Research ArticleElectromagnetic Nondestructive Testing byPerturbation Homotopy Method

Liang Ding1 and Jun Cao2

1 Department of Mathematics Northeast Forestry University No 26 Hexing Road Xiangfang District Harbin 150040 China2 College of Mechanical and Electrical Engineering Northeast Forestry University No 26 Hexing Road Xiangfang DistrictHarbin 150040 China

Correspondence should be addressed to Jun Cao nefudinggmailcom

Received 23 September 2013 Accepted 2 January 2014 Published 5 March 2014

Academic Editor Manuel Ruiz-Galan

Copyright copy 2014 L Ding and J Cao 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

Now electromagnetic nondestructive testing methods have been applied to many fields of engineering But traditionalelectromagnetic methods (usually based on least square and local iteration) just roughly give the information of location scale andquality In this paper we consider inverse electromagnetic problem which is concerned with the estimation of electric conductivityofMaxwellrsquos equations (2D and 3D) A perturbation homotopymethod combined with damping Gauss-Newtonmethods is appliedto the inverse electromagnetic problem This method differs from traditional homotopy method The structure of homotopyfunction is similar to Tikhonov functional Sets of solutions are produced by perturbation for every homotopy parameter 120582 = 120582

119894

119894 = 0 119871 At each iterative step of the algorithm we add stochastic perturbation to numerical solutions The previous solutionand perturbation solution are regarded as the initial value in the next iteration Although the number of solution in set increased itincreased the likelihood of obtaining correct solution Results exhibits clear advantages over damping Gauss-Newton method andtestify that it is an available method especially on aspects of wide convergence and precision

1 Introduction

In this paper we discuss an inverse problem of electro-magnetic field For a given source the direct problem is todetermine the electromagnetic field for the known coefficientwhich has been well studied [1] Our work is devoted to thenumerical solution of the electrical conductivity inversionThis problem is a model problem for many real industrialapplications including mathematical physics atmosphericscience quantum mechanics telemetry nondestructive test-ing andmedical imagingThe first pioneering solution of thefully 3D Maxwellrsquos equation inverse problem was presentedby Eaton more than 15 years ago [2] Yet in spite of thisuntil recently the trial-and-error forward modelling wasalmost the only available tool to interpret the fully 3D EMdataset Today the situation has slightly improved and themethods of unconstrained nonlinear optimization [3] aregaining popularity to address the problem Although manynew theoretical approaches have been applied to this domain[4 5] successfully and the high level of computer hardware is

changing with each passing day the desired numerical solu-tion of the inverse electromagnetic problems is still difficult toresolve for the following reasons (1) The rapid accurate andstable forward numerical solution [6ndash9] is generally requiredby inverse problem it may accelerate convergence of theinverse problem (especially formodels with low conductivitycontrasts) [10 11] but the general stable and accurate forwardsolution is still an open problem (2) The problem has theproperty of ill-posedness and nonlinearity It means that thesolution is not unique any slight noise will lead to huge error(3) In order to get a unique solution which is dependent onthe measured data continuously and stably it is necessaryto investigate a stable regularization theory Therefore it ismeaningful to design a highly efficient numerically stableand globally convergent algorithm to overcome the above-mentioned difficulties

Homotopy method is a global convergence algorithm fornonlinear operator equation It has been applied to inverseproblem in recent decades Han et al first applied homo-topy method to inverse petroleum well-logging problem

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 895159 10 pageshttpdxdoiorg1011552014895159

2 Mathematical Problems in Engineering

Global optimal solution

Stop critrerion 1

NoYes

Stop critrerion 2

YesNo

Input source Sr reference model mref measured datadobs and k = 1 0 = 1205820 le 1205821 le middot middot middot le 120582n = 1

CalculatedJ(mk0)data error r(mk0) = F(mk0) minus dobs

and regularization parameter 120572 = argminGCV(120572)

j = 0

Gauss-Newton iteration mkj+1 = mkj + Q

j = j + 1

mk+10 = mkj and mk+10 = mkj

120575 k = k + 1

Note mkj+1

120575as

perturbation of mkj+1

Nmax is limit of set

Nmax

Figure 1 The sketch map of perturbation homotopy method

in 1991 [12] Afterwards they extended thismethod to generalparameter identification Fu et al gave the numerical solutionof inverse acoustic problem and elastic wave equation withfluid saturated porous media by homotopy method respec-tively [13 14] Vasco applied homotopymethod to geophysicsinverse problem in 1994 [15] He makes use of singularityand bifurcation theory to research inverse problem Dunlavyapplied homotopy optimization method to protein structureprediction in 2005 [16] In summary the homotopy methodhas shown great advantage for solving inverse problemespecially in aspects of enlarging the convergence domainand enhancing stability with noise But homotopy inversionmethod is still far away from perfection and bifurcationand convolution about homotopy curves will appear in theprocess of computation

As mentioned above we will introduce a perturbationhomotopy method for inverse electromagnetic problem Themodel is Maxwellrsquos equations In each iterative step of thealgorithm we add stochastic perturbation to numericalsolutions The previous solution and perturbation solutionare regarded as the initial value in the next iteration Althoughthe number of solution in set increased it increased thelikelihood of obtaining correct solution With the number ofelements in set increasing we require the input value 119873max(maximum number of solutions in a set) to limit the size ofset

The paper is organized as follows We first briefly reviewsome basic ideas about inverse electromagnetic problem and

homotopy In Section 2 we develop perturbation homotopymethod and derive the basic coefficient identification algo-rithm using this method In Section 3 numerical experi-ments are presented which demonstrate the performance ofthe algorithm and indicate the validity of this method Andfinally in Section 4 some conclusions and future directionsare given

2 Electromagnetic Forward Model

The time domain Maxwell equations has the form

nabla times E + 120583120597H120597119905

= 0

nabla timesH minus 120590E minus 120576120597E120597119905

= 119904119903 (119905)

(1)

over the domainΩtimes[0 119879] whereE andH are the electric andmagnetic fields 120590 120576 and 120583 are conductivity dielectric con-stant and permeability respectively and 119904

119903(119905) is transmitting

source Discretizing (1) over time grid [119905119899 119905119899+1

] we get theequations

nabla times E119899+1 + 120591119899120583H119899+1 = 120591

119899120583H119899 in Ω

nabla timesH119899+1 minus (120590 + 120591119899120576)E119899+1 = 119904

119899+1

119903minus 120591119899120576E119899 in Ω

(2)

Mathematical Problems in Engineering 3

05 05

2

4

6

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12

X Y

Z

minus5

minus5

(a)

2

4

6

8

10

12

Z

05

Y minus5

05 X

minus5

(b)

2

4

6

8

10

12

Z

05 X

minus505

Yminus5

(c)

Figure 2 (a) Original model (b) inversion results of damping Gauss-Newton (c) inversion results of perturbation homotopy method

05 05

2

4

6

8

10

12

X Y

Z

0

50

100

150

minus5minus5

(a)

05 05

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4

6

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12

X

Z

0

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150

Y

minus5minus5

(b)

05 05

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XY

Z

0

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150

minus5minus5

(c)

10 20 30

5

10

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0

50

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(d)

10 20 30

5

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0

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(e)

10 20 30

5

10

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0

50

100

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(f)

Figure 3 (a) True model (b) results of damping Gauss-Newton (c) results of perturbation homotopy method (d) cross-section of (a) at119909 = 0 (e) cross-section of (b) at 119909 = 0 (f) cross-section of (c) at 119909 = 0

4 Mathematical Problems in Engineering

05 05

2

4

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120

50

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minus5minus5

Z

XY

(a)

05

2

4

6

8

10

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50

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150

Z

X

05 minus5minus5

Y

(b)

05 05

2

4

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8

10

120

50

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Z

Xminus5

minus5

Y

(c)

10 20 30

2

4

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0

50

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(d)

10 20 30

2

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2

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(f)

Figure 4 (a) True model (b) results of damping Gauss-Newton method (c) results of perturbation homotopy method (d) cross-section of(a) at 119885 = 8 (e) cross-section of (b) at 119885 = 8 (f) cross-section of (c) at 119885 = 8

02

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x (m)

y(m

)

05 10 15 20 252

25

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35

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45

5

Figure 5 Square cavity model

where 120591119899= 1(119905

119899+1minus 119905119899) With discretization we get matrix

equation

119860 (120590) 119906 = 119902 (3)

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2000

t(n

s)

72

144

216

288

36

Trace

minus1000

minus500

Figure 6 Measured data

In the inverse problem we want to recover the model120590 with the measured data Measured data can be written as

119889obs

= 119876119906 + 120578 (4)

Mathematical Problems in Engineering 5

x (m)

y(m

)

05 10 15 20 25

02

04

06

08

10

12

14

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20

222

25

3

35

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45

5

Figure 7 Inversion results of Gauss-Newton method

y(m

)

02

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18

20

22

x (m)05 10 15 20 25

2

25

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35

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45

5

Figure 8 Inversion results of perturbation homotopy method

where 119876 is a projection operator which projects the electro-magnetic fields onto the observation locations and 120578 is themeasurement noise From (3) and (4) we can obtain

119889obs

= 119876119860minus1119902 + 120578 (5)

3 Inversion Framework

31 Damping Gauss-Newton Method For every homotopyparameter 120582 = 120582

119894 119894 = 1 119871 we must solve the

unconstrained optimization problem

min120590

119867(120590 120582) = min120590

12058210038171003817100381710038171003817119865 (120590) minus 119889

obs10038171003817100381710038171003817

2

+ (1 minus 120582)1003817100381710038171003817119882 (120590 minus 120590ref)

10038171003817100381710038172

(6)

This can be transformed to the Euler-Lagrange system

120597119867

120597120590= 119892 (120590) = (1 minus 120582)119882

119879119882(120590 minus 120590ref)

+ 120582119869(120590)119879(119865 (120590) minus 119889

obs)

(7)

Denoting 120573 = (1 minus 120582)120582 we obtain

120573119882119879119882(120590 minus 120590ref) + 119869(120590)

119879(119865 (120590) minus 119889

obs) = 0 (8)

where 120573 played a regularization parameter role In order toavoid the calculation of the second Frechet derivative of 119865 to120590 119865 is linearized by

119865 (120590 + 120575120590) = 119865 (120590) + 119869 (120590) 120575120590 + 119877 (120590 120575120590) (9)

Then the minimization problem is solved iteratively At the119896th iteration we solve

(119869 (120590119879

119896) 119869 (120590119896) + 120573119882

119879119882)120575120590

= 119869 (120590119879

119896) (119889

obsminus 119865 (120590

119896) + 119869 (120590

119896)) minus 120573119882

119879119882(120590119896minus 120590ref)

(10)

to find the perturbation 120575120590 At each iteration we have anoption of solving directly for a perturbation or solving foran updated model To formulate the latter option we write120590119896+1

= 120590119896+ 120575120590 and substitute into (10) to obtain

(119869 (120590119879

119896) 119869 (120590119896) + 120573119882

119879119882)120590119896+1

= 119869 (120590119879

119896) (119889

obsminus 119865 (120590

119896) +119869 (120590

119896) 120590119896) minus 120573119882

119879119882120590ref

(11)

32 A Perturbation Homotopy Method As above mentionedwe can formulate the inverse electromagnetic problem as thefollowing nonlinear operator equation

119865 (120590) = 119889obs

(12)

where 119865 = 119876119860minus1119902 is the forward nonlinear operator

It is very difficult to estimate 120590 in the problem (12) dueto the presence of local minima Therefore we design aglobally convergent algorithm which can overcome the localminima namely perturbation homotopy method to solvethe nonlinear equation (12) For inverse electromagneticproblem we want to solve the minimization problem Given120601 119883 sube 119877

119899rarr 119877 find 120590

lowastisin 119883 sube 119877

119899 such that

120601 (120590lowast) = min120590isin119883

10038171003817100381710038171003817119865 (120590) minus 119889

obs10038171003817100381710038171003817

2

(13)

In order to apply homotopy to PDEoptimization problemwedenote

1206010=1003817100381710038171003817119882 (120590 minus 120590ref)

10038171003817100381710038172

1206011=10038171003817100381710038171003817119865 (120590) minus 119889

obs10038171003817100381710038171003817

2

(14)

where sdot is 2-norm 119882 is positive weight function and120590ref is the known reference model So continuous homotopyfunctional119867 119877

119899+1rarr 119877 can be written as

119867(120590 120582) = 1205821206011+ (1 minus 120582) 120601

0 (15)

where120582 isin [0 1] are homotopy parameters and the homotopystep length Δ

119894 119894 = 0 119871 minus 1 is satisfied with sum

119871minus1

119894=0Δ119894= 1

A general sketch of the homotopy algorithm is as follows

6 Mathematical Problems in Engineeringy

(m)

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)y

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0204060810121416182022

0204060810121416182022

0204060810121416182022

0204060810121416182022

0204060810121416182022

0204060810121416182022

0204060810121416182022

0204060810121416182022

0204060810121416182022

0204060810121416182022

0204060810121416182022

0204060810121416182022

Figure 9 Inversion results with different homotopy parameters (150001 15001 12001 11501 11001 1501 151 16 23 100105 10001005and 100000100005)

(i) Input given minimum norm solution of 1206010 namely1205900= 120590ref 1205820 = 0 and Δ

