1E minus 4

1E minus 5

(a)

0 20 40 60 80 100

001

01

1

Refle

ctan

ce


FVM

120573ct

a0 = 10a0 = 0

a0 = minus10

1E minus 3

1E minus 4

1E minus 5

(b)


Tumor

Liver

z2

z1

(120581a0 120590s0)

(120581a1 120590s1)



119901= 01m



= 20mminus1 and 1205901199040


1 119911


1199111and 119911






Size [m] 1199031

01 02 01 01

1199032

02 01 02 02

Location [m] (11991111 119911

21) (03 045) (03 045) (07 065) (03 045)

(11991112 119911

22) (075 065) (075 065) (035 025) (075 065)


1199041) (10 75) (10 75) (10 75) (40 55)

(1205811198862 120590

1199042) (02 98) (02 98) (02 98) (15 90)

120581a2 = 02mminus1

120590s2 = 98mminus1

120581a0 = 20mminus1

120590s0 = 30mminus1

120581a1 = 10mminus1

120590s1 = 75mminus1

z22

z21

z11 z12

(a)

120581a0 = 20mminus1

120590s0 = 30mminus1 120581a2 = 02mminus1

120590s2 = 98mminus1

120581a1 = 10mminus1

120590s1 = 75mminus1

z22

z21

z11 z12

(b)

120581a0 = 20mminus1

120590s0 = 30mminus1

120581a2 = 02mminus1

120590s2 = 98mminus1

120581a1 = 10mminus1

120590s1 = 75mminus1

z22

z21

z11 z12

(c)

120581a2 = 15mminus1

120590s2 = 90mminus1

120581a0 = 20mminus1

120590s0 = 30mminus1

120581a1 = 40mminus1

120590s1 = 55mminus1

z22

z21

z11 z12

(d)



119910= 100 times 100





0 1 2 3 4 5

Refle

ctan

ce

ct (m)

1E minus 4

1E minus 3

1E minus 5

1E minus 6

1E minus 7

01

001

Case 1

Case 2

Case 3

Case 4

(a)

0 1 2 3 4 5

001

Tran

smitt

ance

ct (m)

1E minus 4

1E minus 3

1E minus 5

1E minus 6

1E minus 7

Case 1

Case 2

Case 3

Case 4

(b)


Point-irradiated

Face-irradiated

120590s1)(120581a1

(120581a0 120590s0)

(120581a0 120590s0)



1= 119911

2= 05m and


1198860= 020mminus1 and





1198860= 020mminus1 and 120590

1199040=




[]1205811198861= 10 120590

1199041= 90 120581

11988611205901199041

PSO PI 10185 91372 1846 1524

FI 09964 89671 0356 0366

SPSO PI 09880 88960 1200 1155

FI 09979 90151 0206 0167

DE-PSO PI 10098 89138 0983 0958

FI 10008 89956 0077 0049


1198861 120590

1199041) = (10mminus1

90mminus1

)

and (1205811198862 120590

1199042) = (45mminus1

55mminus1


11 119911

21 1199031) =

(03 03 02)m and (11991112 119911

22 1199032) = (07 06 015)m The



and 1205901199041



D4

D1

D3

D2

120581a0 = 02mminus1

120590s0 = 03mminus1

120581a1 = 10mminus1

120590s1 = 90mminus1

(a)

D4

D1D2

D3

120581a0 = 02mminus1

120590s0 = 03mminus1

120581a2 = 45mminus1

120590s2 = 55mminus1

120581a1 = 10mminus1

120590s1 = 90mminus1

A

B

(b)




10 (1205811198861 120590

1199041) = (10 90)

(1205811198862 120590

1199042) = (45 55)

PSO (09750 93216) (24962 35738)(74025 59205) (645009 76456)

SPSO (09803 89818) (19718 02021)(54861 03347) (219131 939149)

DE-PSO (09844 88901) (15635 12207)(33674 37704) (251700 314477)

50 (1205811198861 120590

1199041) = (10 90)

(1205811198862 120590

1199042) = (45 55)

PSO (09790 90331) (21025 03679)(64496 75222) (433244 367678)

SPSO (09995 89996) (00478 00040)(35513 46093) (210814 161946)

DE-PSO (10021 90036) (02071 00397)(33049 40275) (265584 267718)

100 (1205811198861 120590

1199041) = (10 90)

(1205811198862 120590

1199042) = (45 55)

PSO (09790 90331) (21025 03679)(6449 75222) (433244 367678)

SPSO (09998 89995) (00234 00054)(35119 48832) (219588 112147)

DE-PSO (10007 90042) (00696 00472)(36868 48677) (180717 114957)



PSO 25476 (1205811198861 120590

1199041) = (10 90) (09985 90019) (01486 00207)

(1205811198862 120590

1199042) = (45 55) (69803 75019) (55118 36398)

SPSO 32775 (1205811198861 120590

1199041) = (10 90) (10000 89996) (00034 00046)

(1205811198862 120590

1199042) = (45 55) (44444 54556) (12349 08077)

DE-PSO 49554 (1205811198861 120590

1199041) = (10 90) (10005 90036) (00544 00397)

(1205811198862 120590

1199042) = (45 55) (45248 54454) (05508 09925)


0 20 40 60 80 100 120

Obj

ect f

unct

ion

Generation

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

PSO SPSO

DE-PSO




119878119898119894(120588

119895) =

120597120588119895

120597119898119894

100381610038161003816100381610038161003816100381610038161003816119898119894=1198980

=

120588119895(119898

0+ 119898

0Δ) minus 120588

119895(119898

0minus 119898

0Δ)

21198980Δ

(19)


1199111 119911




1 119911

2) and size




2) and size

119903 of cases (1199111= 119911

2= 025m 119903 = 025m) (119911

1= 119911

2=

05m 119903 = 025m) and (1199111= 119911

2= 075m 119903 = 025m)


1198860and








1= 119911

2= 119903 = 025m







1199112

119903 1199111

1199112

119903


SPSO 02499 02500 02500 0027 0007 0017

DE-PSO 02500 02500 02500 0015 0005 0007


SPSO 0500 0500 02499 0042 0002 0033

DE-PSO 04999 0500 02499 0019 0006 0021


SPSO 07497 07500 02500 0041 0005 0105

DE-PSO 07498 07500 02498 0026 0005 0067

0 1 2 3 4 5

000

002

ct (m)

minus002

minus004

minus006

minus008

minus010

r = 025m

D1D2

D3D4

z1 = 05m

z2 = 05m

Δz1 = 0005

S z1

(120588)

(a)

0 1 2 3 4 5

0000

0003

minus0003

minus0006

D1D2

D3D4

ct (m)

r = 025m

z1 = 05m

z2 = 05m

Δz2 = 0005

S z2

(120588)

(b)

0 1 2 3 4 5

000

001

002

003

004

005

006

007

ct (m)

minus001

D1D2

D3D4

r = 025m

Δr = 0005

z1 = 05m

z2 = 05m

S r(120588)

(c)





1199112

119903 1199111

1199112

119903


SPSO 02509 02506 02492 0356 0243 0339

DE-PSO 02340 02412 02452 6395 3514 1934


SPSO 05006 05267 02453 0120 5331 1894

DE-PSO 05150 05302 02386 2996 6045 4549


SPSO 07356 07681 02265 1926 2419 9418

DE-PSO 07237 07786 02324 3508 3821 7041

0 1 2 3 4 5

001


Refle

ctan

ce

ct (m)

z1 = 025 z2 = 025 r = 025

1E minus 3

1E minus 4

1E minus 5

1E minus 6



Nomenclature




119873120579 119873












Greeks Symbols









Subscripts


Superscript




Acknowledgments


References























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














1

119888

120597119868119889(119904Ω 119905)

120597119905+120597119868

119889(119904Ω 119905)

120597119904

= minus120573119868119889(119904Ω 119905) + 119878

119888(119904Ω 119905) + 119878

119889(119904Ω 119905)

= minus120573119868119889(119904Ω 119905) + 119878

119905(119904Ω 119905)

(4)

where

119878119888=120590119904

4120587int

4120587

119868119888(119904Ω 119905) Φ (Ω

1015840

Ω) dΩ1015840

119878119889=120590119904

4120587int

4120587

119868119889(119904Ω 119905) Φ (Ω

1015840

Ω) dΩ1015840

(5)

In (4) 119878119905= 119878

119888+ 119878







119868 (119904119908Ω 119905) = 120576

119908119868119887(119904119908 119905)

+1 minus 120576

119908

120587int

nsdotΩ1015840gt0119868 (119904

119908Ω

1015840

119905)10038161003816100381610038161003816n sdotΩ101584010038161003816100381610038161003816 dΩ

1015840

(n sdotΩ) lt 0

(6)

where 119868(119904119908Ω 119905) and 119868(119904

119908Ω

1015840


119908is the emis-


lowast

= 119888119905 into (4) we can get

120597119868119889(119904Ω 119905

lowast

)

120597119905lowast

+120597119868

119889(119904Ω 119905

lowast

)

120597119904=minus120573119868

119889(119904Ω 119905

lowast

) + 119878119905(119904Ω 119905

lowast

)

(7)


119878119905(119904Ω 119905

lowast

) =120590119904

4120587int

4120587

119868119888(119904Ω 119905

lowast

)Φ (Ω0Ω) dΩ1015840

+120590119904

4120587int

4120587

119868119889(119904Ω 119905

lowast

)Φ (Ω1015840

Ω) dΩ1015840

(8)




119868119898

119889119875(119905lowast

) =

1003816100381610038161003816119863119898

119909

1003816100381610038161003816 119868119898

119889119909119906(119905lowast

) 119891119910+10038161003816100381610038161003816119863119898

119910

10038161003816100381610038161003816119868119898

119889119910119906(119905lowast

) 119891119909+ 119878

119898

119875

1003816100381610038161003816119863119898

119909

1003816100381610038161003816 +10038161003816100381610038161003816119863119898

119910

10038161003816100381610038161003816+ (120573Δ119909ΔΩ

119898119861) 119891

119909119891119910

119878119898

119875= Δ119881ΔΩ

119898

119878119898

119905119901119891119909119891119910+119862Δ119881ΔΩ

119898

119861119868119898

119889119875(119905lowast


) 119891119909119891119910

119878119898

119905119875= 119878

119898

119888+ 119878

119898

119889=120590119904

4120587119868119888Φ(Ω

0Ω) +

120590119904

4120587119868119898

119889119875Φ(Ω

119898

Ω) ΔΩ119898

119863119898

119909= int

ΔΩ119898

(n sdot s119898) dΩ(9)


120573Δ119905lowast


and 119868119898119889119910119906



119877 can be

expressed as

120588119879(119897119909 119910 119905

lowast

)

=1

119902in[2120587int

120583gt0

119868119889(119897119909 119910 120583 119905

lowast

) d120583 + 119902119888(119897119909 119910Ω

0 119905lowast

)]

120588119877(0 119910 119905

lowast

) = minus2120587

119902inint

120583lt0

119868119889(0 119910 120583 119905

lowast

) d120583

(10)



3 The Inverse Model





V119894(119905 + 1) = 119908V

119894(119905) + 119888

1sdot 119903

1sdot [P

119894(119905) minus X

119894(119905)]

+ 1198882sdot 119903

2sdot [P

119892(119905) minus X

119894(119905)]

(11)

X119894(119905 + 1) = X

119894(119905) + V

119894(119905 + 1) (12)

where 1198881and 119888





119894(119905) =

[1198831198941(119905) 119883

1198942(119905) sdot sdot sdot 119883in(119905)]


119894(119905) =

[1198811198941(119905) 119881

1198942(119905) sdot sdot sdot 119881in(119905)]


1198941(119905) 119875

1198942(119905) 119875in(119905)]


119892


The object function


x2

x1

lowast

lowast


Xr3

Xr2 Tk

d[Xr2(t) minus Xr3

(t)]

Xr1(t)




2and




T119896(119905) = X

1199031(119905) + 119889 [X

1199032(119905) minus X

1199033(119905)] (13)


1199032(119905)minusX

1199033(119905)] and


2 and 119903

3are



X119894(119905 + 1) = X

119894(119905) + V

119894(119905 + 1) + 120573 (120576DE minus 119865min) (14)

where 120573 = 119889[X1199032(119905) minus X







119894(119905) of each



119894(119905) can

be determined








119865 (a) = 1

2

4

sum

119895=1

int

1199051199042

1199051199041

[120588119895est (a) minus 120588119895mea (a)]

2

d119905 (15)



0 119886

1 119886

119873)T Moreover 119905

1199041and 119905

1199042are start



120588mea = 120588exa + rand ( ) sdot 120590 (16)


120590 =120588exa sdot 1205742576

(17)






119904are set as


Φ(Ω1015840

Ω) = 1 + 1198860(120583120583

1015840

+ 1205781205781015840

+ 1205851205851015840

) (18)

where 1198860= 1 119886

0= 0 and 119886



119909times 119873

119910=


120601= 20times10



Initialization

Output results

Yes

No


Yes


No

t = t + 1

t gt Nt

Update Pi and Pg

Fobj (Pg) lt 120576

(a)

Initialization

Output results

Yes

No


Yes


No


No

i = j

(11) and (12) with w = 0

Pg = Pj

Pg = P998400g

or t gt Nt

t = t + 1

Fobj (Pg) lt 120576



(b)

Initialization


Output results

Yes

No


Yes

No

No

Yes


t = t + 1

t gt Nt

Update Pi and Pg


Ft+1obj le Ft

obj

(c)



0 20 40 60 80 100

Tran

smitt

ance


FVM

120573ct

a0 = 10

a0 = 0


1E minus 4

1E minus 5

(a)

0 20 40 60 80 100

001

01

1

Refle

ctan

ce


FVM

120573ct

a0 = 10a0 = 0

a0 = minus10

1E minus 3

1E minus 4

1E minus 5

(b)


Tumor

Liver

z2

z1

(120581a0 120590s0)

(120581a1 120590s1)



119901= 01m



= 20mminus1 and 1205901199040


1 119911


1199111and 119911






Size [m] 1199031

01 02 01 01

1199032

02 01 02 02

Location [m] (11991111 119911

21) (03 045) (03 045) (07 065) (03 045)

(11991112 119911

22) (075 065) (075 065) (035 025) (075 065)


1199041) (10 75) (10 75) (10 75) (40 55)

(1205811198862 120590

1199042) (02 98) (02 98) (02 98) (15 90)

120581a2 = 02mminus1

120590s2 = 98mminus1

120581a0 = 20mminus1

120590s0 = 30mminus1

120581a1 = 10mminus1

120590s1 = 75mminus1

z22

z21

z11 z12

(a)

120581a0 = 20mminus1

120590s0 = 30mminus1 120581a2 = 02mminus1

120590s2 = 98mminus1

120581a1 = 10mminus1

120590s1 = 75mminus1

z22

z21

z11 z12

(b)

120581a0 = 20mminus1

120590s0 = 30mminus1

120581a2 = 02mminus1

120590s2 = 98mminus1

120581a1 = 10mminus1

120590s1 = 75mminus1

z22

z21

z11 z12

(c)

120581a2 = 15mminus1

120590s2 = 90mminus1

120581a0 = 20mminus1

120590s0 = 30mminus1

120581a1 = 40mminus1

120590s1 = 55mminus1

z22

z21

z11 z12

(d)



119910= 100 times 100





0 1 2 3 4 5

Refle

ctan

ce

ct (m)

1E minus 4

1E minus 3

1E minus 5

1E minus 6

1E minus 7

01

001

Case 1

Case 2

Case 3

Case 4

(a)

0 1 2 3 4 5

001

Tran

smitt

ance

ct (m)

1E minus 4

1E minus 3

1E minus 5

1E minus 6

1E minus 7

Case 1

Case 2

Case 3

Case 4

(b)


Point-irradiated

Face-irradiated

120590s1)(120581a1

(120581a0 120590s0)

(120581a0 120590s0)



1= 119911

2= 05m and


1198860= 020mminus1 and





1198860= 020mminus1 and 120590

1199040=




[]1205811198861= 10 120590

1199041= 90 120581

11988611205901199041

PSO PI 10185 91372 1846 1524

FI 09964 89671 0356 0366

SPSO PI 09880 88960 1200 1155

FI 09979 90151 0206 0167

DE-PSO PI 10098 89138 0983 0958

FI 10008 89956 0077 0049


1198861 120590

1199041) = (10mminus1

90mminus1

)

and (1205811198862 120590

1199042) = (45mminus1

55mminus1


11 119911

21 1199031) =

(03 03 02)m and (11991112 119911

22 1199032) = (07 06 015)m The



and 1205901199041



D4

D1

D3

D2

120581a0 = 02mminus1

120590s0 = 03mminus1

120581a1 = 10mminus1

120590s1 = 90mminus1

(a)

D4

D1D2

D3

120581a0 = 02mminus1

120590s0 = 03mminus1

120581a2 = 45mminus1

120590s2 = 55mminus1

120581a1 = 10mminus1

120590s1 = 90mminus1

A

B

(b)




10 (1205811198861 120590

1199041) = (10 90)

(1205811198862 120590

1199042) = (45 55)

PSO (09750 93216) (24962 35738)(74025 59205) (645009 76456)

SPSO (09803 89818) (19718 02021)(54861 03347) (219131 939149)

DE-PSO (09844 88901) (15635 12207)(33674 37704) (251700 314477)

50 (1205811198861 120590

1199041) = (10 90)

(1205811198862 120590

1199042) = (45 55)

PSO (09790 90331) (21025 03679)(64496 75222) (433244 367678)

SPSO (09995 89996) (00478 00040)(35513 46093) (210814 161946)

DE-PSO (10021 90036) (02071 00397)(33049 40275) (265584 267718)

100 (1205811198861 120590

1199041) = (10 90)

(1205811198862 120590

1199042) = (45 55)

PSO (09790 90331) (21025 03679)(6449 75222) (433244 367678)

SPSO (09998 89995) (00234 00054)(35119 48832) (219588 112147)

DE-PSO (10007 90042) (00696 00472)(36868 48677) (180717 114957)



PSO 25476 (1205811198861 120590

1199041) = (10 90) (09985 90019) (01486 00207)

(1205811198862 120590

1199042) = (45 55) (69803 75019) (55118 36398)

SPSO 32775 (1205811198861 120590

1199041) = (10 90) (10000 89996) (00034 00046)

(1205811198862 120590

1199042) = (45 55) (44444 54556) (12349 08077)

DE-PSO 49554 (1205811198861 120590

1199041) = (10 90) (10005 90036) (00544 00397)

(1205811198862 120590

1199042) = (45 55) (45248 54454) (05508 09925)


0 20 40 60 80 100 120

Obj

ect f

unct

ion

Generation

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

PSO SPSO

DE-PSO




119878119898119894(120588

119895) =

120597120588119895

120597119898119894

100381610038161003816100381610038161003816100381610038161003816119898119894=1198980

=

120588119895(119898

0+ 119898

0Δ) minus 120588

119895(119898

0minus 119898

0Δ)

21198980Δ

(19)


1199111 119911




1 119911

2) and size




2) and size

119903 of cases (1199111= 119911

2= 025m 119903 = 025m) (119911

1= 119911

2=

05m 119903 = 025m) and (1199111= 119911

2= 075m 119903 = 025m)


1198860and








1= 119911

2= 119903 = 025m







1199112

119903 1199111

1199112

119903


SPSO 02499 02500 02500 0027 0007 0017

DE-PSO 02500 02500 02500 0015 0005 0007


SPSO 0500 0500 02499 0042 0002 0033

DE-PSO 04999 0500 02499 0019 0006 0021


SPSO 07497 07500 02500 0041 0005 0105

DE-PSO 07498 07500 02498 0026 0005 0067

0 1 2 3 4 5

000

002

ct (m)

minus002

minus004

minus006

minus008

minus010

r = 025m

D1D2

D3D4

z1 = 05m

z2 = 05m

Δz1 = 0005

S z1

(120588)