119894gt 0 119894 = 1 119871

(ii) Update for 120582119894= 120582119894minus1

+ Δ119894minus1

we use damping Gauss-Newton iteration method compute an approximatesolution 120590

119894to minimize min

120590isin119883119867(120590 120582

119894) by initial

value 120590119894minus1

(iii) Output the iteration will be stopped if the solution 120590

119894

satisfied stopping criterion output 1205901 = 120590119898

The homotopy algorithmrsquos greatest advantage lies in increas-ing region of convergency But in the process of computationthe nonconvergent phenomenon often happens to homotopycurve for example standstill convolution and bifurcation

These difficulties can make the homotopy method quitecomplicated In order to make the homotopy more efficientsets of solutions are produced by perturbation for everyhomotopy parameter 120582 = 120582

119894 119894 = 0 119871 Although the

calculation increased it ensures the likelihood of obtainingaccurate solutionWith different homotopy parameter 120582

119894 the

minima is in the sequence and120590119894is solved by dampingGauss-

Newton method with the initial value 120590119894minus1

The next set oflocal minima (120582 = 120582

119894+1) is solved with the initial values in the

previous set (120582 = 120582119894) which has one or more perturbations of

each of those points in the set We denote the perturbationof solution per(120590) The perturbation stochastically disturbsone or more of the solutions 120590 in set We use a single initial

Mathematical Problems in Engineering 7

02

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x (m)

y(m

)

05 10 15 20 252

25

3

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4

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5

Figure 10 Three square cavities model

20 40 60 80 100

0

500

1000

1500

2000

t(n

s)

72

144

216

288

36

Trace

minus1000

minus500

Figure 11 Measured data

value 1205900for 120582 = 120582

0 the number of solution solved at

each homotopy parameter120582 increases exponentiallyWithoutlimit on the size of the set the number of solutions in theset would be 2

119894 when homotopy iteration stops In order toconstraint on computational complexity it is necessary tolimit the size of the set We denote by 119873max the maximumnumber of solutions in a set If the number of local minimain the set at the current iteration is less than the maximumset size 119873max then every solution in the set is used inthe next iteration otherwise we will choose the preferablesolution An ideal measure of what constitutes the preferablesolutions is regularization functional value of (15) solutionwith the lower regularization functional values is consideredthe preferable The perturbation homotopy algorithm is asfollows

(i) Input given minimum norm solution of 1206010 namely1205900= 120590ref 1205820 = 0119873max ge 1 and Δ

119894gt 0 119894 = 1 119871

(ii) Update for 120582119894

= 120582119894minus1

+ Δ119894minus1

we use damp-ing Gauss-Newton iteration method compute anapproximate solution 120590

119896

119894 119896 = 1 2

119894 to minimizemin120590isin119883

119867(120590 120582119894) by initial value 120590119895

119894minus1and (per(120590

119894minus1))119895

119895 = 1 2119894

(iii) Order the distinct solutions among 120590119896

119894 119896 = 1 2

119894from best to worst as 1205901

119894 1205902119894 and discard any worst

solution when the number of elements in 119883 largerthan119873max

(iv) Output the iteration will be stopped if the solution120590119898

119894 1 le 119898 le 2

119894 is the best solution in the set andsatisfied stopping criterion output 1205901 = 120590

119898

The sketch map of the algorithm is shown in Figure 1

4 Numerical Simulation

In this section we exhibit clear advantages of perturbationhomotopy method over traditional damping Gauss-Newtonmethod in the previous three experiments In the fourthexperiment we analysis the influence of homotopy parameteron inversion results In the fifth experiment we apply ouralgorithm to more complicated model and confirm that thisalgorithm is stable In the last experiment we solve a simplepractice data In the first three 3D experiments we selecthomotopy parameter 120582

119894= 11989410 119894 = 0 10 119873max = 2 and

internal Gauss-Newton iteration 5 with each homotopy stepin our numerical experiments The space domain is Ω Theinverse problem is considered on a uniform 32 times 32times16 gridThe transmitter source is 119904(119905) = 119905

2119890minus120572119905 sin(120596

0119905) where 120572 =

1205960radic3 120596

0= 2120587119891

0 and 119891

0= 100MHz is the central

frequency In the last three 2D experiments 119873max = 4 andinternal Gauss-Newton iteration is 10 The inversion spaceΩ = 25m times 25m space step Δ119909 = Δ119910 = 005m samplingtime 119905 = 36 ns time step 120591 = 18 ns and the transmitter sourceis Ricker wavelet with 900MHz central frequency

41 HomogeneousModel Wefirst test a simple homogeneousmodel where permittivity 120576 = 56064 times 10

minus12 Fm perme-ability 120583 = 0 and relative conductivity 120590 = minus52983 BothdampingGauss-Newtonmethod and perturbation homotopymethod select the same relative initial value 120590 = minus48993Iterations of damping Gauss-Newton method are five timeselapsed time is 1230426 seconds homotopy algorithmrsquos outeriterations 120582

119894 119894 = 1 10 are ten times inner local

iterations (damping Gauss-Newton method) are three timesand elapsed time is 1405536 seconds Figure 2 gives resultsof true model damping Gauss-Newton and perturbationhomotopy method The relative error 120590inv minus 120590true120590trueof (a) and (b) in Figure 2 is 00516 and of (a) and (c) is00830 where 120590inv and 120590true are inversion solution and truemodel respectively From Figure 2 we can see that wheninitial value is close to original conductivity both methodshave satisfactory result but perturbation homotopy is notmore efficient

42 One Circle Cavity Model The relative conductivity ofcavity is minus32775 other parameters have the same value asin the above model Both methods select the same initialvalue 120590 = minus16094 Elapsed time of damping Gauss-Newtonmethod and perturbation homotopy method is 11120536seconds and 12578436 seconds respectively The relativeerror 120590inv minus 120590true120590true of (a) and (b) in Figure 3 is

8 Mathematical Problems in Engineeringy

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0204060810121416182022

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0204060810121416182022

Figure 12 Inversion results with different SNR measured data (1 dB 3 dB 5 dB 8 dB 10 dB 12 dB 15 dB 20 dB and 30 dB)

2535 and of (a) and (c) is 961 Although elapsed timeof perturbation homotopy is ten times more than dampingGauss-Newton method when initial value is far away fromtrue value the latter is inefficient compared to the former

43 Two Cavities Model The conductivity of two cavities isequal other parameters have the same value as in Section 42Both methods select the same relative initial value 120590 =

minus08775 Elapsed time of damping Gauss-Newton methodand perturbation homotopy method is 11080546 secondsand 15458768 seconds respectively Figure 4 gives results oforiginal model damping Gauss-Newton and perturbationhomotopy method The relative error 120590inv minus 120590true120590true of(a) and (b) in Figure 4 is 4935 and of (a) and (c) is 1285When the initial value is further away from original valuedamping Gauss-Newton method is invalid

44 One Square Cavity 2D Model The vertex coordinate ofsquare cavity is 12m 04m 13m 04m 12m 05m 13m05m respectively The original model structure schemeis shown in Figure 5 Inversion results by Gauss-Newtonmethod are shown in Figure 7 Measured data is shownin Figure 6 With the influence of local minima Figure 7

roughly describes the information of location and scale ofsquare cavity but the quality information error is hugeBecause traditional Tikhonov regularization would smooththe inversion solution the solution always appears blurringphenomenon at boundary Inversion results by perturbationhomotopy method with different homotopy parameter areshown in Figure 8 the relative error between inversion andoriginal model is 455 We select homotopy parameter as150001 15001 12001 11501 11001 1501 151 16 23100105 10001005 and 100000100005 From Figure 9 wecan conclude that the rate of convergence is fast but whenhomotopy parameter is close to 1 numerical solution isdiverging so the optimal solution is not at 120582 = 1

45 Three Square Cavities 2D Model The original modelstructure scheme is shown in Figure 10 Measured data isshown in Figure 11 In order to demonstrate that the methodis stable with noise we add different Gauss white noise tomeasured data The SNR (signal-to-noise ratio) is 1 dB 3 dB5 dB 8 dB 10 dB 12 dB 15 dB 20 dB and 30 dB respectivelyFrom Figure 12 we can see that with noise decrease numer-ical solution is convergent to original model So our methodis stable

Mathematical Problems in Engineering 9

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Figure 13 Original model

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minus1000

minus1500

minus500

Figure 14 Practice measured data

46 Practice Data Inversion The practice measured data isprovided by Wuhan Changjiang Engineering GeophysicalExploration amp Testing Co LtdChina The testing object isa concrete member with three circle rebar in it The centre ofcircle is 14m 06m 22m 04m 34m 07m respectivelyradius is 01mThe original model structure scheme is shownin Figure 13 Practice measured data is shown in Figure 14Inversion result is shown in Figure 15 The relative error is1575 Before the inversion it is necessary to preprocess thedata because practice measured data has large noise

5 Conclusion

This paper has addressed the perturbation homotopymethodfor solving inverse electromagnetic problem Perturbationhomotopy coupled with damping Gauss-Newton methodshas been used in the experiments Results show that wheninitial value is far away from true model local convergentGauss-Newton method just roughly gives the informationof location and scale the quality has huge error with true

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x (m)05 10 15 20 25

Figure 15 Inversion results

model Perturbation homotopy method is a widely conver-gent algorithm it could give amore accurate inversion resultseven if initial value is far away from the original value Whenadding Gauss white noise to measured data we can see thatwith noise decrease numerical solution is convergence tooriginal model so numerical results support that ourmethodis stable Traditional homotopy theory tells us that optimumsolution is determined at 120582 = 1 but numerical resultsshow that the solution will diverge when 120582 is close to 1because of regularization parameter impact In a word thismethod has clear advantages over damping Gauss-Newtonmethod it is an effective method especially on aspects ofwide convergence computational efficiency and precisionand has broad application prospects

Conflict of Interests

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

Acknowledgments

This paper is supported by Natural Science Foundation ofChina Project no 41304093 the Fundamental ResearchFunds for the Central Universities Project no DL12BB20and the Educational Commission of Heilongjiang Provinceof China no 12533013

References

[1] D Colton and R Kress Inverse Acoustic and ElectromagneticScattering Theory Springer New York NY USA 1998

[2] P A Eaton ldquo3-D electromagnetic inversion using integralequationsrdquo Geophysical Prospecting vol 37 pp 407ndash426 1989

[3] J Nocedal and S Wright Numerical Optimization SpringerNew York NY USA 1999

[4] G A Newman S Recher B Tezkan and F M Neubauerldquo3D inversion of a scalar radio magnetotelluric field data setrdquoGeophysics vol 68 no 3 pp 791ndash802 2003

10 Mathematical Problems in Engineering

[5] E Haber ldquoQuasi-Newton methods for large-scale electromag-netic inverse problemsrdquo Inverse Problems vol 21 no 1 pp 305ndash323 2005

[6] M S Zhdanov S Fang and G Hursan ldquoElectromagneticinversion using quasi-linear approximationrdquo Geophysics vol65 no 5 pp 1501ndash1513 2000

[7] C Torres-Verdın and T M Habashy ldquoRapid numerical sim-ulation of axisymmetric single-well induction data using theextended Born approximationrdquo Radio Science vol 36 no 6 pp1287ndash1306 2001

[8] H-W Tseng K H Lee and A Becker ldquo3D interpretation ofelectromagnetic data using a modified extended Born approxi-mationrdquo Geophysics vol 68 no 1 pp 127ndash137 2003

[9] Z Zhang ldquo3D resistivity mapping of airborne EM datardquoGeophysics vol 68 no 6 pp 1896ndash1905 2003

[10] M S Zhdanov Geophysical Inverse Theory and RegularizationProblems Elsevier Amsterdam The Netherlands 2002

[11] M S Zhdanov Geophysical Electromagnetic Theory and Meth-ods Elsevier Amsterdam The Netherlands 2009

[12] H B Han Bo K Z Kuang Zheng and L J-Q Liu Jia-Qi ldquoAmonotonous homotopy method for solving the resistivities ofthe earthrdquo Acta Geophysica Sinica vol 34 no 4 pp 517ndash5221991

[13] H S Fu BHan andGQGai ldquoAwaveletmultiscale-homotopymethod for the inverse problem of two-dimensional acousticwave equationrdquoAppliedMathematics andComputation vol 190no 1 pp 576ndash582 2007

[14] Y He and B Han ldquoA wavelet adaptive-homotopy method forinverse problem in the fluid-saturated porous mediardquo AppliedMathematics andComputation vol 208 no 1 pp 189ndash196 2009

[15] D W Vasco ldquoSingularity and branching a path-following for-malism for geophysical inverse problemsrdquo Geophysical JournalInternational vol 119 no 3 pp 809ndash830 1994

[16] D M Dunlavy D P OrsquoLeary D Klimov and D ThirumalaildquoHOPE a homotopy optimizationmethod for protein structurepredictionrdquo Journal of Computational Biology vol 12 no 10 pp1275ndash1288 2005

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

Global optimal solution

Stop critrerion 1

NoYes

Stop critrerion 2

YesNo

Input source Sr reference model mref measured datadobs and k = 1 0 = 1205820 le 1205821 le middot middot middot le 120582n = 1

CalculatedJ(mk0)data error r(mk0) = F(mk0) minus dobs

and regularization parameter 120572 = argminGCV(120572)

j = 0

Gauss-Newton iteration mkj+1 = mkj + Q

j = j + 1

mk+10 = mkj and mk+10 = mkj

120575 k = k + 1

Note mkj+1

120575as

perturbation of mkj+1

Nmax is limit of set

Nmax

Figure 1 The sketch map of perturbation homotopy method

in 1991 [12] Afterwards they extended thismethod to generalparameter identification Fu et al gave the numerical solutionof inverse acoustic problem and elastic wave equation withfluid saturated porous media by homotopy method respec-tively [13 14] Vasco applied homotopymethod to geophysicsinverse problem in 1994 [15] He makes use of singularityand bifurcation theory to research inverse problem Dunlavyapplied homotopy optimization method to protein structureprediction in 2005 [16] In summary the homotopy methodhas shown great advantage for solving inverse problemespecially in aspects of enlarging the convergence domainand enhancing stability with noise But homotopy inversionmethod is still far away from perfection and bifurcationand convolution about homotopy curves will appear in theprocess of computation

As mentioned above we will introduce a perturbationhomotopy method for inverse electromagnetic problem Themodel is Maxwellrsquos equations In each iterative step of thealgorithm we add stochastic perturbation to numericalsolutions The previous solution and perturbation solutionare regarded as the initial value in the next iteration Althoughthe number of solution in set increased it increased thelikelihood of obtaining correct solution With the number ofelements in set increasing we require the input value 119873max(maximum number of solutions in a set) to limit the size ofset

The paper is organized as follows We first briefly reviewsome basic ideas about inverse electromagnetic problem and

homotopy In Section 2 we develop perturbation homotopymethod and derive the basic coefficient identification algo-rithm using this method In Section 3 numerical experi-ments are presented which demonstrate the performance ofthe algorithm and indicate the validity of this method Andfinally in Section 4 some conclusions and future directionsare given

2 Electromagnetic Forward Model

The time domain Maxwell equations has the form

nabla times E + 120583120597H120597119905

= 0

nabla timesH minus 120590E minus 120576120597E120597119905

= 119904119903 (119905)

(1)

over the domainΩtimes[0 119879] whereE andH are the electric andmagnetic fields 120590 120576 and 120583 are conductivity dielectric con-stant and permeability respectively and 119904

119903(119905) is transmitting

source Discretizing (1) over time grid [119905119899 119905119899+1

] we get theequations

nabla times E119899+1 + 120591119899120583H119899+1 = 120591

119899120583H119899 in Ω

nabla timesH119899+1 minus (120590 + 120591119899120576)E119899+1 = 119904

119899+1

119903minus 120591119899120576E119899 in Ω

(2)

Mathematical Problems in Engineering 3

05 05

2

4

6

8

10

12

X Y

Z

minus5

minus5

(a)

2

4

6

8

10

12

Z

05

Y minus5

05 X

minus5

(b)

2

4

6

8

10

12

Z

05 X

minus505

Yminus5

(c)

Figure 2 (a) Original model (b) inversion results of damping Gauss-Newton (c) inversion results of perturbation homotopy method

05 05

2

4

6

8

10

12

X Y

Z

0

50

100

150

minus5minus5

(a)

05 05

2

4

6

8

10

12

X

Z

0

50

100

150

Y

minus5minus5

(b)

05 05

2

4

6

8

10

12

XY

Z

0

50

100

150

minus5minus5

(c)

10 20 30

5

10

15

20

25

30

0

50

100

150

(d)

10 20 30

5

10

15

20

25

30

0

50

100

150

(e)

10 20 30

5

10

15

20

25

30

0

50

100

150

(f)

Figure 3 (a) True model (b) results of damping Gauss-Newton (c) results of perturbation homotopy method (d) cross-section of (a) at119909 = 0 (e) cross-section of (b) at 119909 = 0 (f) cross-section of (c) at 119909 = 0

4 Mathematical Problems in Engineering

05 05

2

4

6

8

10

120

50

100

150

minus5minus5

Z

XY

(a)

05

2

4

6

8

10

120

50

100

150

Z

X

05 minus5minus5

Y

(b)

05 05

2

4

6

8

10

120

50

100

150

Z

Xminus5

minus5

Y

(c)

10 20 30

2

4

6

8

10

12

14

16

0

50

100

150

(d)

10 20 30

2

4

6

8

10

12

14

16

0

50

100

150

(e)

10 20 30

2

4

6

8

10

12

14

16

0

50

100

150

(f)

Figure 4 (a) True model (b) results of damping Gauss-Newton method (c) results of perturbation homotopy method (d) cross-section of(a) at 119885 = 8 (e) cross-section of (b) at 119885 = 8 (f) cross-section of (c) at 119885 = 8

02

04

06

08

10

12

14

16

18

20

22

x (m)

y(m

)

05 10 15 20 252

25

3

35

4

45

5

Figure 5 Square cavity model

where 120591119899= 1(119905

119899+1minus 119905119899) With discretization we get matrix

equation

119860 (120590) 119906 = 119902 (3)

20 40 60 80 100

0

500

1000

1500

2000

t(n

s)

72

144

216

288

36

Trace

minus1000

minus500

Figure 6 Measured data

In the inverse problem we want to recover the model120590 with the measured data Measured data can be written as

119889obs

= 119876119906 + 120578 (4)

Mathematical Problems in Engineering 5

x (m)

y(m

)

05 10 15 20 25

02

04

06

08

10

12

14

16

18

20

222

25

3

35

4

45

5

Figure 7 Inversion results of Gauss-Newton method

y(m

)

02

04

06

08

10

12

14

16

18

20

22

x (m)05 10 15 20 25

2

25

3

35

4

45

5

Figure 8 Inversion results of perturbation homotopy method

where 119876 is a projection operator which projects the electro-magnetic fields onto the observation locations and 120578 is themeasurement noise From (3) and (4) we can obtain

119889obs

= 119876119860minus1119902 + 120578 (5)

3 Inversion Framework

31 Damping Gauss-Newton Method For every homotopyparameter 120582 = 120582

119894 119894 = 1 119871 we must solve the

unconstrained optimization problem

min120590

119867(120590 120582) = min120590

12058210038171003817100381710038171003817119865 (120590) minus 119889

obs10038171003817100381710038171003817

2

+ (1 minus 120582)1003817100381710038171003817119882 (120590 minus 120590ref)

10038171003817100381710038172

(6)

This can be transformed to the Euler-Lagrange system

120597119867

120597120590= 119892 (120590) = (1 minus 120582)119882

119879119882(120590 minus 120590ref)

+ 120582119869(120590)119879(119865 (120590) minus 119889

obs)

(7)

Denoting 120573 = (1 minus 120582)120582 we obtain

120573119882119879119882(120590 minus 120590ref) + 119869(120590)

119879(119865 (120590) minus 119889

obs) = 0 (8)

where 120573 played a regularization parameter role In order toavoid the calculation of the second Frechet derivative of 119865 to120590 119865 is linearized by

119865 (120590 + 120575120590) = 119865 (120590) + 119869 (120590) 120575120590 + 119877 (120590 120575120590) (9)

Then the minimization problem is solved iteratively At the119896th iteration we solve

(119869 (120590119879

119896) 119869 (120590119896) + 120573119882

119879119882)120575120590

= 119869 (120590119879

119896) (119889

obsminus 119865 (120590

119896) + 119869 (120590

119896)) minus 120573119882

119879119882(120590119896minus 120590ref)

(10)

to find the perturbation 120575120590 At each iteration we have anoption of solving directly for a perturbation or solving foran updated model To formulate the latter option we write120590119896+1

= 120590119896+ 120575120590 and substitute into (10) to obtain

(119869 (120590119879

119896) 119869 (120590119896) + 120573119882

119879119882)120590119896+1

= 119869 (120590119879

119896) (119889

obsminus 119865 (120590

119896) +119869 (120590

119896) 120590119896) minus 120573119882

119879119882120590ref

(11)

32 A Perturbation Homotopy Method As above mentionedwe can formulate the inverse electromagnetic problem as thefollowing nonlinear operator equation

119865 (120590) = 119889obs

(12)

where 119865 = 119876119860minus1119902 is the forward nonlinear operator

It is very difficult to estimate 120590 in the problem (12) dueto the presence of local minima Therefore we design aglobally convergent algorithm which can overcome the localminima namely perturbation homotopy method to solvethe nonlinear equation (12) For inverse electromagneticproblem we want to solve the minimization problem Given120601 119883 sube 119877

119899rarr 119877 find 120590

lowastisin 119883 sube 119877

119899 such that

120601 (120590lowast) = min120590isin119883

10038171003817100381710038171003817119865 (120590) minus 119889

obs10038171003817100381710038171003817

2

(13)

In order to apply homotopy to PDEoptimization problemwedenote

1206010=1003817100381710038171003817119882 (120590 minus 120590ref)

10038171003817100381710038172

1206011=10038171003817100381710038171003817119865 (120590) minus 119889

obs10038171003817100381710038171003817

2

(14)

where sdot is 2-norm 119882 is positive weight function and120590ref is the known reference model So continuous homotopyfunctional119867 119877

119899+1rarr 119877 can be written as

119867(120590 120582) = 1205821206011+ (1 minus 120582) 120601

0 (15)

where120582 isin [0 1] are homotopy parameters and the homotopystep length Δ

119894 119894 = 0 119871 minus 1 is satisfied with sum

119871minus1

119894=0Δ119894= 1

A general sketch of the homotopy algorithm is as follows

6 Mathematical Problems in Engineeringy

(m)

y(m

)

x (m)05 10 15 20 25

x (m)05 10 15 20 25

2

25

3

35

4

45

5

2

25

3

35

4

45

5

2

25

3

35

4

45

5

2

25

3

35

4

45

5

2

25

3

35

4

45

5

2

25

3

35

4

45

5

2

25

3

35

4

45

5

2

25

3

35

4

45

5

2

25

3

35

4

45

5

2

25

3

35

4

45

5

2

25

3

35

4

45

5

2

25

3

35

4

45

5

y(m

)

x (m)05 10 15 20 25

y(m

)y

(m)

x (m)05 10 15 20 25

x (m)05 10 15 20 25

y(m

)

x (m)05 10 15 20 25

y(m

)

x (m)05 10 15 20 25

y(m

)

x (m)05 10 15 20 25

y(m

)

x (m)05 10 15 20 25

y(m

)

x (m)05 10 15 20 25

y(m

)

x (m)05 10 15 20 25

y(m

)

x (m)05 10 15 20 25

0204060810121416182022

0204060810121416182022

0204060810121416182022

0204060810121416182022

0204060810121416182022

0204060810121416182022

0204060810121416182022

0204060810121416182022

0204060810121416182022

0204060810121416182022

0204060810121416182022

0204060810121416182022

Figure 9 Inversion results with different homotopy parameters (150001 15001 12001 11501 11001 1501 151 16 23 100105 10001005and 100000100005)

(i) Input given minimum norm solution of 1206010 namely1205900= 120590ref 1205820 = 0 and Δ

119894gt 0 119894 = 1 119871

(ii) Update for 120582119894= 120582119894minus1

+ Δ119894minus1

we use damping Gauss-Newton iteration method compute an approximatesolution 120590

119894to minimize min

120590isin119883119867(120590 120582

119894) by initial

value 120590119894minus1

(iii) Output the iteration will be stopped if the solution 120590

119894

satisfied stopping criterion output 1205901 = 120590119898

The homotopy algorithmrsquos greatest advantage lies in increas-ing region of convergency But in the process of computationthe nonconvergent phenomenon often happens to homotopycurve for example standstill convolution and bifurcation

These difficulties can make the homotopy method quitecomplicated In order to make the homotopy more efficientsets of solutions are produced by perturbation for everyhomotopy parameter 120582 = 120582

119894 119894 = 0 119871 Although the

calculation increased it ensures the likelihood of obtainingaccurate solutionWith different homotopy parameter 120582

119894 the

minima is in the sequence and120590119894is solved by dampingGauss-

Newton method with the initial value 120590119894minus1

The next set oflocal minima (120582 = 120582

119894+1) is solved with the initial values in the

previous set (120582 = 120582119894) which has one or more perturbations of

each of those points in the set We denote the perturbationof solution per(120590) The perturbation stochastically disturbsone or more of the solutions 120590 in set We use a single initial

Mathematical Problems in Engineering 7

02

04

06

08

10

12

14

16

18

20

22

x (m)

y(m

)

05 10 15 20 252

25

3

35

4

45

5

Figure 10 Three square cavities model

20 40 60 80 100

0

500

1000

1500

2000

t(n

s)

72

144

216

288

36

Trace

minus1000

minus500

Figure 11 Measured data

value 1205900for 120582 = 120582

0 the number of solution solved at

each homotopy parameter120582 increases exponentiallyWithoutlimit on the size of the set the number of solutions in theset would be 2

119894 when homotopy iteration stops In order toconstraint on computational complexity it is necessary tolimit the size of the set We denote by 119873max the maximumnumber of solutions in a set If the number of local minimain the set at the current iteration is less than the maximumset size 119873max then every solution in the set is used inthe next iteration otherwise we will choose the preferablesolution An ideal measure of what constitutes the preferablesolutions is regularization functional value of (15) solutionwith the lower regularization functional values is consideredthe preferable The perturbation homotopy algorithm is asfollows