(a)

0 1 2 3 4 5

0000

0003

minus0003

minus0006

D1D2

D3D4

ct (m)

r = 025m

z1 = 05m

z2 = 05m

Δz2 = 0005

S z2

(120588)

(b)

0 1 2 3 4 5

000

001

002

003

004

005

006

007

ct (m)

minus001

D1D2

D3D4

r = 025m

Δr = 0005

z1 = 05m

z2 = 05m

S r(120588)

(c)





1199112

119903 1199111

1199112

119903


SPSO 02509 02506 02492 0356 0243 0339

DE-PSO 02340 02412 02452 6395 3514 1934


SPSO 05006 05267 02453 0120 5331 1894

DE-PSO 05150 05302 02386 2996 6045 4549


SPSO 07356 07681 02265 1926 2419 9418

DE-PSO 07237 07786 02324 3508 3821 7041

0 1 2 3 4 5

001


Refle

ctan

ce

ct (m)

z1 = 025 z2 = 025 r = 025

1E minus 3

1E minus 4

1E minus 5

1E minus 6



Nomenclature




119873120579 119873












Greeks Symbols









Subscripts


Superscript




Acknowledgments


References























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












V119894(119905 + 1) = 119908V

119894(119905) + 119888

1sdot 119903

1sdot [P

119894(119905) minus X

119894(119905)]

+ 1198882sdot 119903

2sdot [P

119892(119905) minus X

119894(119905)]

(11)

X119894(119905 + 1) = X

119894(119905) + V

119894(119905 + 1) (12)

where 1198881and 119888





119894(119905) =

[1198831198941(119905) 119883

1198942(119905) sdot sdot sdot 119883in(119905)]


119894(119905) =

[1198811198941(119905) 119881

1198942(119905) sdot sdot sdot 119881in(119905)]


1198941(119905) 119875

1198942(119905) 119875in(119905)]


119892


The object function


x2

x1

lowast

lowast


Xr3

Xr2 Tk

d[Xr2(t) minus Xr3

(t)]

Xr1(t)




2and




T119896(119905) = X

1199031(119905) + 119889 [X

1199032(119905) minus X

1199033(119905)] (13)


1199032(119905)minusX

1199033(119905)] and


2 and 119903

3are



X119894(119905 + 1) = X

119894(119905) + V

119894(119905 + 1) + 120573 (120576DE minus 119865min) (14)

where 120573 = 119889[X1199032(119905) minus X







119894(119905) of each



119894(119905) can

be determined








119865 (a) = 1

2

4

sum

119895=1

int

1199051199042

1199051199041

[120588119895est (a) minus 120588119895mea (a)]

2

d119905 (15)



0 119886

1 119886

119873)T Moreover 119905

1199041and 119905

1199042are start



120588mea = 120588exa + rand ( ) sdot 120590 (16)


120590 =120588exa sdot 1205742576

(17)






119904are set as


Φ(Ω1015840

Ω) = 1 + 1198860(120583120583

1015840

+ 1205781205781015840

+ 1205851205851015840

) (18)

where 1198860= 1 119886

0= 0 and 119886



119909times 119873

119910=


120601= 20times10



Initialization

Output results

Yes

No


Yes


No

t = t + 1

t gt Nt

Update Pi and Pg

Fobj (Pg) lt 120576

(a)

Initialization

Output results

Yes

No


Yes


No


No

i = j

(11) and (12) with w = 0

Pg = Pj

Pg = P998400g

or t gt Nt

t = t + 1

Fobj (Pg) lt 120576



(b)

Initialization


Output results

Yes

No


Yes

No

No

Yes


t = t + 1

t gt Nt

Update Pi and Pg


Ft+1obj le Ft

obj

(c)



0 20 40 60 80 100

Tran

smitt

ance


FVM

120573ct

a0 = 10

a0 = 0


1E minus 4

1E minus 5

(a)

0 20 40 60 80 100

001

01

1

Refle

ctan

ce


FVM

120573ct

a0 = 10a0 = 0

a0 = minus10

1E minus 3

1E minus 4

1E minus 5

(b)


Tumor

Liver

z2

z1

(120581a0 120590s0)

(120581a1 120590s1)



119901= 01m



= 20mminus1 and 1205901199040


1 119911


1199111and 119911






Size [m] 1199031

01 02 01 01

1199032

02 01 02 02

Location [m] (11991111 119911

21) (03 045) (03 045) (07 065) (03 045)

(11991112 119911

22) (075 065) (075 065) (035 025) (075 065)


1199041) (10 75) (10 75) (10 75) (40 55)

(1205811198862 120590

1199042) (02 98) (02 98) (02 98) (15 90)

120581a2 = 02mminus1

120590s2 = 98mminus1

120581a0 = 20mminus1

120590s0 = 30mminus1

120581a1 = 10mminus1

120590s1 = 75mminus1

z22

z21

z11 z12

(a)

120581a0 = 20mminus1

120590s0 = 30mminus1 120581a2 = 02mminus1

120590s2 = 98mminus1

120581a1 = 10mminus1

120590s1 = 75mminus1

z22

z21

z11 z12

(b)

120581a0 = 20mminus1

120590s0 = 30mminus1

120581a2 = 02mminus1

120590s2 = 98mminus1

120581a1 = 10mminus1

120590s1 = 75mminus1

z22

z21

z11 z12

(c)

120581a2 = 15mminus1

120590s2 = 90mminus1

120581a0 = 20mminus1

120590s0 = 30mminus1

120581a1 = 40mminus1

120590s1 = 55mminus1

z22

z21

z11 z12

(d)



119910= 100 times 100





0 1 2 3 4 5

Refle

ctan

ce

ct (m)

1E minus 4

1E minus 3

1E minus 5

1E minus 6

1E minus 7

01

001

Case 1

Case 2

Case 3

Case 4

(a)

0 1 2 3 4 5

001

Tran

smitt

ance

ct (m)

1E minus 4

1E minus 3

1E minus 5

1E minus 6

1E minus 7

Case 1

Case 2

Case 3

Case 4

(b)


Point-irradiated

Face-irradiated

120590s1)(120581a1

(120581a0 120590s0)

(120581a0 120590s0)



1= 119911

2= 05m and


1198860= 020mminus1 and





1198860= 020mminus1 and 120590

1199040=




[]1205811198861= 10 120590

1199041= 90 120581

11988611205901199041

PSO PI 10185 91372 1846 1524

FI 09964 89671 0356 0366

SPSO PI 09880 88960 1200 1155

FI 09979 90151 0206 0167

DE-PSO PI 10098 89138 0983 0958

FI 10008 89956 0077 0049


1198861 120590

1199041) = (10mminus1

90mminus1

)

and (1205811198862 120590

1199042) = (45mminus1

55mminus1


11 119911

21 1199031) =

(03 03 02)m and (11991112 119911

22 1199032) = (07 06 015)m The



and 1205901199041



D4

D1

D3

D2

120581a0 = 02mminus1

120590s0 = 03mminus1

120581a1 = 10mminus1

120590s1 = 90mminus1

(a)

D4

D1D2

D3

120581a0 = 02mminus1

120590s0 = 03mminus1

120581a2 = 45mminus1

120590s2 = 55mminus1

120581a1 = 10mminus1

120590s1 = 90mminus1

A

B

(b)




10 (1205811198861 120590

1199041) = (10 90)

(1205811198862 120590

1199042) = (45 55)

PSO (09750 93216) (24962 35738)(74025 59205) (645009 76456)

SPSO (09803 89818) (19718 02021)(54861 03347) (219131 939149)

DE-PSO (09844 88901) (15635 12207)(33674 37704) (251700 314477)

50 (1205811198861 120590

1199041) = (10 90)

(1205811198862 120590

1199042) = (45 55)

PSO (09790 90331) (21025 03679)(64496 75222) (433244 367678)

SPSO (09995 89996) (00478 00040)(35513 46093) (210814 161946)

DE-PSO (10021 90036) (02071 00397)(33049 40275) (265584 267718)

100 (1205811198861 120590

1199041) = (10 90)

(1205811198862 120590

1199042) = (45 55)

PSO (09790 90331) (21025 03679)(6449 75222) (433244 367678)

SPSO (09998 89995) (00234 00054)(35119 48832) (219588 112147)

DE-PSO (10007 90042) (00696 00472)(36868 48677) (180717 114957)



PSO 25476 (1205811198861 120590

1199041) = (10 90) (09985 90019) (01486 00207)

(1205811198862 120590

1199042) = (45 55) (69803 75019) (55118 36398)

SPSO 32775 (1205811198861 120590

1199041) = (10 90) (10000 89996) (00034 00046)

(1205811198862 120590

1199042) = (45 55) (44444 54556) (12349 08077)

DE-PSO 49554 (1205811198861 120590

1199041) = (10 90) (10005 90036) (00544 00397)

(1205811198862 120590

1199042) = (45 55) (45248 54454) (05508 09925)


0 20 40 60 80 100 120

Obj

ect f

unct

ion

Generation

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

PSO SPSO

DE-PSO




119878119898119894(120588

119895) =

120597120588119895

120597119898119894

100381610038161003816100381610038161003816100381610038161003816119898119894=1198980

=

120588119895(119898

0+ 119898

0Δ) minus 120588

119895(119898

0minus 119898

0Δ)

21198980Δ

(19)


1199111 119911




1 119911

2) and size




2) and size

119903 of cases (1199111= 119911

2= 025m 119903 = 025m) (119911

1= 119911

2=

05m 119903 = 025m) and (1199111= 119911

2= 075m 119903 = 025m)