(i) Input given minimum norm solution of 1206010 namely1205900= 120590ref 1205820 = 0119873max ge 1 and Δ

119894gt 0 119894 = 1 119871

(ii) Update for 120582119894

= 120582119894minus1

+ Δ119894minus1

we use damp-ing Gauss-Newton iteration method compute anapproximate solution 120590

119896

119894 119896 = 1 2

119894 to minimizemin120590isin119883

119867(120590 120582119894) by initial value 120590119895

119894minus1and (per(120590

119894minus1))119895

119895 = 1 2119894

(iii) Order the distinct solutions among 120590119896

119894 119896 = 1 2

119894from best to worst as 1205901

119894 1205902119894 and discard any worst

solution when the number of elements in 119883 largerthan119873max

(iv) Output the iteration will be stopped if the solution120590119898

119894 1 le 119898 le 2

119894 is the best solution in the set andsatisfied stopping criterion output 1205901 = 120590

119898

The sketch map of the algorithm is shown in Figure 1

4 Numerical Simulation

In this section we exhibit clear advantages of perturbationhomotopy method over traditional damping Gauss-Newtonmethod in the previous three experiments In the fourthexperiment we analysis the influence of homotopy parameteron inversion results In the fifth experiment we apply ouralgorithm to more complicated model and confirm that thisalgorithm is stable In the last experiment we solve a simplepractice data In the first three 3D experiments we selecthomotopy parameter 120582

119894= 11989410 119894 = 0 10 119873max = 2 and

internal Gauss-Newton iteration 5 with each homotopy stepin our numerical experiments The space domain is Ω Theinverse problem is considered on a uniform 32 times 32times16 gridThe transmitter source is 119904(119905) = 119905

2119890minus120572119905 sin(120596

0119905) where 120572 =

1205960radic3 120596

0= 2120587119891

0 and 119891

0= 100MHz is the central

frequency In the last three 2D experiments 119873max = 4 andinternal Gauss-Newton iteration is 10 The inversion spaceΩ = 25m times 25m space step Δ119909 = Δ119910 = 005m samplingtime 119905 = 36 ns time step 120591 = 18 ns and the transmitter sourceis Ricker wavelet with 900MHz central frequency

41 HomogeneousModel Wefirst test a simple homogeneousmodel where permittivity 120576 = 56064 times 10

minus12 Fm perme-ability 120583 = 0 and relative conductivity 120590 = minus52983 BothdampingGauss-Newtonmethod and perturbation homotopymethod select the same relative initial value 120590 = minus48993Iterations of damping Gauss-Newton method are five timeselapsed time is 1230426 seconds homotopy algorithmrsquos outeriterations 120582

119894 119894 = 1 10 are ten times inner local

iterations (damping Gauss-Newton method) are three timesand elapsed time is 1405536 seconds Figure 2 gives resultsof true model damping Gauss-Newton and perturbationhomotopy method The relative error 120590inv minus 120590true120590trueof (a) and (b) in Figure 2 is 00516 and of (a) and (c) is00830 where 120590inv and 120590true are inversion solution and truemodel respectively From Figure 2 we can see that wheninitial value is close to original conductivity both methodshave satisfactory result but perturbation homotopy is notmore efficient

42 One Circle Cavity Model The relative conductivity ofcavity is minus32775 other parameters have the same value asin the above model Both methods select the same initialvalue 120590 = minus16094 Elapsed time of damping Gauss-Newtonmethod and perturbation homotopy method is 11120536seconds and 12578436 seconds respectively The relativeerror 120590inv minus 120590true120590true of (a) and (b) in Figure 3 is

8 Mathematical Problems in Engineeringy

(m)

x (m)05 10 15 20 25

2

25

3

35

4

45

5

2

25

3

35

4

45

5

2

25

3

35

4

45

5

2

25

3

35

4

45

5

2

25

3

35

4

45

5

2

25

3

35

4

45

5

2

25

3

35

4

45

5

2

25

3

35

4

45

5

2

25

3

35

4

45

5

0204060810121416182022

x (m)05 10 15 20 25

y(m

)

0204060810121416182022

x (m)05 10 15 20 25

y(m

)

0204060810121416182022

x (m)05 10 15 20 25

y(m

)

0204060810121416182022

x (m)05 10 15 20 25

y(m

)

0204060810121416182022

x (m)05 10 15 20 25

y(m

)

0204060810121416182022

x (m)05 10 15 20 25

y(m

)

0204060810121416182022

y(m

)

x (m)05 10 15 20 25

0204060810121416182022

y(m

)

x (m)05 10 15 20 25

0204060810121416182022

Figure 12 Inversion results with different SNR measured data (1 dB 3 dB 5 dB 8 dB 10 dB 12 dB 15 dB 20 dB and 30 dB)

2535 and of (a) and (c) is 961 Although elapsed timeof perturbation homotopy is ten times more than dampingGauss-Newton method when initial value is far away fromtrue value the latter is inefficient compared to the former

43 Two Cavities Model The conductivity of two cavities isequal other parameters have the same value as in Section 42Both methods select the same relative initial value 120590 =

minus08775 Elapsed time of damping Gauss-Newton methodand perturbation homotopy method is 11080546 secondsand 15458768 seconds respectively Figure 4 gives results oforiginal model damping Gauss-Newton and perturbationhomotopy method The relative error 120590inv minus 120590true120590true of(a) and (b) in Figure 4 is 4935 and of (a) and (c) is 1285When the initial value is further away from original valuedamping Gauss-Newton method is invalid

44 One Square Cavity 2D Model The vertex coordinate ofsquare cavity is 12m 04m 13m 04m 12m 05m 13m05m respectively The original model structure schemeis shown in Figure 5 Inversion results by Gauss-Newtonmethod are shown in Figure 7 Measured data is shownin Figure 6 With the influence of local minima Figure 7

roughly describes the information of location and scale ofsquare cavity but the quality information error is hugeBecause traditional Tikhonov regularization would smooththe inversion solution the solution always appears blurringphenomenon at boundary Inversion results by perturbationhomotopy method with different homotopy parameter areshown in Figure 8 the relative error between inversion andoriginal model is 455 We select homotopy parameter as150001 15001 12001 11501 11001 1501 151 16 23100105 10001005 and 100000100005 From Figure 9 wecan conclude that the rate of convergence is fast but whenhomotopy parameter is close to 1 numerical solution isdiverging so the optimal solution is not at 120582 = 1

45 Three Square Cavities 2D Model The original modelstructure scheme is shown in Figure 10 Measured data isshown in Figure 11 In order to demonstrate that the methodis stable with noise we add different Gauss white noise tomeasured data The SNR (signal-to-noise ratio) is 1 dB 3 dB5 dB 8 dB 10 dB 12 dB 15 dB 20 dB and 30 dB respectivelyFrom Figure 12 we can see that with noise decrease numer-ical solution is convergent to original model So our methodis stable

Mathematical Problems in Engineering 9

1 2 3 4 5

50

100

150

200

250

300

350

400

450

500

y(m

)

x (m)

01

02

03

04

05

06

07

08

09

10

Figure 13 Original model

10 20 30 40 50 60 70 80 90 100

3

6

9

12

15

0

500

1000

1500

t(n

s)

Trace

minus1000

minus1500

minus500

Figure 14 Practice measured data

46 Practice Data Inversion The practice measured data isprovided by Wuhan Changjiang Engineering GeophysicalExploration amp Testing Co LtdChina The testing object isa concrete member with three circle rebar in it The centre ofcircle is 14m 06m 22m 04m 34m 07m respectivelyradius is 01mThe original model structure scheme is shownin Figure 13 Practice measured data is shown in Figure 14Inversion result is shown in Figure 15 The relative error is1575 Before the inversion it is necessary to preprocess thedata because practice measured data has large noise

5 Conclusion

This paper has addressed the perturbation homotopymethodfor solving inverse electromagnetic problem Perturbationhomotopy coupled with damping Gauss-Newton methodshas been used in the experiments Results show that wheninitial value is far away from true model local convergentGauss-Newton method just roughly gives the informationof location and scale the quality has huge error with true

50

100

150

200

250

300

350

400

450

500

y(m

)

01

02

03

04

05

06

07

08

09

10

x (m)05 10 15 20 25

Figure 15 Inversion results

model Perturbation homotopy method is a widely conver-gent algorithm it could give amore accurate inversion resultseven if initial value is far away from the original value Whenadding Gauss white noise to measured data we can see thatwith noise decrease numerical solution is convergence tooriginal model so numerical results support that ourmethodis stable Traditional homotopy theory tells us that optimumsolution is determined at 120582 = 1 but numerical resultsshow that the solution will diverge when 120582 is close to 1because of regularization parameter impact In a word thismethod has clear advantages over damping Gauss-Newtonmethod it is an effective method especially on aspects ofwide convergence computational efficiency and precisionand has broad application prospects

Conflict of Interests

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

Acknowledgments

This paper is supported by Natural Science Foundation ofChina Project no 41304093 the Fundamental ResearchFunds for the Central Universities Project no DL12BB20and the Educational Commission of Heilongjiang Provinceof China no 12533013

References

[1] D Colton and R Kress Inverse Acoustic and ElectromagneticScattering Theory Springer New York NY USA 1998

[2] P A Eaton ldquo3-D electromagnetic inversion using integralequationsrdquo Geophysical Prospecting vol 37 pp 407ndash426 1989

[3] J Nocedal and S Wright Numerical Optimization SpringerNew York NY USA 1999

[4] G A Newman S Recher B Tezkan and F M Neubauerldquo3D inversion of a scalar radio magnetotelluric field data setrdquoGeophysics vol 68 no 3 pp 791ndash802 2003

10 Mathematical Problems in Engineering

[5] E Haber ldquoQuasi-Newton methods for large-scale electromag-netic inverse problemsrdquo Inverse Problems vol 21 no 1 pp 305ndash323 2005

[6] M S Zhdanov S Fang and G Hursan ldquoElectromagneticinversion using quasi-linear approximationrdquo Geophysics vol65 no 5 pp 1501ndash1513 2000

[7] C Torres-Verdın and T M Habashy ldquoRapid numerical sim-ulation of axisymmetric single-well induction data using theextended Born approximationrdquo Radio Science vol 36 no 6 pp1287ndash1306 2001

[8] H-W Tseng K H Lee and A Becker ldquo3D interpretation ofelectromagnetic data using a modified extended Born approxi-mationrdquo Geophysics vol 68 no 1 pp 127ndash137 2003

[9] Z Zhang ldquo3D resistivity mapping of airborne EM datardquoGeophysics vol 68 no 6 pp 1896ndash1905 2003

[10] M S Zhdanov Geophysical Inverse Theory and RegularizationProblems Elsevier Amsterdam The Netherlands 2002

[11] M S Zhdanov Geophysical Electromagnetic Theory and Meth-ods Elsevier Amsterdam The Netherlands 2009

[12] H B Han Bo K Z Kuang Zheng and L J-Q Liu Jia-Qi ldquoAmonotonous homotopy method for solving the resistivities ofthe earthrdquo Acta Geophysica Sinica vol 34 no 4 pp 517ndash5221991

[13] H S Fu BHan andGQGai ldquoAwaveletmultiscale-homotopymethod for the inverse problem of two-dimensional acousticwave equationrdquoAppliedMathematics andComputation vol 190no 1 pp 576ndash582 2007

[14] Y He and B Han ldquoA wavelet adaptive-homotopy method forinverse problem in the fluid-saturated porous mediardquo AppliedMathematics andComputation vol 208 no 1 pp 189ndash196 2009

[15] D W Vasco ldquoSingularity and branching a path-following for-malism for geophysical inverse problemsrdquo Geophysical JournalInternational vol 119 no 3 pp 809ndash830 1994

[16] D M Dunlavy D P OrsquoLeary D Klimov and D ThirumalaildquoHOPE a homotopy optimizationmethod for protein structurepredictionrdquo Journal of Computational Biology vol 12 no 10 pp1275ndash1288 2005

Submit your manuscripts athttpwwwhindawicom

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

05 05

2

4

6

8

10

12

X Y

Z

minus5

minus5

(a)

2

4

6

8

10

12

Z

05

Y minus5

05 X

minus5

(b)

2

4

6

8

10

12

Z

05 X

minus505

Yminus5

(c)

Figure 2 (a) Original model (b) inversion results of damping Gauss-Newton (c) inversion results of perturbation homotopy method

05 05

2

4

6

8

10

12

X Y

Z

0

50

100

150

minus5minus5

(a)

05 05

2

4

6

8

10

12

X

Z

0

50

100

150

Y

minus5minus5

(b)

05 05

2

4

6

8

10

12

XY

Z

0

50

100

150

minus5minus5

(c)

10 20 30

5

10

15

20

25

30

0

50

100

150

(d)

10 20 30

5

10

15

20

25

30

0

50

100

150

(e)

10 20 30

5

10

15

20

25

30

0

50

100

150

(f)

Figure 3 (a) True model (b) results of damping Gauss-Newton (c) results of perturbation homotopy method (d) cross-section of (a) at119909 = 0 (e) cross-section of (b) at 119909 = 0 (f) cross-section of (c) at 119909 = 0

4 Mathematical Problems in Engineering

05 05

2

4

6

8

10

120

50

100

150

minus5minus5

Z

XY

(a)

05

2

4

6

8

10

120

50

100

150

Z

X

05 minus5minus5

Y

(b)

05 05

2

4

6

8

10

120

50

100

150

Z

Xminus5

minus5

Y

(c)

10 20 30

2

4

6

8

10

12

14

16

0

50

100

150

(d)

10 20 30

2

4

6

8

10

12

14

16

0

50

100

150

(e)

10 20 30

2

4

6

8

10

12

14

16

0

50

100

150

(f)

Figure 4 (a) True model (b) results of damping Gauss-Newton method (c) results of perturbation homotopy method (d) cross-section of(a) at 119885 = 8 (e) cross-section of (b) at 119885 = 8 (f) cross-section of (c) at 119885 = 8

02

04

06

08

10

12

14

16

18

20

22

x (m)

y(m

)

05 10 15 20 252

25

3

35

4

45

5

Figure 5 Square cavity model

where 120591119899= 1(119905

119899+1minus 119905119899) With discretization we get matrix

equation

119860 (120590) 119906 = 119902 (3)

20 40 60 80 100

0

500

1000

1500

2000

t(n

s)

72

144

216

288

36

Trace

minus1000

minus500

Figure 6 Measured data

In the inverse problem we want to recover the model120590 with the measured data Measured data can be written as

119889obs

= 119876119906 + 120578 (4)

Mathematical Problems in Engineering 5

x (m)

y(m

)

05 10 15 20 25

02

04

06

08

10

12

14

16

18

20

222

25

3

35

4

45

5

Figure 7 Inversion results of Gauss-Newton method

y(m

)

02

04

06

08

10

12

14

16

18

20

22

x (m)05 10 15 20 25

2

25

3

35

4

45

5

Figure 8 Inversion results of perturbation homotopy method

where 119876 is a projection operator which projects the electro-magnetic fields onto the observation locations and 120578 is themeasurement noise From (3) and (4) we can obtain

119889obs

= 119876119860minus1119902 + 120578 (5)

3 Inversion Framework

31 Damping Gauss-Newton Method For every homotopyparameter 120582 = 120582

119894 119894 = 1 119871 we must solve the

unconstrained optimization problem

min120590

119867(120590 120582) = min120590

12058210038171003817100381710038171003817119865 (120590) minus 119889

obs10038171003817100381710038171003817

2

+ (1 minus 120582)1003817100381710038171003817119882 (120590 minus 120590ref)

10038171003817100381710038172

(6)

This can be transformed to the Euler-Lagrange system

120597119867

120597120590= 119892 (120590) = (1 minus 120582)119882

119879119882(120590 minus 120590ref)

+ 120582119869(120590)119879(119865 (120590) minus 119889

obs)

(7)

Denoting 120573 = (1 minus 120582)120582 we obtain

120573119882119879119882(120590 minus 120590ref) + 119869(120590)

119879(119865 (120590) minus 119889

obs) = 0 (8)

where 120573 played a regularization parameter role In order toavoid the calculation of the second Frechet derivative of 119865 to120590 119865 is linearized by

119865 (120590 + 120575120590) = 119865 (120590) + 119869 (120590) 120575120590 + 119877 (120590 120575120590) (9)

Then the minimization problem is solved iteratively At the119896th iteration we solve

(119869 (120590119879

119896) 119869 (120590119896) + 120573119882

119879119882)120575120590

= 119869 (120590119879

119896) (119889

obsminus 119865 (120590

119896) + 119869 (120590

119896)) minus 120573119882

119879119882(120590119896minus 120590ref)

(10)

to find the perturbation 120575120590 At each iteration we have anoption of solving directly for a perturbation or solving foran updated model To formulate the latter option we write120590119896+1

= 120590119896+ 120575120590 and substitute into (10) to obtain

(119869 (120590119879

119896) 119869 (120590119896) + 120573119882

119879119882)120590119896+1

= 119869 (120590119879

119896) (119889

obsminus 119865 (120590

119896) +119869 (120590

119896) 120590119896) minus 120573119882

119879119882120590ref

(11)

32 A Perturbation Homotopy Method As above mentionedwe can formulate the inverse electromagnetic problem as thefollowing nonlinear operator equation

119865 (120590) = 119889obs

(12)

where 119865 = 119876119860minus1119902 is the forward nonlinear operator

It is very difficult to estimate 120590 in the problem (12) dueto the presence of local minima Therefore we design aglobally convergent algorithm which can overcome the localminima namely perturbation homotopy method to solvethe nonlinear equation (12) For inverse electromagneticproblem we want to solve the minimization problem Given120601 119883 sube 119877

119899rarr 119877 find 120590

lowastisin 119883 sube 119877

119899 such that

120601 (120590lowast) = min120590isin119883

10038171003817100381710038171003817119865 (120590) minus 119889

obs10038171003817100381710038171003817

2

(13)

In order to apply homotopy to PDEoptimization problemwedenote

1206010=1003817100381710038171003817119882 (120590 minus 120590ref)

10038171003817100381710038172

1206011=10038171003817100381710038171003817119865 (120590) minus 119889

obs10038171003817100381710038171003817

2

(14)

where sdot is 2-norm 119882 is positive weight function and120590ref is the known reference model So continuous homotopyfunctional119867 119877

119899+1rarr 119877 can be written as

119867(120590 120582) = 1205821206011+ (1 minus 120582) 120601

0 (15)

where120582 isin [0 1] are homotopy parameters and the homotopystep length Δ

119894 119894 = 0 119871 minus 1 is satisfied with sum

119871minus1

119894=0Δ119894= 1

A general sketch of the homotopy algorithm is as follows

6 Mathematical Problems in Engineeringy

(m)

y(m

)

x (m)05 10 15 20 25

x (m)05 10 15 20 25

2

25

3

35

4

45

5

2

25

3

35

4

45

5

2

25

3

35

4

45

5

2

25

3

35

4

45

5

2

25

3

35

4

45

5

2

25

3

35

4

45

5

2

25

3

35

4

45

5

2

25

3

35

4

45

5

2

25

3

35

4

45

5

2

25

3

35

4

45

5

2

25

3

35

4

45

5

2

25

3

35

4

45

5

y(m

)

x (m)05 10 15 20 25

y(m

)y

(m)

x (m)05 10 15 20 25

x (m)05 10 15 20 25

y(m

)

x (m)05 10 15 20 25

y(m

)

x (m)05 10 15 20 25

y(m

)

x (m)05 10 15 20 25

y(m

)

x (m)05 10 15 20 25

y(m

)

x (m)05 10 15 20 25

y(m

)

x (m)05 10 15 20 25

y(m

)

x (m)05 10 15 20 25

0204060810121416182022

0204060810121416182022

0204060810121416182022

0204060810121416182022

0204060810121416182022

0204060810121416182022

0204060810121416182022

0204060810121416182022

0204060810121416182022

0204060810121416182022

0204060810121416182022

0204060810121416182022

Figure 9 Inversion results with different homotopy parameters (150001 15001 12001 11501 11001 1501 151 16 23 100105 10001005and 100000100005)

(i) Input given minimum norm solution of 1206010 namely1205900= 120590ref 1205820 = 0 and Δ

119894gt 0 119894 = 1 119871

(ii) Update for 120582119894= 120582119894minus1

+ Δ119894minus1

we use damping Gauss-Newton iteration method compute an approximatesolution 120590

119894to minimize min

120590isin119883119867(120590 120582

119894) by initial

value 120590119894minus1

(iii) Output the iteration will be stopped if the solution 120590

119894

satisfied stopping criterion output 1205901 = 120590119898

The homotopy algorithmrsquos greatest advantage lies in increas-ing region of convergency But in the process of computationthe nonconvergent phenomenon often happens to homotopycurve for example standstill convolution and bifurcation

These difficulties can make the homotopy method quitecomplicated In order to make the homotopy more efficientsets of solutions are produced by perturbation for everyhomotopy parameter 120582 = 120582

119894 119894 = 0 119871 Although the

calculation increased it ensures the likelihood of obtainingaccurate solutionWith different homotopy parameter 120582

119894 the

minima is in the sequence and120590119894is solved by dampingGauss-

Newton method with the initial value 120590119894minus1

The next set oflocal minima (120582 = 120582

119894+1) is solved with the initial values in the

previous set (120582 = 120582119894) which has one or more perturbations of

each of those points in the set We denote the perturbationof solution per(120590) The perturbation stochastically disturbsone or more of the solutions 120590 in set We use a single initial

Mathematical Problems in Engineering 7

02

04

06

08

10

12

14

16

18

20

22

x (m)

y(m

)

05 10 15 20 252

25

3

35

4

45

5

Figure 10 Three square cavities model

20 40 60 80 100

0

500

1000

1500

2000

t(n

s)

72

144

216

288

36

Trace

minus1000

minus500

Figure 11 Measured data

value 1205900for 120582 = 120582

0 the number of solution solved at

each homotopy parameter120582 increases exponentiallyWithoutlimit on the size of the set the number of solutions in theset would be 2

119894 when homotopy iteration stops In order toconstraint on computational complexity it is necessary tolimit the size of the set We denote by 119873max the maximumnumber of solutions in a set If the number of local minimain the set at the current iteration is less than the maximumset size 119873max then every solution in the set is used inthe next iteration otherwise we will choose the preferablesolution An ideal measure of what constitutes the preferablesolutions is regularization functional value of (15) solutionwith the lower regularization functional values is consideredthe preferable The perturbation homotopy algorithm is asfollows

(i) Input given minimum norm solution of 1206010 namely1205900= 120590ref 1205820 = 0119873max ge 1 and Δ

119894gt 0 119894 = 1 119871

(ii) Update for 120582119894

= 120582119894minus1

+ Δ119894minus1

we use damp-ing Gauss-Newton iteration method compute anapproximate solution 120590

119896

119894 119896 = 1 2

119894 to minimizemin120590isin119883

119867(120590 120582119894) by initial value 120590119895

119894minus1and (per(120590

119894minus1))119895

119895 = 1 2119894

(iii) Order the distinct solutions among 120590119896

119894 119896 = 1 2

119894from best to worst as 1205901

119894 1205902119894 and discard any worst

solution when the number of elements in 119883 largerthan119873max

(iv) Output the iteration will be stopped if the solution120590119898

119894 1 le 119898 le 2

119894 is the best solution in the set andsatisfied stopping criterion output 1205901 = 120590

119898

The sketch map of the algorithm is shown in Figure 1

4 Numerical Simulation

In this section we exhibit clear advantages of perturbationhomotopy method over traditional damping Gauss-Newtonmethod in the previous three experiments In the fourthexperiment we analysis the influence of homotopy parameteron inversion results In the fifth experiment we apply ouralgorithm to more complicated model and confirm that thisalgorithm is stable In the last experiment we solve a simplepractice data In the first three 3D experiments we selecthomotopy parameter 120582

119894= 11989410 119894 = 0 10 119873max = 2 and

internal Gauss-Newton iteration 5 with each homotopy stepin our numerical experiments The space domain is Ω Theinverse problem is considered on a uniform 32 times 32times16 gridThe transmitter source is 119904(119905) = 119905

2119890minus120572119905 sin(120596

0119905) where 120572 =

1205960radic3 120596

0= 2120587119891

0 and 119891

0= 100MHz is the central

frequency In the last three 2D experiments 119873max = 4 andinternal Gauss-Newton iteration is 10 The inversion spaceΩ = 25m times 25m space step Δ119909 = Δ119910 = 005m samplingtime 119905 = 36 ns time step 120591 = 18 ns and the transmitter sourceis Ricker wavelet with 900MHz central frequency

41 HomogeneousModel Wefirst test a simple homogeneousmodel where permittivity 120576 = 56064 times 10

minus12 Fm perme-ability 120583 = 0 and relative conductivity 120590 = minus52983 BothdampingGauss-Newtonmethod and perturbation homotopymethod select the same relative initial value 120590 = minus48993Iterations of damping Gauss-Newton method are five timeselapsed time is 1230426 seconds homotopy algorithmrsquos outeriterations 120582

119894 119894 = 1 10 are ten times inner local

iterations (damping Gauss-Newton method) are three timesand elapsed time is 1405536 seconds Figure 2 gives resultsof true model damping Gauss-Newton and perturbationhomotopy method The relative error 120590inv minus 120590true120590trueof (a) and (b) in Figure 2 is 00516 and of (a) and (c) is00830 where 120590inv and 120590true are inversion solution and truemodel respectively From Figure 2 we can see that wheninitial value is close to original conductivity both methodshave satisfactory result but perturbation homotopy is notmore efficient

42 One Circle Cavity Model The relative conductivity ofcavity is minus32775 other parameters have the same value asin the above model Both methods select the same initialvalue 120590 = minus16094 Elapsed time of damping Gauss-Newtonmethod and perturbation homotopy method is 11120536seconds and 12578436 seconds respectively The relativeerror 120590inv minus 120590true120590true of (a) and (b) in Figure 3 is

8 Mathematical Problems in Engineeringy

(m)

x (m)05 10 15 20 25

2

25

3

35

4

45

5

2

25

3

35

4

45

5

2

25

3

35

4

45

5

2

25

3

35

4

45

5

2

25

3

35

4

45

5

2

25

3

35

4

45

5

2

25

3

35

4

45

5

2

25

3

35

4

45

5

2

25

3

35

4

45

5

0204060810121416182022

x (m)05 10 15 20 25

y(m

)