1198860and








1= 119911

2= 119903 = 025m







1199112

119903 1199111

1199112

119903


SPSO 02499 02500 02500 0027 0007 0017

DE-PSO 02500 02500 02500 0015 0005 0007


SPSO 0500 0500 02499 0042 0002 0033

DE-PSO 04999 0500 02499 0019 0006 0021


SPSO 07497 07500 02500 0041 0005 0105

DE-PSO 07498 07500 02498 0026 0005 0067

0 1 2 3 4 5

000

002

ct (m)

minus002

minus004

minus006

minus008

minus010

r = 025m

D1D2

D3D4

z1 = 05m

z2 = 05m

Δz1 = 0005

S z1

(120588)

(a)

0 1 2 3 4 5

0000

0003

minus0003

minus0006

D1D2

D3D4

ct (m)

r = 025m

z1 = 05m

z2 = 05m

Δz2 = 0005

S z2

(120588)

(b)

0 1 2 3 4 5

000

001

002

003

004

005

006

007

ct (m)

minus001

D1D2

D3D4

r = 025m

Δr = 0005

z1 = 05m

z2 = 05m

S r(120588)

(c)





1199112

119903 1199111

1199112

119903


SPSO 02509 02506 02492 0356 0243 0339

DE-PSO 02340 02412 02452 6395 3514 1934


SPSO 05006 05267 02453 0120 5331 1894

DE-PSO 05150 05302 02386 2996 6045 4549


SPSO 07356 07681 02265 1926 2419 9418

DE-PSO 07237 07786 02324 3508 3821 7041

0 1 2 3 4 5

001


Refle

ctan

ce

ct (m)

z1 = 025 z2 = 025 r = 025

1E minus 3

1E minus 4

1E minus 5

1E minus 6



Nomenclature




119873120579 119873












Greeks Symbols









Subscripts


Superscript




Acknowledgments


References























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119894(119905) of each



119894(119905) can

be determined








119865 (a) = 1

2

4

sum

119895=1

int

1199051199042

1199051199041

[120588119895est (a) minus 120588119895mea (a)]

2

d119905 (15)



0 119886

1 119886

119873)T Moreover 119905

1199041and 119905

1199042are start



120588mea = 120588exa + rand ( ) sdot 120590 (16)


120590 =120588exa sdot 1205742576

(17)






119904are set as


Φ(Ω1015840

Ω) = 1 + 1198860(120583120583

1015840

+ 1205781205781015840

+ 1205851205851015840

) (18)

where 1198860= 1 119886

0= 0 and 119886



119909times 119873

119910=


120601= 20times10



Initialization

Output results

Yes

No


Yes


No

t = t + 1

t gt Nt

Update Pi and Pg

Fobj (Pg) lt 120576

(a)

Initialization

Output results

Yes

No


Yes


No


No

i = j

(11) and (12) with w = 0

Pg = Pj

Pg = P998400g

or t gt Nt

t = t + 1

Fobj (Pg) lt 120576



(b)

Initialization


Output results

Yes

No


Yes

No

No

Yes


t = t + 1

t gt Nt

Update Pi and Pg


Ft+1obj le Ft

obj

(c)



0 20 40 60 80 100

Tran

smitt

ance


FVM

120573ct

a0 = 10

a0 = 0


1E minus 4

1E minus 5

(a)

0 20 40 60 80 100

001

01

1

Refle

ctan

ce


FVM

120573ct

a0 = 10a0 = 0

a0 = minus10

1E minus 3

1E minus 4

1E minus 5

(b)


Tumor

Liver

z2

z1

(120581a0 120590s0)

(120581a1 120590s1)



119901= 01m



= 20mminus1 and 1205901199040


1 119911


1199111and 119911






Size [m] 1199031

01 02 01 01

1199032

02 01 02 02

Location [m] (11991111 119911

21) (03 045) (03 045) (07 065) (03 045)

(11991112 119911

22) (075 065) (075 065) (035 025) (075 065)


1199041) (10 75) (10 75) (10 75) (40 55)

(1205811198862 120590

1199042) (02 98) (02 98) (02 98) (15 90)

120581a2 = 02mminus1

120590s2 = 98mminus1

120581a0 = 20mminus1

120590s0 = 30mminus1

120581a1 = 10mminus1

120590s1 = 75mminus1

z22

z21

z11 z12

(a)

120581a0 = 20mminus1

120590s0 = 30mminus1 120581a2 = 02mminus1

120590s2 = 98mminus1

120581a1 = 10mminus1

120590s1 = 75mminus1

z22

z21

z11 z12

(b)

120581a0 = 20mminus1

120590s0 = 30mminus1

120581a2 = 02mminus1

120590s2 = 98mminus1

120581a1 = 10mminus1

120590s1 = 75mminus1

z22

z21

z11 z12

(c)

120581a2 = 15mminus1

120590s2 = 90mminus1

120581a0 = 20mminus1

120590s0 = 30mminus1

120581a1 = 40mminus1

120590s1 = 55mminus1

z22

z21

z11 z12

(d)



119910= 100 times 100





0 1 2 3 4 5

Refle

ctan

ce

ct (m)

1E minus 4

1E minus 3

1E minus 5

1E minus 6

1E minus 7

01

001

Case 1

Case 2

Case 3

Case 4

(a)

0 1 2 3 4 5

001

Tran

smitt

ance

ct (m)

1E minus 4

1E minus 3

1E minus 5

1E minus 6

1E minus 7

Case 1

Case 2

Case 3

Case 4

(b)


Point-irradiated

Face-irradiated

120590s1)(120581a1

(120581a0 120590s0)

(120581a0 120590s0)



1= 119911

2= 05m and


1198860= 020mminus1 and





1198860= 020mminus1 and 120590

1199040=




[]1205811198861= 10 120590

1199041= 90 120581

11988611205901199041

PSO PI 10185 91372 1846 1524

FI 09964 89671 0356 0366

SPSO PI 09880 88960 1200 1155

FI 09979 90151 0206 0167

DE-PSO PI 10098 89138 0983 0958

FI 10008 89956 0077 0049


1198861 120590

1199041) = (10mminus1

90mminus1

)

and (1205811198862 120590

1199042) = (45mminus1

55mminus1


11 119911

21 1199031) =

(03 03 02)m and (11991112 119911

22 1199032) = (07 06 015)m The



and 1205901199041



D4

D1

D3

D2

120581a0 = 02mminus1

120590s0 = 03mminus1

120581a1 = 10mminus1

120590s1 = 90mminus1

(a)

D4

D1D2

D3

120581a0 = 02mminus1

120590s0 = 03mminus1

120581a2 = 45mminus1

120590s2 = 55mminus1

120581a1 = 10mminus1

120590s1 = 90mminus1

A

B

(b)




10 (1205811198861 120590

1199041) = (10 90)

(1205811198862 120590

1199042) = (45 55)

PSO (09750 93216) (24962 35738)(74025 59205) (645009 76456)

SPSO (09803 89818) (19718 02021)(54861 03347) (219131 939149)

DE-PSO (09844 88901) (15635 12207)(33674 37704) (251700 314477)

50 (1205811198861 120590

1199041) = (10 90)

(1205811198862 120590

1199042) = (45 55)

PSO (09790 90331) (21025 03679)(64496 75222) (433244 367678)

SPSO (09995 89996) (00478 00040)(35513 46093) (210814 161946)

DE-PSO (10021 90036) (02071 00397)(33049 40275) (265584 267718)

100 (1205811198861 120590

1199041) = (10 90)

(1205811198862 120590

1199042) = (45 55)

PSO (09790 90331) (21025 03679)(6449 75222) (433244 367678)

SPSO (09998 89995) (00234 00054)(35119 48832) (219588 112147)

DE-PSO (10007 90042) (00696 00472)(36868 48677) (180717 114957)



PSO 25476 (1205811198861 120590

1199041) = (10 90) (09985 90019) (01486 00207)

(1205811198862 120590

1199042) = (45 55) (69803 75019) (55118 36398)

SPSO 32775 (1205811198861 120590

1199041) = (10 90) (10000 89996) (00034 00046)

(1205811198862 120590

1199042) = (45 55) (44444 54556) (12349 08077)

DE-PSO 49554 (1205811198861 120590

1199041) = (10 90) (10005 90036) (00544 00397)

(1205811198862 120590

1199042) = (45 55) (45248 54454) (05508 09925)


0 20 40 60 80 100 120

Obj

ect f

unct

ion

Generation

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

PSO SPSO

DE-PSO




119878119898119894(120588

119895) =

120597120588119895

120597119898119894

100381610038161003816100381610038161003816100381610038161003816119898119894=1198980

=

120588119895(119898

0+ 119898

0Δ) minus 120588

119895(119898

0minus 119898

0Δ)

21198980Δ

(19)


1199111 119911




1 119911

2) and size




2) and size

119903 of cases (1199111= 119911

2= 025m 119903 = 025m) (119911

1= 119911

2=

05m 119903 = 025m) and (1199111= 119911

2= 075m 119903 = 025m)


1198860and








1= 119911

2= 119903 = 025m







1199112

119903 1199111

1199112

119903


SPSO 02499 02500 02500 0027 0007 0017

DE-PSO 02500 02500 02500 0015 0005 0007


SPSO 0500 0500 02499 0042 0002 0033

DE-PSO 04999 0500 02499 0019 0006 0021


SPSO 07497 07500 02500 0041 0005 0105

DE-PSO 07498 07500 02498 0026 0005 0067

0 1 2 3 4 5

000

002

ct (m)

minus002

minus004

minus006

minus008

minus010

r = 025m

D1D2

D3D4

z1 = 05m

z2 = 05m

Δz1 = 0005

S z1

(120588)

(a)

0 1 2 3 4 5

0000

0003

minus0003

minus0006

D1D2

D3D4

ct (m)

r = 025m

z1 = 05m

z2 = 05m

Δz2 = 0005

S z2

(120588)

(b)