0204060810121416182022

x (m)05 10 15 20 25

y(m

)

0204060810121416182022

x (m)05 10 15 20 25

y(m

)

0204060810121416182022

x (m)05 10 15 20 25

y(m

)

0204060810121416182022

x (m)05 10 15 20 25

y(m

)

0204060810121416182022

x (m)05 10 15 20 25

y(m

)

0204060810121416182022

y(m

)

x (m)05 10 15 20 25

0204060810121416182022

y(m

)

x (m)05 10 15 20 25

0204060810121416182022

Figure 12 Inversion results with different SNR measured data (1 dB 3 dB 5 dB 8 dB 10 dB 12 dB 15 dB 20 dB and 30 dB)

2535 and of (a) and (c) is 961 Although elapsed timeof perturbation homotopy is ten times more than dampingGauss-Newton method when initial value is far away fromtrue value the latter is inefficient compared to the former

43 Two Cavities Model The conductivity of two cavities isequal other parameters have the same value as in Section 42Both methods select the same relative initial value 120590 =

minus08775 Elapsed time of damping Gauss-Newton methodand perturbation homotopy method is 11080546 secondsand 15458768 seconds respectively Figure 4 gives results oforiginal model damping Gauss-Newton and perturbationhomotopy method The relative error 120590inv minus 120590true120590true of(a) and (b) in Figure 4 is 4935 and of (a) and (c) is 1285When the initial value is further away from original valuedamping Gauss-Newton method is invalid

44 One Square Cavity 2D Model The vertex coordinate ofsquare cavity is 12m 04m 13m 04m 12m 05m 13m05m respectively The original model structure schemeis shown in Figure 5 Inversion results by Gauss-Newtonmethod are shown in Figure 7 Measured data is shownin Figure 6 With the influence of local minima Figure 7

roughly describes the information of location and scale ofsquare cavity but the quality information error is hugeBecause traditional Tikhonov regularization would smooththe inversion solution the solution always appears blurringphenomenon at boundary Inversion results by perturbationhomotopy method with different homotopy parameter areshown in Figure 8 the relative error between inversion andoriginal model is 455 We select homotopy parameter as150001 15001 12001 11501 11001 1501 151 16 23100105 10001005 and 100000100005 From Figure 9 wecan conclude that the rate of convergence is fast but whenhomotopy parameter is close to 1 numerical solution isdiverging so the optimal solution is not at 120582 = 1

45 Three Square Cavities 2D Model The original modelstructure scheme is shown in Figure 10 Measured data isshown in Figure 11 In order to demonstrate that the methodis stable with noise we add different Gauss white noise tomeasured data The SNR (signal-to-noise ratio) is 1 dB 3 dB5 dB 8 dB 10 dB 12 dB 15 dB 20 dB and 30 dB respectivelyFrom Figure 12 we can see that with noise decrease numer-ical solution is convergent to original model So our methodis stable

Mathematical Problems in Engineering 9

1 2 3 4 5

50

100

150

200

250

300

350

400

450

500

y(m

)

x (m)

01

02

03

04

05

06

07

08

09

10

Figure 13 Original model

10 20 30 40 50 60 70 80 90 100

3

6

9

12

15

0

500

1000

1500

t(n

s)

Trace

minus1000

minus1500

minus500

Figure 14 Practice measured data

46 Practice Data Inversion The practice measured data isprovided by Wuhan Changjiang Engineering GeophysicalExploration amp Testing Co LtdChina The testing object isa concrete member with three circle rebar in it The centre ofcircle is 14m 06m 22m 04m 34m 07m respectivelyradius is 01mThe original model structure scheme is shownin Figure 13 Practice measured data is shown in Figure 14Inversion result is shown in Figure 15 The relative error is1575 Before the inversion it is necessary to preprocess thedata because practice measured data has large noise

5 Conclusion

This paper has addressed the perturbation homotopymethodfor solving inverse electromagnetic problem Perturbationhomotopy coupled with damping Gauss-Newton methodshas been used in the experiments Results show that wheninitial value is far away from true model local convergentGauss-Newton method just roughly gives the informationof location and scale the quality has huge error with true

50

100

150

200

250

300

350

400

450

500

y(m

)

01

02

03

04

05

06

07

08

09

10

x (m)05 10 15 20 25

Figure 15 Inversion results

model Perturbation homotopy method is a widely conver-gent algorithm it could give amore accurate inversion resultseven if initial value is far away from the original value Whenadding Gauss white noise to measured data we can see thatwith noise decrease numerical solution is convergence tooriginal model so numerical results support that ourmethodis stable Traditional homotopy theory tells us that optimumsolution is determined at 120582 = 1 but numerical resultsshow that the solution will diverge when 120582 is close to 1because of regularization parameter impact In a word thismethod has clear advantages over damping Gauss-Newtonmethod it is an effective method especially on aspects ofwide convergence computational efficiency and precisionand has broad application prospects

Conflict of Interests

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

Acknowledgments

This paper is supported by Natural Science Foundation ofChina Project no 41304093 the Fundamental ResearchFunds for the Central Universities Project no DL12BB20and the Educational Commission of Heilongjiang Provinceof China no 12533013

References

[1] D Colton and R Kress Inverse Acoustic and ElectromagneticScattering Theory Springer New York NY USA 1998

[2] P A Eaton ldquo3-D electromagnetic inversion using integralequationsrdquo Geophysical Prospecting vol 37 pp 407ndash426 1989

[3] J Nocedal and S Wright Numerical Optimization SpringerNew York NY USA 1999

[4] G A Newman S Recher B Tezkan and F M Neubauerldquo3D inversion of a scalar radio magnetotelluric field data setrdquoGeophysics vol 68 no 3 pp 791ndash802 2003

10 Mathematical Problems in Engineering

[5] E Haber ldquoQuasi-Newton methods for large-scale electromag-netic inverse problemsrdquo Inverse Problems vol 21 no 1 pp 305ndash323 2005

[6] M S Zhdanov S Fang and G Hursan ldquoElectromagneticinversion using quasi-linear approximationrdquo Geophysics vol65 no 5 pp 1501ndash1513 2000

[7] C Torres-Verdın and T M Habashy ldquoRapid numerical sim-ulation of axisymmetric single-well induction data using theextended Born approximationrdquo Radio Science vol 36 no 6 pp1287ndash1306 2001

[8] H-W Tseng K H Lee and A Becker ldquo3D interpretation ofelectromagnetic data using a modified extended Born approxi-mationrdquo Geophysics vol 68 no 1 pp 127ndash137 2003

[9] Z Zhang ldquo3D resistivity mapping of airborne EM datardquoGeophysics vol 68 no 6 pp 1896ndash1905 2003

[10] M S Zhdanov Geophysical Inverse Theory and RegularizationProblems Elsevier Amsterdam The Netherlands 2002

[11] M S Zhdanov Geophysical Electromagnetic Theory and Meth-ods Elsevier Amsterdam The Netherlands 2009

[12] H B Han Bo K Z Kuang Zheng and L J-Q Liu Jia-Qi ldquoAmonotonous homotopy method for solving the resistivities ofthe earthrdquo Acta Geophysica Sinica vol 34 no 4 pp 517ndash5221991

[13] H S Fu BHan andGQGai ldquoAwaveletmultiscale-homotopymethod for the inverse problem of two-dimensional acousticwave equationrdquoAppliedMathematics andComputation vol 190no 1 pp 576ndash582 2007

[14] Y He and B Han ldquoA wavelet adaptive-homotopy method forinverse problem in the fluid-saturated porous mediardquo AppliedMathematics andComputation vol 208 no 1 pp 189ndash196 2009

[15] D W Vasco ldquoSingularity and branching a path-following for-malism for geophysical inverse problemsrdquo Geophysical JournalInternational vol 119 no 3 pp 809ndash830 1994

[16] D M Dunlavy D P OrsquoLeary D Klimov and D ThirumalaildquoHOPE a homotopy optimizationmethod for protein structurepredictionrdquo Journal of Computational Biology vol 12 no 10 pp1275ndash1288 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

05 05

2

4

6

8

10

120

50

100

150

minus5minus5

Z

XY

(a)

05

2

4

6

8

10

120

50

100

150

Z

X

05 minus5minus5

Y

(b)

05 05

2

4

6

8

10

120

50

100

150

Z

Xminus5

minus5

Y

(c)

10 20 30

2

4

6

8

10

12

14

16

0

50

100

150

(d)

10 20 30

2

4

6

8

10

12

14

16

0

50

100

150

(e)

10 20 30

2

4

6

8

10

12

14

16

0

50

100

150

(f)

Figure 4 (a) True model (b) results of damping Gauss-Newton method (c) results of perturbation homotopy method (d) cross-section of(a) at 119885 = 8 (e) cross-section of (b) at 119885 = 8 (f) cross-section of (c) at 119885 = 8

02

04

06

08

10

12

14

16

18

20

22

x (m)

y(m

)

05 10 15 20 252

25

3

35

4

45

5

Figure 5 Square cavity model

where 120591119899= 1(119905

119899+1minus 119905119899) With discretization we get matrix

equation

119860 (120590) 119906 = 119902 (3)

20 40 60 80 100

0

500

1000

1500

2000

t(n

s)

72

144

216

288

36

Trace

minus1000

minus500

Figure 6 Measured data

In the inverse problem we want to recover the model120590 with the measured data Measured data can be written as

119889obs

= 119876119906 + 120578 (4)

Mathematical Problems in Engineering 5

x (m)

y(m

)

05 10 15 20 25

02

04

06

08

10

12

14

16

18

20

222

25

3

35

4

45

5

Figure 7 Inversion results of Gauss-Newton method

y(m

)

02

04

06

08

10

12

14

16

18

20

22

x (m)05 10 15 20 25

2

25

3

35

4

45

5

Figure 8 Inversion results of perturbation homotopy method

where 119876 is a projection operator which projects the electro-magnetic fields onto the observation locations and 120578 is themeasurement noise From (3) and (4) we can obtain

119889obs

= 119876119860minus1119902 + 120578 (5)

3 Inversion Framework

31 Damping Gauss-Newton Method For every homotopyparameter 120582 = 120582

119894 119894 = 1 119871 we must solve the

unconstrained optimization problem

min120590

119867(120590 120582) = min120590

12058210038171003817100381710038171003817119865 (120590) minus 119889

obs10038171003817100381710038171003817

2

+ (1 minus 120582)1003817100381710038171003817119882 (120590 minus 120590ref)

10038171003817100381710038172

(6)

This can be transformed to the Euler-Lagrange system

120597119867

120597120590= 119892 (120590) = (1 minus 120582)119882

119879119882(120590 minus 120590ref)

+ 120582119869(120590)119879(119865 (120590) minus 119889

obs)

(7)

Denoting 120573 = (1 minus 120582)120582 we obtain

120573119882119879119882(120590 minus 120590ref) + 119869(120590)

119879(119865 (120590) minus 119889

obs) = 0 (8)

where 120573 played a regularization parameter role In order toavoid the calculation of the second Frechet derivative of 119865 to120590 119865 is linearized by

119865 (120590 + 120575120590) = 119865 (120590) + 119869 (120590) 120575120590 + 119877 (120590 120575120590) (9)

Then the minimization problem is solved iteratively At the119896th iteration we solve

(119869 (120590119879

119896) 119869 (120590119896) + 120573119882

119879119882)120575120590

= 119869 (120590119879

119896) (119889

obsminus 119865 (120590

119896) + 119869 (120590

119896)) minus 120573119882

119879119882(120590119896minus 120590ref)

(10)

to find the perturbation 120575120590 At each iteration we have anoption of solving directly for a perturbation or solving foran updated model To formulate the latter option we write120590119896+1

= 120590119896+ 120575120590 and substitute into (10) to obtain

(119869 (120590119879

119896) 119869 (120590119896) + 120573119882

119879119882)120590119896+1

= 119869 (120590119879

119896) (119889

obsminus 119865 (120590

119896) +119869 (120590

119896) 120590119896) minus 120573119882

119879119882120590ref

(11)

32 A Perturbation Homotopy Method As above mentionedwe can formulate the inverse electromagnetic problem as thefollowing nonlinear operator equation

119865 (120590) = 119889obs

(12)

where 119865 = 119876119860minus1119902 is the forward nonlinear operator

It is very difficult to estimate 120590 in the problem (12) dueto the presence of local minima Therefore we design aglobally convergent algorithm which can overcome the localminima namely perturbation homotopy method to solvethe nonlinear equation (12) For inverse electromagneticproblem we want to solve the minimization problem Given120601 119883 sube 119877

119899rarr 119877 find 120590

lowastisin 119883 sube 119877

119899 such that

120601 (120590lowast) = min120590isin119883

10038171003817100381710038171003817119865 (120590) minus 119889

obs10038171003817100381710038171003817

2

(13)

In order to apply homotopy to PDEoptimization problemwedenote

1206010=1003817100381710038171003817119882 (120590 minus 120590ref)