0 1 2 3 4 5

000

001

002

003

004

005

006

007

ct (m)

minus001

D1D2

D3D4

r = 025m

Δr = 0005

z1 = 05m

z2 = 05m

S r(120588)

(c)





1199112

119903 1199111

1199112

119903


SPSO 02509 02506 02492 0356 0243 0339

DE-PSO 02340 02412 02452 6395 3514 1934


SPSO 05006 05267 02453 0120 5331 1894

DE-PSO 05150 05302 02386 2996 6045 4549


SPSO 07356 07681 02265 1926 2419 9418

DE-PSO 07237 07786 02324 3508 3821 7041

0 1 2 3 4 5

001


Refle

ctan

ce

ct (m)

z1 = 025 z2 = 025 r = 025

1E minus 3

1E minus 4

1E minus 5

1E minus 6



Nomenclature




119873120579 119873












Greeks Symbols









Subscripts


Superscript




Acknowledgments


References























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Initialization

Output results

Yes

No


Yes


No

t = t + 1

t gt Nt

Update Pi and Pg

Fobj (Pg) lt 120576

(a)

Initialization

Output results

Yes

No


Yes


No


No

i = j

(11) and (12) with w = 0

Pg = Pj

Pg = P998400g

or t gt Nt

t = t + 1

Fobj (Pg) lt 120576



(b)

Initialization


Output results

Yes

No


Yes

No

No

Yes


t = t + 1

t gt Nt

Update Pi and Pg


Ft+1obj le Ft

obj

(c)



0 20 40 60 80 100

Tran

smitt

ance


FVM

120573ct

a0 = 10

a0 = 0


1E minus 4

1E minus 5

(a)

0 20 40 60 80 100

001

01

1

Refle

ctan

ce


FVM

120573ct

a0 = 10a0 = 0

a0 = minus10

1E minus 3

1E minus 4

1E minus 5

(b)


Tumor

Liver

z2

z1

(120581a0 120590s0)

(120581a1 120590s1)



119901= 01m



= 20mminus1 and 1205901199040


1 119911


1199111and 119911






Size [m] 1199031

01 02 01 01

1199032

02 01 02 02

Location [m] (11991111 119911

21) (03 045) (03 045) (07 065) (03 045)

(11991112 119911

22) (075 065) (075 065) (035 025) (075 065)


1199041) (10 75) (10 75) (10 75) (40 55)

(1205811198862 120590

1199042) (02 98) (02 98) (02 98) (15 90)

120581a2 = 02mminus1

120590s2 = 98mminus1

120581a0 = 20mminus1

120590s0 = 30mminus1

120581a1 = 10mminus1

120590s1 = 75mminus1

z22

z21

z11 z12

(a)

120581a0 = 20mminus1

120590s0 = 30mminus1 120581a2 = 02mminus1

120590s2 = 98mminus1

120581a1 = 10mminus1

120590s1 = 75mminus1

z22

z21

z11 z12

(b)

120581a0 = 20mminus1

120590s0 = 30mminus1

120581a2 = 02mminus1

120590s2 = 98mminus1

120581a1 = 10mminus1

120590s1 = 75mminus1

z22

z21

z11 z12

(c)

120581a2 = 15mminus1

120590s2 = 90mminus1

120581a0 = 20mminus1

120590s0 = 30mminus1

120581a1 = 40mminus1

120590s1 = 55mminus1

z22

z21

z11 z12

(d)



119910= 100 times 100





0 1 2 3 4 5

Refle

ctan

ce

ct (m)

1E minus 4

1E minus 3

1E minus 5

1E minus 6

1E minus 7

01

001

Case 1

Case 2

Case 3

Case 4

(a)

0 1 2 3 4 5

001

Tran

smitt

ance

ct (m)

1E minus 4

1E minus 3

1E minus 5

1E minus 6

1E minus 7

Case 1

Case 2

Case 3

Case 4

(b)


Point-irradiated

Face-irradiated

120590s1)(120581a1

(120581a0 120590s0)

(120581a0 120590s0)



1= 119911

2= 05m and


1198860= 020mminus1 and





1198860= 020mminus1 and 120590

1199040=




[]1205811198861= 10 120590

1199041= 90 120581

11988611205901199041

PSO PI 10185 91372 1846 1524

FI 09964 89671 0356 0366

SPSO PI 09880 88960 1200 1155

FI 09979 90151 0206 0167

DE-PSO PI 10098 89138 0983 0958

FI 10008 89956 0077 0049


1198861 120590

1199041) = (10mminus1

90mminus1

)

and (1205811198862 120590

1199042) = (45mminus1

55mminus1


11 119911

21 1199031) =

(03 03 02)m and (11991112 119911

22 1199032) = (07 06 015)m The



and 1205901199041



D4

D1

D3

D2

120581a0 = 02mminus1

120590s0 = 03mminus1

120581a1 = 10mminus1

120590s1 = 90mminus1

(a)

D4

D1D2

D3

120581a0 = 02mminus1

120590s0 = 03mminus1

120581a2 = 45mminus1

120590s2 = 55mminus1

120581a1 = 10mminus1

120590s1 = 90mminus1

A

B

(b)




10 (1205811198861 120590

1199041) = (10 90)

(1205811198862 120590

1199042) = (45 55)

PSO (09750 93216) (24962 35738)(74025 59205) (645009 76456)

SPSO (09803 89818) (19718 02021)(54861 03347) (219131 939149)

DE-PSO (09844 88901) (15635 12207)(33674 37704) (251700 314477)

50 (1205811198861 120590

1199041) = (10 90)

(1205811198862 120590

1199042) = (45 55)

PSO (09790 90331) (21025 03679)(64496 75222) (433244 367678)

SPSO (09995 89996) (00478 00040)(35513 46093) (210814 161946)

DE-PSO (10021 90036) (02071 00397)(33049 40275) (265584 267718)

100 (1205811198861 120590

1199041) = (10 90)

(1205811198862 120590

1199042) = (45 55)

PSO (09790 90331) (21025 03679)(6449 75222) (433244 367678)

SPSO (09998 89995) (00234 00054)(35119 48832) (219588 112147)

DE-PSO (10007 90042) (00696 00472)(36868 48677) (180717 114957)



PSO 25476 (1205811198861 120590

1199041) = (10 90) (09985 90019) (01486 00207)

(1205811198862 120590

1199042) = (45 55) (69803 75019) (55118 36398)

SPSO 32775 (1205811198861 120590

1199041) = (10 90) (10000 89996) (00034 00046)

(1205811198862 120590

1199042) = (45 55) (44444 54556) (12349 08077)

DE-PSO 49554 (1205811198861 120590

1199041) = (10 90) (10005 90036) (00544 00397)

(1205811198862 120590

1199042) = (45 55) (45248 54454) (05508 09925)


0 20 40 60 80 100 120

Obj

ect f

unct

ion

Generation

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

PSO SPSO

DE-PSO




119878119898119894(120588

119895) =

120597120588119895

120597119898119894

100381610038161003816100381610038161003816100381610038161003816119898119894=1198980

=

120588119895(119898

0+ 119898

0Δ) minus 120588

119895(119898

0minus 119898

0Δ)

21198980Δ

(19)


1199111 119911




1 119911

2) and size




2) and size

119903 of cases (1199111= 119911

2= 025m 119903 = 025m) (119911

1= 119911

2=

05m 119903 = 025m) and (1199111= 119911

2= 075m 119903 = 025m)


1198860and








1= 119911

2= 119903 = 025m







1199112

119903 1199111

1199112

119903


SPSO 02499 02500 02500 0027 0007 0017

DE-PSO 02500 02500 02500 0015 0005 0007


SPSO 0500 0500 02499 0042 0002 0033

DE-PSO 04999 0500 02499 0019 0006 0021


SPSO 07497 07500 02500 0041 0005 0105

DE-PSO 07498 07500 02498 0026 0005 0067

0 1 2 3 4 5

000

002

ct (m)

minus002

minus004

minus006

minus008

minus010

r = 025m

D1D2

D3D4

z1 = 05m

z2 = 05m

Δz1 = 0005

S z1

(120588)

(a)

0 1 2 3 4 5

0000

0003

minus0003

minus0006

D1D2

D3D4

ct (m)

r = 025m

z1 = 05m

z2 = 05m

Δz2 = 0005

S z2

(120588)

(b)

0 1 2 3 4 5

000

001

002

003

004

005

006

007

ct (m)

minus001

D1D2

D3D4

r = 025m

Δr = 0005

z1 = 05m

z2 = 05m

S r(120588)

(c)





1199112

119903 1199111

1199112

119903


SPSO 02509 02506 02492 0356 0243 0339

DE-PSO 02340 02412 02452 6395 3514 1934


SPSO 05006 05267 02453 0120 5331 1894

DE-PSO 05150 05302 02386 2996 6045 4549


SPSO 07356 07681 02265 1926 2419 9418

DE-PSO 07237 07786 02324 3508 3821 7041

0 1 2 3 4 5

001


Refle

ctan

ce

ct (m)

z1 = 025 z2 = 025 r = 025

1E minus 3

1E minus 4

1E minus 5

1E minus 6



Nomenclature




119873120579 119873












Greeks Symbols









Subscripts


Superscript




Acknowledgments


References























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 20 40 60 80 100

Tran

smitt

ance


FVM

120573ct

a0 = 10

a0 = 0


1E minus 4

1E minus 5

(a)

0 20 40 60 80 100

001

01

1

Refle

ctan

ce


FVM

120573ct

a0 = 10a0 = 0

a0 = minus10

1E minus 3

1E minus 4

1E minus 5

(b)


Tumor

Liver

z2

z1

(120581a0 120590s0)

(120581a1 120590s1)



119901= 01m



= 20mminus1 and 1205901199040


1 119911


1199111and 119911






Size [m] 1199031

01 02 01 01

1199032

02 01 02 02

Location [m] (11991111 119911

21) (03 045) (03 045) (07 065) (03 045)

(11991112 119911

22) (075 065) (075 065) (035 025) (075 065)


1199041) (10 75) (10 75) (10 75) (40 55)

(1205811198862 120590

1199042) (02 98) (02 98) (02 98) (15 90)

120581a2 = 02mminus1

120590s2 = 98mminus1

120581a0 = 20mminus1

120590s0 = 30mminus1

120581a1 = 10mminus1

120590s1 = 75mminus1

z22

z21

z11 z12

(a)

120581a0 = 20mminus1

120590s0 = 30mminus1 120581a2 = 02mminus1

120590s2 = 98mminus1

120581a1 = 10mminus1

120590s1 = 75mminus1

z22

z21

z11 z12

(b)

120581a0 = 20mminus1

120590s0 = 30mminus1

120581a2 = 02mminus1

120590s2 = 98mminus1

120581a1 = 10mminus1

120590s1 = 75mminus1

z22

z21

z11 z12

(c)

120581a2 = 15mminus1

120590s2 = 90mminus1

120581a0 = 20mminus1

120590s0 = 30mminus1

120581a1 = 40mminus1

120590s1 = 55mminus1

z22

z21

z11 z12

(d)



119910= 100 times 100





0 1 2 3 4 5

Refle

ctan

ce

ct (m)

1E minus 4

1E minus 3

1E minus 5

1E minus 6

1E minus 7

01

001

Case 1

Case 2

Case 3

Case 4

(a)

0 1 2 3 4 5

001

Tran

smitt

ance

ct (m)

1E minus 4

1E minus 3

1E minus 5

1E minus 6

1E minus 7

Case 1

Case 2

Case 3

Case 4

(b)


Point-irradiated

Face-irradiated

120590s1)(120581a1

(120581a0 120590s0)

(120581a0 120590s0)



1= 119911

2= 05m and


1198860= 020mminus1 and





1198860= 020mminus1 and 120590

1199040=




[]1205811198861= 10 120590

1199041= 90 120581

11988611205901199041

PSO PI 10185 91372 1846 1524

FI 09964 89671 0356 0366

SPSO PI 09880 88960 1200 1155

FI 09979 90151 0206 0167

DE-PSO PI 10098 89138 0983 0958

FI 10008 89956 0077 0049


1198861 120590

1199041) = (10mminus1

90mminus1

)

and (1205811198862 120590

1199042) = (45mminus1

55mminus1


11 119911

21 1199031) =

(03 03 02)m and (11991112 119911

22 1199032) = (07 06 015)m The



and 1205901199041



D4

D1

D3

D2

120581a0 = 02mminus1

120590s0 = 03mminus1

120581a1 = 10mminus1

120590s1 = 90mminus1

(a)

D4

D1D2

D3

120581a0 = 02mminus1

120590s0 = 03mminus1

120581a2 = 45mminus1

120590s2 = 55mminus1

120581a1 = 10mminus1

120590s1 = 90mminus1

A

B

(b)




10 (1205811198861 120590

1199041) = (10 90)

(1205811198862 120590

1199042) = (45 55)

PSO (09750 93216) (24962 35738)(74025 59205) (645009 76456)

SPSO (09803 89818) (19718 02021)(54861 03347) (219131 939149)

DE-PSO (09844 88901) (15635 12207)(33674 37704) (251700 314477)

50 (1205811198861 120590

1199041) = (10 90)

(1205811198862 120590

1199042) = (45 55)

PSO (09790 90331) (21025 03679)(64496 75222) (433244 367678)

SPSO (09995 89996) (00478 00040)(35513 46093) (210814 161946)

DE-PSO (10021 90036) (02071 00397)(33049 40275) (265584 267718)

100 (1205811198861 120590

1199041) = (10 90)

(1205811198862 120590

1199042) = (45 55)

PSO (09790 90331) (21025 03679)(6449 75222) (433244 367678)

SPSO (09998 89995) (00234 00054)(35119 48832) (219588 112147)

DE-PSO (10007 90042) (00696 00472)(36868 48677) (180717 114957)



PSO 25476 (1205811198861 120590

1199041) = (10 90) (09985 90019) (01486 00207)

(1205811198862 120590

1199042) = (45 55) (69803 75019) (55118 36398)

SPSO 32775 (1205811198861 120590

1199041) = (10 90) (10000 89996) (00034 00046)

(1205811198862 120590

1199042) = (45 55) (44444 54556) (12349 08077)

DE-PSO 49554 (1205811198861 120590

1199041) = (10 90) (10005 90036) (00544 00397)

(1205811198862 120590

1199042) = (45 55) (45248 54454) (05508 09925)


0 20 40 60 80 100 120

Obj

ect f

unct

ion

Generation

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

PSO SPSO

DE-PSO




119878119898119894(120588

119895) =

120597120588119895

120597119898119894

100381610038161003816100381610038161003816100381610038161003816119898119894=1198980

=

120588119895(119898

0+ 119898

0Δ) minus 120588

119895(119898

0minus 119898

0Δ)

21198980Δ

(19)


1199111 119911




1 119911

2) and size




2) and size

119903 of cases (1199111= 119911

2= 025m 119903 = 025m) (119911

1= 119911

2=

05m 119903 = 025m) and (1199111= 119911

2= 075m 119903 = 025m)


1198860and








1= 119911

2= 119903 = 025m







1199112

119903 1199111

1199112

119903


SPSO 02499 02500 02500 0027 0007 0017

DE-PSO 02500 02500 02500 0015 0005 0007


SPSO 0500 0500 02499 0042 0002 0033

DE-PSO 04999 0500 02499 0019 0006 0021


SPSO 07497 07500 02500 0041 0005 0105

DE-PSO 07498 07500 02498 0026 0005 0067

0 1 2 3 4 5

000

002

ct (m)

minus002

minus004

minus006

minus008

minus010

r = 025m

D1D2

D3D4

z1 = 05m

z2 = 05m

Δz1 = 0005

S z1

(120588)

(a)

0 1 2 3 4 5

0000

0003

minus0003

minus0006

D1D2

D3D4

ct (m)

r = 025m

z1 = 05m

z2 = 05m

Δz2 = 0005

S z2

(120588)

(b)

0 1 2 3 4 5

000

001

002

003

004

005

006

007

ct (m)

minus001

D1D2

D3D4

r = 025m

Δr = 0005

z1 = 05m

z2 = 05m

S r(120588)

(c)





1199112

119903 1199111

1199112

119903


SPSO 02509 02506 02492 0356 0243 0339

DE-PSO 02340 02412 02452 6395 3514 1934


SPSO 05006 05267 02453 0120 5331 1894

DE-PSO 05150 05302 02386 2996 6045 4549


SPSO 07356 07681 02265 1926 2419 9418

DE-PSO 07237 07786 02324 3508 3821 7041

0 1 2 3 4 5

001


Refle

ctan

ce

ct (m)

z1 = 025 z2 = 025 r = 025

1E minus 3

1E minus 4

1E minus 5

1E minus 6



Nomenclature




119873120579 119873












Greeks Symbols









Subscripts


Superscript




Acknowledgments


References























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












Size [m] 1199031

01 02 01 01

1199032

02 01 02 02

Location [m] (11991111 119911

21) (03 045) (03 045) (07 065) (03 045)

(11991112 119911

22) (075 065) (075 065) (035 025) (075 065)


1199041) (10 75) (10 75) (10 75) (40 55)

(1205811198862 120590

1199042) (02 98) (02 98) (02 98) (15 90)

120581a2 = 02mminus1

120590s2 = 98mminus1

120581a0 = 20mminus1

120590s0 = 30mminus1

120581a1 = 10mminus1

120590s1 = 75mminus1

z22

z21

z11 z12

(a)

120581a0 = 20mminus1

120590s0 = 30mminus1 120581a2 = 02mminus1

120590s2 = 98mminus1

120581a1 = 10mminus1

120590s1 = 75mminus1

z22

z21

z11 z12

(b)

120581a0 = 20mminus1

120590s0 = 30mminus1

120581a2 = 02mminus1

120590s2 = 98mminus1

120581a1 = 10mminus1

120590s1 = 75mminus1

z22

z21

z11 z12

(c)

120581a2 = 15mminus1

120590s2 = 90mminus1

120581a0 = 20mminus1

120590s0 = 30mminus1

120581a1 = 40mminus1

120590s1 = 55mminus1

z22

z21

z11 z12

(d)



119910= 100 times 100





0 1 2 3 4 5

Refle

ctan

ce

ct (m)

1E minus 4

1E minus 3

1E minus 5

1E minus 6

1E minus 7

01

001

Case 1

Case 2

Case 3

Case 4

(a)

0 1 2 3 4 5

001

Tran

smitt

ance

ct (m)

1E minus 4

1E minus 3

1E minus 5

1E minus 6

1E minus 7

Case 1

Case 2

Case 3

Case 4

(b)


Point-irradiated

Face-irradiated

120590s1)(120581a1

(120581a0 120590s0)

(120581a0 120590s0)



1= 119911

2= 05m and


1198860= 020mminus1 and





1198860= 020mminus1 and 120590

1199040=




[]1205811198861= 10 120590

1199041= 90 120581

11988611205901199041

PSO PI 10185 91372 1846 1524

FI 09964 89671 0356 0366

SPSO PI 09880 88960 1200 1155

FI 09979 90151 0206 0167

DE-PSO PI 10098 89138 0983 0958

FI 10008 89956 0077 0049


1198861 120590

1199041) = (10mminus1

90mminus1

)

and (1205811198862 120590

1199042) = (45mminus1

55mminus1


11 119911

21 1199031) =

(03 03 02)m and (11991112 119911

22 1199032) = (07 06 015)m The



and 1205901199041



D4

D1

D3

D2

120581a0 = 02mminus1

120590s0 = 03mminus1

120581a1 = 10mminus1

120590s1 = 90mminus1

(a)

D4

D1D2

D3

120581a0 = 02mminus1

120590s0 = 03mminus1

120581a2 = 45mminus1

120590s2 = 55mminus1

120581a1 = 10mminus1

120590s1 = 90mminus1

A

B

(b)