10038171003817100381710038172

1206011=10038171003817100381710038171003817119865 (120590) minus 119889

obs10038171003817100381710038171003817

2

(14)

where sdot is 2-norm 119882 is positive weight function and120590ref is the known reference model So continuous homotopyfunctional119867 119877

119899+1rarr 119877 can be written as

119867(120590 120582) = 1205821206011+ (1 minus 120582) 120601

0 (15)

where120582 isin [0 1] are homotopy parameters and the homotopystep length Δ

119894 119894 = 0 119871 minus 1 is satisfied with sum

119871minus1

119894=0Δ119894= 1

A general sketch of the homotopy algorithm is as follows

6 Mathematical Problems in Engineeringy

(m)

y(m

)

x (m)05 10 15 20 25

x (m)05 10 15 20 25

2

25

3

35

4

45

5

2

25

3

35

4

45

5

2

25

3

35

4

45

5

2

25

3

35

4

45

5

2

25

3

35

4

45

5

2

25

3

35

4

45

5

2

25

3

35

4

45

5

2

25

3

35

4

45

5

2

25

3

35

4

45

5

2

25

3

35

4

45

5

2

25

3

35

4

45

5

2

25

3

35

4

45

5

y(m

)

x (m)05 10 15 20 25

y(m

)y

(m)

x (m)05 10 15 20 25

x (m)05 10 15 20 25

y(m

)

x (m)05 10 15 20 25

y(m

)

x (m)05 10 15 20 25

y(m

)

x (m)05 10 15 20 25

y(m

)

x (m)05 10 15 20 25

y(m

)

x (m)05 10 15 20 25

y(m

)

x (m)05 10 15 20 25

y(m

)

x (m)05 10 15 20 25

0204060810121416182022

0204060810121416182022

0204060810121416182022

0204060810121416182022

0204060810121416182022

0204060810121416182022

0204060810121416182022

0204060810121416182022

0204060810121416182022

0204060810121416182022

0204060810121416182022

0204060810121416182022

Figure 9 Inversion results with different homotopy parameters (150001 15001 12001 11501 11001 1501 151 16 23 100105 10001005and 100000100005)

(i) Input given minimum norm solution of 1206010 namely1205900= 120590ref 1205820 = 0 and Δ

119894gt 0 119894 = 1 119871

(ii) Update for 120582119894= 120582119894minus1

+ Δ119894minus1

we use damping Gauss-Newton iteration method compute an approximatesolution 120590

119894to minimize min

120590isin119883119867(120590 120582

119894) by initial

value 120590119894minus1

(iii) Output the iteration will be stopped if the solution 120590

119894

satisfied stopping criterion output 1205901 = 120590119898

The homotopy algorithmrsquos greatest advantage lies in increas-ing region of convergency But in the process of computationthe nonconvergent phenomenon often happens to homotopycurve for example standstill convolution and bifurcation

These difficulties can make the homotopy method quitecomplicated In order to make the homotopy more efficientsets of solutions are produced by perturbation for everyhomotopy parameter 120582 = 120582

119894 119894 = 0 119871 Although the

calculation increased it ensures the likelihood of obtainingaccurate solutionWith different homotopy parameter 120582

119894 the

minima is in the sequence and120590119894is solved by dampingGauss-

Newton method with the initial value 120590119894minus1

The next set oflocal minima (120582 = 120582

119894+1) is solved with the initial values in the

previous set (120582 = 120582119894) which has one or more perturbations of

each of those points in the set We denote the perturbationof solution per(120590) The perturbation stochastically disturbsone or more of the solutions 120590 in set We use a single initial

Mathematical Problems in Engineering 7

02

04

06

08

10

12

14

16

18

20

22

x (m)

y(m

)

05 10 15 20 252

25

3

35

4

45

5

Figure 10 Three square cavities model

20 40 60 80 100

0

500

1000

1500

2000

t(n

s)

72

144

216

288

36

Trace

minus1000

minus500

Figure 11 Measured data

value 1205900for 120582 = 120582

0 the number of solution solved at

each homotopy parameter120582 increases exponentiallyWithoutlimit on the size of the set the number of solutions in theset would be 2

119894 when homotopy iteration stops In order toconstraint on computational complexity it is necessary tolimit the size of the set We denote by 119873max the maximumnumber of solutions in a set If the number of local minimain the set at the current iteration is less than the maximumset size 119873max then every solution in the set is used inthe next iteration otherwise we will choose the preferablesolution An ideal measure of what constitutes the preferablesolutions is regularization functional value of (15) solutionwith the lower regularization functional values is consideredthe preferable The perturbation homotopy algorithm is asfollows

(i) Input given minimum norm solution of 1206010 namely1205900= 120590ref 1205820 = 0119873max ge 1 and Δ

119894gt 0 119894 = 1 119871

(ii) Update for 120582119894

= 120582119894minus1

+ Δ119894minus1

we use damp-ing Gauss-Newton iteration method compute anapproximate solution 120590

119896

119894 119896 = 1 2

119894 to minimizemin120590isin119883

119867(120590 120582119894) by initial value 120590119895

119894minus1and (per(120590

119894minus1))119895

119895 = 1 2119894

(iii) Order the distinct solutions among 120590119896

119894 119896 = 1 2

119894from best to worst as 1205901

119894 1205902119894 and discard any worst

solution when the number of elements in 119883 largerthan119873max

(iv) Output the iteration will be stopped if the solution120590119898

119894 1 le 119898 le 2

119894 is the best solution in the set andsatisfied stopping criterion output 1205901 = 120590

119898

The sketch map of the algorithm is shown in Figure 1

4 Numerical Simulation

In this section we exhibit clear advantages of perturbationhomotopy method over traditional damping Gauss-Newtonmethod in the previous three experiments In the fourthexperiment we analysis the influence of homotopy parameteron inversion results In the fifth experiment we apply ouralgorithm to more complicated model and confirm that thisalgorithm is stable In the last experiment we solve a simplepractice data In the first three 3D experiments we selecthomotopy parameter 120582

119894= 11989410 119894 = 0 10 119873max = 2 and

internal Gauss-Newton iteration 5 with each homotopy stepin our numerical experiments The space domain is Ω Theinverse problem is considered on a uniform 32 times 32times16 gridThe transmitter source is 119904(119905) = 119905

2119890minus120572119905 sin(120596

0119905) where 120572 =

1205960radic3 120596

0= 2120587119891

0 and 119891

0= 100MHz is the central

frequency In the last three 2D experiments 119873max = 4 andinternal Gauss-Newton iteration is 10 The inversion spaceΩ = 25m times 25m space step Δ119909 = Δ119910 = 005m samplingtime 119905 = 36 ns time step 120591 = 18 ns and the transmitter sourceis Ricker wavelet with 900MHz central frequency

41 HomogeneousModel Wefirst test a simple homogeneousmodel where permittivity 120576 = 56064 times 10

minus12 Fm perme-ability 120583 = 0 and relative conductivity 120590 = minus52983 BothdampingGauss-Newtonmethod and perturbation homotopymethod select the same relative initial value 120590 = minus48993Iterations of damping Gauss-Newton method are five timeselapsed time is 1230426 seconds homotopy algorithmrsquos outeriterations 120582

119894 119894 = 1 10 are ten times inner local

iterations (damping Gauss-Newton method) are three timesand elapsed time is 1405536 seconds Figure 2 gives resultsof true model damping Gauss-Newton and perturbationhomotopy method The relative error 120590inv minus 120590true120590trueof (a) and (b) in Figure 2 is 00516 and of (a) and (c) is00830 where 120590inv and 120590true are inversion solution and truemodel respectively From Figure 2 we can see that wheninitial value is close to original conductivity both methodshave satisfactory result but perturbation homotopy is notmore efficient

42 One Circle Cavity Model The relative conductivity ofcavity is minus32775 other parameters have the same value asin the above model Both methods select the same initialvalue 120590 = minus16094 Elapsed time of damping Gauss-Newtonmethod and perturbation homotopy method is 11120536seconds and 12578436 seconds respectively The relativeerror 120590inv minus 120590true120590true of (a) and (b) in Figure 3 is

8 Mathematical Problems in Engineeringy

(m)

x (m)05 10 15 20 25

2

25

3

35

4

45

5

2

25

3

35

4

45

5

2

25

3

35

4

45

5

2

25

3

35

4

45

5

2

25

3

35

4

45

5

2

25

3

35

4

45

5

2

25

3

35

4

45

5

2

25

3

35

4

45

5

2

25

3

35

4

45

5

0204060810121416182022

x (m)05 10 15 20 25

y(m

)

0204060810121416182022

x (m)05 10 15 20 25

y(m

)

0204060810121416182022

x (m)05 10 15 20 25

y(m

)

0204060810121416182022

x (m)05 10 15 20 25

y(m

)

0204060810121416182022

x (m)05 10 15 20 25

y(m

)

0204060810121416182022

x (m)05 10 15 20 25

y(m

)

0204060810121416182022

y(m

)

x (m)05 10 15 20 25

0204060810121416182022

y(m

)

x (m)05 10 15 20 25

0204060810121416182022

Figure 12 Inversion results with different SNR measured data (1 dB 3 dB 5 dB 8 dB 10 dB 12 dB 15 dB 20 dB and 30 dB)

2535 and of (a) and (c) is 961 Although elapsed timeof perturbation homotopy is ten times more than dampingGauss-Newton method when initial value is far away fromtrue value the latter is inefficient compared to the former

43 Two Cavities Model The conductivity of two cavities isequal other parameters have the same value as in Section 42Both methods select the same relative initial value 120590 =

minus08775 Elapsed time of damping Gauss-Newton methodand perturbation homotopy method is 11080546 secondsand 15458768 seconds respectively Figure 4 gives results oforiginal model damping Gauss-Newton and perturbationhomotopy method The relative error 120590inv minus 120590true120590true of(a) and (b) in Figure 4 is 4935 and of (a) and (c) is 1285When the initial value is further away from original valuedamping Gauss-Newton method is invalid

44 One Square Cavity 2D Model The vertex coordinate ofsquare cavity is 12m 04m 13m 04m 12m 05m 13m05m respectively The original model structure schemeis shown in Figure 5 Inversion results by Gauss-Newtonmethod are shown in Figure 7 Measured data is shownin Figure 6 With the influence of local minima Figure 7

roughly describes the information of location and scale ofsquare cavity but the quality information error is hugeBecause traditional Tikhonov regularization would smooththe inversion solution the solution always appears blurringphenomenon at boundary Inversion results by perturbationhomotopy method with different homotopy parameter areshown in Figure 8 the relative error between inversion andoriginal model is 455 We select homotopy parameter as150001 15001 12001 11501 11001 1501 151 16 23100105 10001005 and 100000100005 From Figure 9 wecan conclude that the rate of convergence is fast but whenhomotopy parameter is close to 1 numerical solution isdiverging so the optimal solution is not at 120582 = 1

45 Three Square Cavities 2D Model The original modelstructure scheme is shown in Figure 10 Measured data isshown in Figure 11 In order to demonstrate that the methodis stable with noise we add different Gauss white noise tomeasured data The SNR (signal-to-noise ratio) is 1 dB 3 dB5 dB 8 dB 10 dB 12 dB 15 dB 20 dB and 30 dB respectivelyFrom Figure 12 we can see that with noise decrease numer-ical solution is convergent to original model So our methodis stable

Mathematical Problems in Engineering 9

1 2 3 4 5

50

100

150

200

250

300

350

400

450

500

y(m

)

x (m)

01

02

03

04

05

06

07

08

09

10

Figure 13 Original model

10 20 30 40 50 60 70 80 90 100

3

6

9

12

15

0

500

1000

1500

t(n

s)

Trace

minus1000

minus1500

minus500

Figure 14 Practice measured data

46 Practice Data Inversion The practice measured data isprovided by Wuhan Changjiang Engineering GeophysicalExploration amp Testing Co LtdChina The testing object isa concrete member with three circle rebar in it The centre ofcircle is 14m 06m 22m 04m 34m 07m respectivelyradius is 01mThe original model structure scheme is shownin Figure 13 Practice measured data is shown in Figure 14Inversion result is shown in Figure 15 The relative error is1575 Before the inversion it is necessary to preprocess thedata because practice measured data has large noise

5 Conclusion

This paper has addressed the perturbation homotopymethodfor solving inverse electromagnetic problem Perturbationhomotopy coupled with damping Gauss-Newton methodshas been used in the experiments Results show that wheninitial value is far away from true model local convergentGauss-Newton method just roughly gives the informationof location and scale the quality has huge error with true

50

100

150

200

250

300

350

400

450

500

y(m

)

01

02

03

04

05

06

07

08

09

10

x (m)05 10 15 20 25

Figure 15 Inversion results

model Perturbation homotopy method is a widely conver-gent algorithm it could give amore accurate inversion resultseven if initial value is far away from the original value Whenadding Gauss white noise to measured data we can see thatwith noise decrease numerical solution is convergence tooriginal model so numerical results support that ourmethodis stable Traditional homotopy theory tells us that optimumsolution is determined at 120582 = 1 but numerical resultsshow that the solution will diverge when 120582 is close to 1because of regularization parameter impact In a word thismethod has clear advantages over damping Gauss-Newtonmethod it is an effective method especially on aspects ofwide convergence computational efficiency and precisionand has broad application prospects