10 (1205811198861 120590

1199041) = (10 90)

(1205811198862 120590

1199042) = (45 55)

PSO (09750 93216) (24962 35738)(74025 59205) (645009 76456)

SPSO (09803 89818) (19718 02021)(54861 03347) (219131 939149)

DE-PSO (09844 88901) (15635 12207)(33674 37704) (251700 314477)

50 (1205811198861 120590

1199041) = (10 90)

(1205811198862 120590

1199042) = (45 55)

PSO (09790 90331) (21025 03679)(64496 75222) (433244 367678)

SPSO (09995 89996) (00478 00040)(35513 46093) (210814 161946)

DE-PSO (10021 90036) (02071 00397)(33049 40275) (265584 267718)

100 (1205811198861 120590

1199041) = (10 90)

(1205811198862 120590

1199042) = (45 55)

PSO (09790 90331) (21025 03679)(6449 75222) (433244 367678)

SPSO (09998 89995) (00234 00054)(35119 48832) (219588 112147)

DE-PSO (10007 90042) (00696 00472)(36868 48677) (180717 114957)



PSO 25476 (1205811198861 120590

1199041) = (10 90) (09985 90019) (01486 00207)

(1205811198862 120590

1199042) = (45 55) (69803 75019) (55118 36398)

SPSO 32775 (1205811198861 120590

1199041) = (10 90) (10000 89996) (00034 00046)

(1205811198862 120590

1199042) = (45 55) (44444 54556) (12349 08077)

DE-PSO 49554 (1205811198861 120590

1199041) = (10 90) (10005 90036) (00544 00397)

(1205811198862 120590

1199042) = (45 55) (45248 54454) (05508 09925)


0 20 40 60 80 100 120

Obj

ect f

unct

ion

Generation

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

PSO SPSO

DE-PSO




119878119898119894(120588

119895) =

120597120588119895

120597119898119894

100381610038161003816100381610038161003816100381610038161003816119898119894=1198980

=

120588119895(119898

0+ 119898

0Δ) minus 120588

119895(119898

0minus 119898

0Δ)

21198980Δ

(19)


1199111 119911




1 119911

2) and size




2) and size

119903 of cases (1199111= 119911

2= 025m 119903 = 025m) (119911

1= 119911

2=

05m 119903 = 025m) and (1199111= 119911

2= 075m 119903 = 025m)


1198860and








1= 119911

2= 119903 = 025m







1199112

119903 1199111

1199112

119903


SPSO 02499 02500 02500 0027 0007 0017

DE-PSO 02500 02500 02500 0015 0005 0007


SPSO 0500 0500 02499 0042 0002 0033

DE-PSO 04999 0500 02499 0019 0006 0021


SPSO 07497 07500 02500 0041 0005 0105

DE-PSO 07498 07500 02498 0026 0005 0067

0 1 2 3 4 5

000

002

ct (m)

minus002

minus004

minus006

minus008

minus010

r = 025m

D1D2

D3D4

z1 = 05m

z2 = 05m

Δz1 = 0005

S z1

(120588)

(a)

0 1 2 3 4 5

0000

0003

minus0003

minus0006

D1D2

D3D4

ct (m)

r = 025m

z1 = 05m

z2 = 05m

Δz2 = 0005

S z2

(120588)

(b)

0 1 2 3 4 5

000

001

002

003

004

005

006

007

ct (m)

minus001

D1D2

D3D4

r = 025m

Δr = 0005

z1 = 05m

z2 = 05m

S r(120588)

(c)





1199112

119903 1199111

1199112

119903


SPSO 02509 02506 02492 0356 0243 0339

DE-PSO 02340 02412 02452 6395 3514 1934


SPSO 05006 05267 02453 0120 5331 1894

DE-PSO 05150 05302 02386 2996 6045 4549


SPSO 07356 07681 02265 1926 2419 9418

DE-PSO 07237 07786 02324 3508 3821 7041

0 1 2 3 4 5

001


Refle

ctan

ce

ct (m)

z1 = 025 z2 = 025 r = 025

1E minus 3

1E minus 4

1E minus 5

1E minus 6



Nomenclature




119873120579 119873












Greeks Symbols









Subscripts


Superscript




Acknowledgments


References























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 1 2 3 4 5

Refle

ctan

ce

ct (m)

1E minus 4

1E minus 3

1E minus 5

1E minus 6

1E minus 7

01

001

Case 1

Case 2

Case 3

Case 4

(a)

0 1 2 3 4 5

001

Tran

smitt

ance

ct (m)

1E minus 4

1E minus 3

1E minus 5

1E minus 6

1E minus 7

Case 1

Case 2

Case 3

Case 4

(b)


Point-irradiated

Face-irradiated

120590s1)(120581a1

(120581a0 120590s0)

(120581a0 120590s0)



1= 119911

2= 05m and


1198860= 020mminus1 and





1198860= 020mminus1 and 120590

1199040=




[]1205811198861= 10 120590

1199041= 90 120581

11988611205901199041

PSO PI 10185 91372 1846 1524

FI 09964 89671 0356 0366

SPSO PI 09880 88960 1200 1155

FI 09979 90151 0206 0167

DE-PSO PI 10098 89138 0983 0958

FI 10008 89956 0077 0049


1198861 120590

1199041) = (10mminus1

90mminus1

)

and (1205811198862 120590

1199042) = (45mminus1

55mminus1


11 119911

21 1199031) =

(03 03 02)m and (11991112 119911

22 1199032) = (07 06 015)m The



and 1205901199041



D4

D1

D3

D2

120581a0 = 02mminus1

120590s0 = 03mminus1

120581a1 = 10mminus1

120590s1 = 90mminus1

(a)

D4

D1D2

D3

120581a0 = 02mminus1

120590s0 = 03mminus1

120581a2 = 45mminus1

120590s2 = 55mminus1

120581a1 = 10mminus1

120590s1 = 90mminus1

A

B

(b)




10 (1205811198861 120590

1199041) = (10 90)

(1205811198862 120590

1199042) = (45 55)

PSO (09750 93216) (24962 35738)(74025 59205) (645009 76456)

SPSO (09803 89818) (19718 02021)(54861 03347) (219131 939149)

DE-PSO (09844 88901) (15635 12207)(33674 37704) (251700 314477)

50 (1205811198861 120590

1199041) = (10 90)

(1205811198862 120590

1199042) = (45 55)

PSO (09790 90331) (21025 03679)(64496 75222) (433244 367678)

SPSO (09995 89996) (00478 00040)(35513 46093) (210814 161946)

DE-PSO (10021 90036) (02071 00397)(33049 40275) (265584 267718)

100 (1205811198861 120590

1199041) = (10 90)

(1205811198862 120590

1199042) = (45 55)

PSO (09790 90331) (21025 03679)(6449 75222) (433244 367678)

SPSO (09998 89995) (00234 00054)(35119 48832) (219588 112147)

DE-PSO (10007 90042) (00696 00472)(36868 48677) (180717 114957)



PSO 25476 (1205811198861 120590

1199041) = (10 90) (09985 90019) (01486 00207)

(1205811198862 120590

1199042) = (45 55) (69803 75019) (55118 36398)

SPSO 32775 (1205811198861 120590

1199041) = (10 90) (10000 89996) (00034 00046)

(1205811198862 120590

1199042) = (45 55) (44444 54556) (12349 08077)

DE-PSO 49554 (1205811198861 120590

1199041) = (10 90) (10005 90036) (00544 00397)

(1205811198862 120590

1199042) = (45 55) (45248 54454) (05508 09925)


0 20 40 60 80 100 120

Obj

ect f

unct

ion

Generation

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

PSO SPSO

DE-PSO




119878119898119894(120588

119895) =

120597120588119895

120597119898119894

100381610038161003816100381610038161003816100381610038161003816119898119894=1198980

=

120588119895(119898

0+ 119898

0Δ) minus 120588

119895(119898

0minus 119898

0Δ)

21198980Δ

(19)


1199111 119911




1 119911

2) and size




2) and size

119903 of cases (1199111= 119911

2= 025m 119903 = 025m) (119911

1= 119911

2=

05m 119903 = 025m) and (1199111= 119911

2= 075m 119903 = 025m)


1198860and








1= 119911

2= 119903 = 025m







1199112

119903 1199111

1199112

119903


SPSO 02499 02500 02500 0027 0007 0017

DE-PSO 02500 02500 02500 0015 0005 0007


SPSO 0500 0500 02499 0042 0002 0033

DE-PSO 04999 0500 02499 0019 0006 0021


SPSO 07497 07500 02500 0041 0005 0105

DE-PSO 07498 07500 02498 0026 0005 0067

0 1 2 3 4 5

000

002

ct (m)

minus002

minus004

minus006

minus008

minus010

r = 025m

D1D2

D3D4

z1 = 05m

z2 = 05m

Δz1 = 0005

S z1

(120588)

(a)

0 1 2 3 4 5

0000

0003

minus0003

minus0006

D1D2

D3D4

ct (m)

r = 025m

z1 = 05m

z2 = 05m

Δz2 = 0005

S z2

(120588)

(b)

0 1 2 3 4 5

000

001

002

003

004

005

006

007

ct (m)

minus001

D1D2

D3D4

r = 025m

Δr = 0005

z1 = 05m

z2 = 05m

S r(120588)

(c)





1199112

119903 1199111

1199112

119903


SPSO 02509 02506 02492 0356 0243 0339

DE-PSO 02340 02412 02452 6395 3514 1934


SPSO 05006 05267 02453 0120 5331 1894

DE-PSO 05150 05302 02386 2996 6045 4549


SPSO 07356 07681 02265 1926 2419 9418

DE-PSO 07237 07786 02324 3508 3821 7041

0 1 2 3 4 5

001


Refle

ctan

ce

ct (m)

z1 = 025 z2 = 025 r = 025

1E minus 3

1E minus 4

1E minus 5

1E minus 6



Nomenclature




119873120579 119873












Greeks Symbols









Subscripts


Superscript




Acknowledgments


References























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










D4

D1

D3

D2

120581a0 = 02mminus1

120590s0 = 03mminus1

120581a1 = 10mminus1

120590s1 = 90mminus1

(a)

D4

D1D2

D3

120581a0 = 02mminus1

120590s0 = 03mminus1

120581a2 = 45mminus1

120590s2 = 55mminus1

120581a1 = 10mminus1

120590s1 = 90mminus1

A

B

(b)




10 (1205811198861 120590

1199041) = (10 90)

(1205811198862 120590

1199042) = (45 55)

PSO (09750 93216) (24962 35738)(74025 59205) (645009 76456)

SPSO (09803 89818) (19718 02021)(54861 03347) (219131 939149)

DE-PSO (09844 88901) (15635 12207)(33674 37704) (251700 314477)

50 (1205811198861 120590

1199041) = (10 90)

(1205811198862 120590

1199042) = (45 55)

PSO (09790 90331) (21025 03679)(64496 75222) (433244 367678)

SPSO (09995 89996) (00478 00040)(35513 46093) (210814 161946)

DE-PSO (10021 90036) (02071 00397)(33049 40275) (265584 267718)

100 (1205811198861 120590

1199041) = (10 90)

(1205811198862 120590

1199042) = (45 55)

PSO (09790 90331) (21025 03679)(6449 75222) (433244 367678)

SPSO (09998 89995) (00234 00054)(35119 48832) (219588 112147)

DE-PSO (10007 90042) (00696 00472)(36868 48677) (180717 114957)



PSO 25476 (1205811198861 120590

1199041) = (10 90) (09985 90019) (01486 00207)

(1205811198862 120590

1199042) = (45 55) (69803 75019) (55118 36398)

SPSO 32775 (1205811198861 120590

1199041) = (10 90) (10000 89996) (00034 00046)

(1205811198862 120590

1199042) = (45 55) (44444 54556) (12349 08077)

DE-PSO 49554 (1205811198861 120590

1199041) = (10 90) (10005 90036) (00544 00397)

(1205811198862 120590

1199042) = (45 55) (45248 54454) (05508 09925)


0 20 40 60 80 100 120

Obj

ect f

unct

ion

Generation

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

PSO SPSO

DE-PSO




119878119898119894(120588

119895) =

120597120588119895

120597119898119894

100381610038161003816100381610038161003816100381610038161003816119898119894=1198980

=

120588119895(119898

0+ 119898

0Δ) minus 120588

119895(119898

0minus 119898

0Δ)

21198980Δ

(19)


1199111 119911




1 119911

2) and size




2) and size

119903 of cases (1199111= 119911

2= 025m 119903 = 025m) (119911

1= 119911

2=

05m 119903 = 025m) and (1199111= 119911

2= 075m 119903 = 025m)


1198860and








1= 119911

2= 119903 = 025m







1199112

119903 1199111

1199112

119903


SPSO 02499 02500 02500 0027 0007 0017

DE-PSO 02500 02500 02500 0015 0005 0007


SPSO 0500 0500 02499 0042 0002 0033

DE-PSO 04999 0500 02499 0019 0006 0021


SPSO 07497 07500 02500 0041 0005 0105

DE-PSO 07498 07500 02498 0026 0005 0067

0 1 2 3 4 5

000

002

ct (m)

minus002

minus004

minus006

minus008

minus010

r = 025m

D1D2

D3D4

z1 = 05m

z2 = 05m

Δz1 = 0005

S z1

(120588)

(a)

0 1 2 3 4 5

0000

0003

minus0003

minus0006

D1D2

D3D4

ct (m)

r = 025m

z1 = 05m

z2 = 05m

Δz2 = 0005

S z2

(120588)

(b)

0 1 2 3 4 5

000

001

002

003

004

005

006

007

ct (m)

minus001

D1D2

D3D4

r = 025m

Δr = 0005

z1 = 05m

z2 = 05m

S r(120588)

(c)





1199112

119903 1199111

1199112

119903


SPSO 02509 02506 02492 0356 0243 0339

DE-PSO 02340 02412 02452 6395 3514 1934


SPSO 05006 05267 02453 0120 5331 1894

DE-PSO 05150 05302 02386 2996 6045 4549


SPSO 07356 07681 02265 1926 2419 9418

DE-PSO 07237 07786 02324 3508 3821 7041

0 1 2 3 4 5

001


Refle

ctan

ce

ct (m)

z1 = 025 z2 = 025 r = 025

1E minus 3

1E minus 4

1E minus 5

1E minus 6



Nomenclature




119873120579 119873












Greeks Symbols









Subscripts


Superscript




Acknowledgments


References























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 20 40 60 80 100 120

Obj

ect f

unct

ion

Generation

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

PSO SPSO

DE-PSO




119878119898119894(120588

119895) =

120597120588119895

120597119898119894

100381610038161003816100381610038161003816100381610038161003816119898119894=1198980

=

120588119895(119898

0+ 119898

0Δ) minus 120588

119895(119898

0minus 119898

0Δ)

21198980Δ

(19)


1199111 119911




1 119911

2) and size




2) and size

119903 of cases (1199111= 119911

2= 025m 119903 = 025m) (119911

1= 119911

2=

05m 119903 = 025m) and (1199111= 119911

2= 075m 119903 = 025m)


1198860and








1= 119911

2= 119903 = 025m







1199112

119903 1199111

1199112

119903


SPSO 02499 02500 02500 0027 0007 0017

DE-PSO 02500 02500 02500 0015 0005 0007


SPSO 0500 0500 02499 0042 0002 0033

DE-PSO 04999 0500 02499 0019 0006 0021


SPSO 07497 07500 02500 0041 0005 0105

DE-PSO 07498 07500 02498 0026 0005 0067

0 1 2 3 4 5

000

002

ct (m)

minus002

minus004

minus006

minus008

minus010

r = 025m

D1D2

D3D4

z1 = 05m

z2 = 05m

Δz1 = 0005

S z1

(120588)

(a)

0 1 2 3 4 5

0000

0003

minus0003

minus0006

D1D2

D3D4

ct (m)

r = 025m

z1 = 05m

z2 = 05m

Δz2 = 0005

S z2

(120588)

(b)

0 1 2 3 4 5

000

001

002

003

004

005

006

007

ct (m)

minus001

D1D2

D3D4

r = 025m

Δr = 0005

z1 = 05m

z2 = 05m

S r(120588)

(c)





1199112

119903 1199111

1199112

119903


SPSO 02509 02506 02492 0356 0243 0339

DE-PSO 02340 02412 02452 6395 3514 1934


SPSO 05006 05267 02453 0120 5331 1894

DE-PSO 05150 05302 02386 2996 6045 4549


SPSO 07356 07681 02265 1926 2419 9418

DE-PSO 07237 07786 02324 3508 3821 7041

0 1 2 3 4 5

001


Refle

ctan

ce

ct (m)

z1 = 025 z2 = 025 r = 025

1E minus 3

1E minus 4

1E minus 5

1E minus 6



Nomenclature




119873120579 119873












Greeks Symbols









Subscripts


Superscript




Acknowledgments


References























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












1199112

119903 1199111

1199112

119903


SPSO 02499 02500 02500 0027 0007 0017

DE-PSO 02500 02500 02500 0015 0005 0007


SPSO 0500 0500 02499 0042 0002 0033

DE-PSO 04999 0500 02499 0019 0006 0021


SPSO 07497 07500 02500 0041 0005 0105

DE-PSO 07498 07500 02498 0026 0005 0067

0 1 2 3 4 5

000

002

ct (m)

minus002

minus004

minus006

minus008

minus010

r = 025m

D1D2

D3D4

z1 = 05m

z2 = 05m

Δz1 = 0005

S z1

(120588)

(a)

0 1 2 3 4 5

0000

0003

minus0003

minus0006

D1D2

D3D4

ct (m)

r = 025m

z1 = 05m

z2 = 05m

Δz2 = 0005

S z2

(120588)

(b)

0 1 2 3 4 5

000

001

002

003

004

005

006

007

ct (m)

minus001

D1D2

D3D4

r = 025m

Δr = 0005

z1 = 05m

z2 = 05m

S r(120588)

(c)





1199112

119903 1199111

1199112

119903


SPSO 02509 02506 02492 0356 0243 0339

DE-PSO 02340 02412 02452 6395 3514 1934


SPSO 05006 05267 02453 0120 5331 1894

DE-PSO 05150 05302 02386 2996 6045 4549


SPSO 07356 07681 02265 1926 2419 9418

DE-PSO 07237 07786 02324 3508 3821 7041

0 1 2 3 4 5

001


Refle

ctan

ce

ct (m)

z1 = 025 z2 = 025 r = 025

1E minus 3

1E minus 4

1E minus 5

1E minus 6



Nomenclature




119873120579 119873












Greeks Symbols









Subscripts


Superscript




Acknowledgments


References























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












1199112

119903 1199111

1199112

119903


SPSO 02509 02506 02492 0356 0243 0339

DE-PSO 02340 02412 02452 6395 3514 1934


SPSO 05006 05267 02453 0120 5331 1894

DE-PSO 05150 05302 02386 2996 6045 4549


SPSO 07356 07681 02265 1926 2419 9418

DE-PSO 07237 07786 02324 3508 3821 7041

0 1 2 3 4 5

001


Refle

ctan

ce

ct (m)

z1 = 025 z2 = 025 r = 025

1E minus 3

1E minus 4

1E minus 5

1E minus 6



Nomenclature




119873120579 119873












Greeks Symbols









Subscripts


Superscript




Acknowledgments


References























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














Subscripts


Superscript




Acknowledgments


References























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article Inverse Transient Radiative Analysis in Two …downloads.hindawi.com/journals/mpe/2015/680823.pdf · 2019. 7. 31. · laser machining, and medical diagnostics, to