Conflict of Interests

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

Acknowledgments

This paper is supported by Natural Science Foundation ofChina Project no 41304093 the Fundamental ResearchFunds for the Central Universities Project no DL12BB20and the Educational Commission of Heilongjiang Provinceof China no 12533013

References

[1] D Colton and R Kress Inverse Acoustic and ElectromagneticScattering Theory Springer New York NY USA 1998

[2] P A Eaton ldquo3-D electromagnetic inversion using integralequationsrdquo Geophysical Prospecting vol 37 pp 407ndash426 1989

[3] J Nocedal and S Wright Numerical Optimization SpringerNew York NY USA 1999

[4] G A Newman S Recher B Tezkan and F M Neubauerldquo3D inversion of a scalar radio magnetotelluric field data setrdquoGeophysics vol 68 no 3 pp 791ndash802 2003

10 Mathematical Problems in Engineering

[5] E Haber ldquoQuasi-Newton methods for large-scale electromag-netic inverse problemsrdquo Inverse Problems vol 21 no 1 pp 305ndash323 2005

[6] M S Zhdanov S Fang and G Hursan ldquoElectromagneticinversion using quasi-linear approximationrdquo Geophysics vol65 no 5 pp 1501ndash1513 2000

[7] C Torres-Verdın and T M Habashy ldquoRapid numerical sim-ulation of axisymmetric single-well induction data using theextended Born approximationrdquo Radio Science vol 36 no 6 pp1287ndash1306 2001

[8] H-W Tseng K H Lee and A Becker ldquo3D interpretation ofelectromagnetic data using a modified extended Born approxi-mationrdquo Geophysics vol 68 no 1 pp 127ndash137 2003

[9] Z Zhang ldquo3D resistivity mapping of airborne EM datardquoGeophysics vol 68 no 6 pp 1896ndash1905 2003

[10] M S Zhdanov Geophysical Inverse Theory and RegularizationProblems Elsevier Amsterdam The Netherlands 2002

[11] M S Zhdanov Geophysical Electromagnetic Theory and Meth-ods Elsevier Amsterdam The Netherlands 2009

[12] H B Han Bo K Z Kuang Zheng and L J-Q Liu Jia-Qi ldquoAmonotonous homotopy method for solving the resistivities ofthe earthrdquo Acta Geophysica Sinica vol 34 no 4 pp 517ndash5221991

[13] H S Fu BHan andGQGai ldquoAwaveletmultiscale-homotopymethod for the inverse problem of two-dimensional acousticwave equationrdquoAppliedMathematics andComputation vol 190no 1 pp 576ndash582 2007

[14] Y He and B Han ldquoA wavelet adaptive-homotopy method forinverse problem in the fluid-saturated porous mediardquo AppliedMathematics andComputation vol 208 no 1 pp 189ndash196 2009

[15] D W Vasco ldquoSingularity and branching a path-following for-malism for geophysical inverse problemsrdquo Geophysical JournalInternational vol 119 no 3 pp 809ndash830 1994

[16] D M Dunlavy D P OrsquoLeary D Klimov and D ThirumalaildquoHOPE a homotopy optimizationmethod for protein structurepredictionrdquo Journal of Computational Biology vol 12 no 10 pp1275ndash1288 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

x (m)

y(m

)

05 10 15 20 25

02

04

06

08

10

12

14

16

18

20

222

25

3

35

4

45

5

Figure 7 Inversion results of Gauss-Newton method

y(m

)

02

04

06

08

10

12

14

16

18

20

22

x (m)05 10 15 20 25

2

25

3

35

4

45

5

Figure 8 Inversion results of perturbation homotopy method

where 119876 is a projection operator which projects the electro-magnetic fields onto the observation locations and 120578 is themeasurement noise From (3) and (4) we can obtain

119889obs

= 119876119860minus1119902 + 120578 (5)

3 Inversion Framework

31 Damping Gauss-Newton Method For every homotopyparameter 120582 = 120582

119894 119894 = 1 119871 we must solve the

unconstrained optimization problem

min120590

119867(120590 120582) = min120590

12058210038171003817100381710038171003817119865 (120590) minus 119889

obs10038171003817100381710038171003817

2

+ (1 minus 120582)1003817100381710038171003817119882 (120590 minus 120590ref)

10038171003817100381710038172

(6)

This can be transformed to the Euler-Lagrange system

120597119867

120597120590= 119892 (120590) = (1 minus 120582)119882

119879119882(120590 minus 120590ref)

+ 120582119869(120590)119879(119865 (120590) minus 119889

obs)

(7)

Denoting 120573 = (1 minus 120582)120582 we obtain

120573119882119879119882(120590 minus 120590ref) + 119869(120590)

119879(119865 (120590) minus 119889

obs) = 0 (8)

where 120573 played a regularization parameter role In order toavoid the calculation of the second Frechet derivative of 119865 to120590 119865 is linearized by

119865 (120590 + 120575120590) = 119865 (120590) + 119869 (120590) 120575120590 + 119877 (120590 120575120590) (9)

Then the minimization problem is solved iteratively At the119896th iteration we solve

(119869 (120590119879

119896) 119869 (120590119896) + 120573119882

119879119882)120575120590

= 119869 (120590119879

119896) (119889

obsminus 119865 (120590

119896) + 119869 (120590

119896)) minus 120573119882

119879119882(120590119896minus 120590ref)

(10)

to find the perturbation 120575120590 At each iteration we have anoption of solving directly for a perturbation or solving foran updated model To formulate the latter option we write120590119896+1

= 120590119896+ 120575120590 and substitute into (10) to obtain

(119869 (120590119879

119896) 119869 (120590119896) + 120573119882

119879119882)120590119896+1

= 119869 (120590119879

119896) (119889

obsminus 119865 (120590

119896) +119869 (120590

119896) 120590119896) minus 120573119882

119879119882120590ref

(11)

32 A Perturbation Homotopy Method As above mentionedwe can formulate the inverse electromagnetic problem as thefollowing nonlinear operator equation

119865 (120590) = 119889obs

(12)

where 119865 = 119876119860minus1119902 is the forward nonlinear operator

It is very difficult to estimate 120590 in the problem (12) dueto the presence of local minima Therefore we design aglobally convergent algorithm which can overcome the localminima namely perturbation homotopy method to solvethe nonlinear equation (12) For inverse electromagneticproblem we want to solve the minimization problem Given120601 119883 sube 119877

119899rarr 119877 find 120590

lowastisin 119883 sube 119877

119899 such that

120601 (120590lowast) = min120590isin119883

10038171003817100381710038171003817119865 (120590) minus 119889

obs10038171003817100381710038171003817

2

(13)

In order to apply homotopy to PDEoptimization problemwedenote

1206010=1003817100381710038171003817119882 (120590 minus 120590ref)

10038171003817100381710038172

1206011=10038171003817100381710038171003817119865 (120590) minus 119889

obs10038171003817100381710038171003817

2

(14)

where sdot is 2-norm 119882 is positive weight function and120590ref is the known reference model So continuous homotopyfunctional119867 119877

119899+1rarr 119877 can be written as

119867(120590 120582) = 1205821206011+ (1 minus 120582) 120601

0 (15)

where120582 isin [0 1] are homotopy parameters and the homotopystep length Δ

119894 119894 = 0 119871 minus 1 is satisfied with sum

119871minus1

119894=0Δ119894= 1

A general sketch of the homotopy algorithm is as follows

6 Mathematical Problems in Engineeringy

(m)

y(m

)

x (m)05 10 15 20 25

x (m)05 10 15 20 25

2

25

3

35

4

45

5

2

25

3

35

4

45

5

2

25

3

35

4

45

5

2

25

3

35

4

45

5

2

25

3

35

4

45

5

2

25

3

35

4

45

5

2

25

3

35

4

45

5

2

25

3

35

4

45

5

2

25

3

35

4

45

5

2

25

3

35

4

45

5

2

25

3

35

4

45

5

2

25

3

35

4

45

5

y(m

)

x (m)05 10 15 20 25

y(m

)y

(m)

x (m)05 10 15 20 25

x (m)05 10 15 20 25

y(m

)

x (m)05 10 15 20 25

y(m

)

x (m)05 10 15 20 25

y(m

)

x (m)05 10 15 20 25

y(m

)

x (m)05 10 15 20 25

y(m

)

x (m)05 10 15 20 25

y(m

)

x (m)05 10 15 20 25

y(m

)

x (m)05 10 15 20 25

0204060810121416182022

0204060810121416182022

0204060810121416182022

0204060810121416182022

0204060810121416182022

0204060810121416182022

0204060810121416182022

0204060810121416182022

0204060810121416182022

0204060810121416182022

0204060810121416182022

0204060810121416182022

Figure 9 Inversion results with different homotopy parameters (150001 15001 12001 11501 11001 1501 151 16 23 100105 10001005and 100000100005)

(i) Input given minimum norm solution of 1206010 namely1205900= 120590ref 1205820 = 0 and Δ

119894gt 0 119894 = 1 119871

(ii) Update for 120582119894= 120582119894minus1

+ Δ119894minus1

we use damping Gauss-Newton iteration method compute an approximatesolution 120590

119894to minimize min

120590isin119883119867(120590 120582

119894) by initial

value 120590119894minus1

(iii) Output the iteration will be stopped if the solution 120590

119894

satisfied stopping criterion output 1205901 = 120590119898

The homotopy algorithmrsquos greatest advantage lies in increas-ing region of convergency But in the process of computationthe nonconvergent phenomenon often happens to homotopycurve for example standstill convolution and bifurcation

These difficulties can make the homotopy method quitecomplicated In order to make the homotopy more efficientsets of solutions are produced by perturbation for everyhomotopy parameter 120582 = 120582

119894 119894 = 0 119871 Although the

calculation increased it ensures the likelihood of obtainingaccurate solutionWith different homotopy parameter 120582

119894 the

minima is in the sequence and120590119894is solved by dampingGauss-

Newton method with the initial value 120590119894minus1

The next set oflocal minima (120582 = 120582

119894+1) is solved with the initial values in the

previous set (120582 = 120582119894) which has one or more perturbations of

each of those points in the set We denote the perturbationof solution per(120590) The perturbation stochastically disturbsone or more of the solutions 120590 in set We use a single initial

Mathematical Problems in Engineering 7

02

04

06

08

10

12

14

16

18

20

22

x (m)

y(m

)

05 10 15 20 252

25

3

35

4

45

5

Figure 10 Three square cavities model

20 40 60 80 100

0

500

1000

1500

2000

t(n

s)

72

144

216

288

36

Trace

minus1000

minus500

Figure 11 Measured data

value 1205900for 120582 = 120582

0 the number of solution solved at

each homotopy parameter120582 increases exponentiallyWithoutlimit on the size of the set the number of solutions in theset would be 2

119894 when homotopy iteration stops In order toconstraint on computational complexity it is necessary tolimit the size of the set We denote by 119873max the maximumnumber of solutions in a set If the number of local minimain the set at the current iteration is less than the maximumset size 119873max then every solution in the set is used inthe next iteration otherwise we will choose the preferablesolution An ideal measure of what constitutes the preferablesolutions is regularization functional value of (15) solutionwith the lower regularization functional values is consideredthe preferable The perturbation homotopy algorithm is asfollows

(i) Input given minimum norm solution of 1206010 namely1205900= 120590ref 1205820 = 0119873max ge 1 and Δ

119894gt 0 119894 = 1 119871

(ii) Update for 120582119894

= 120582119894minus1

+ Δ119894minus1

we use damp-ing Gauss-Newton iteration method compute anapproximate solution 120590

119896

119894 119896 = 1 2

119894 to minimizemin120590isin119883

119867(120590 120582119894) by initial value 120590119895

119894minus1and (per(120590

119894minus1))119895

119895 = 1 2119894

(iii) Order the distinct solutions among 120590119896

119894 119896 = 1 2

119894from best to worst as 1205901

119894 1205902119894 and discard any worst

solution when the number of elements in 119883 largerthan119873max

(iv) Output the iteration will be stopped if the solution120590119898

119894 1 le 119898 le 2

119894 is the best solution in the set andsatisfied stopping criterion output 1205901 = 120590

119898

The sketch map of the algorithm is shown in Figure 1

4 Numerical Simulation

In this section we exhibit clear advantages of perturbationhomotopy method over traditional damping Gauss-Newtonmethod in the previous three experiments In the fourthexperiment we analysis the influence of homotopy parameteron inversion results In the fifth experiment we apply ouralgorithm to more complicated model and confirm that thisalgorithm is stable In the last experiment we solve a simplepractice data In the first three 3D experiments we selecthomotopy parameter 120582

119894= 11989410 119894 = 0 10 119873max = 2 and

internal Gauss-Newton iteration 5 with each homotopy stepin our numerical experiments The space domain is Ω Theinverse problem is considered on a uniform 32 times 32times16 gridThe transmitter source is 119904(119905) = 119905

2119890minus120572119905 sin(120596

0119905) where 120572 =

1205960radic3 120596

0= 2120587119891

0 and 119891

0= 100MHz is the central

frequency In the last three 2D experiments 119873max = 4 andinternal Gauss-Newton iteration is 10 The inversion spaceΩ = 25m times 25m space step Δ119909 = Δ119910 = 005m samplingtime 119905 = 36 ns time step 120591 = 18 ns and the transmitter sourceis Ricker wavelet with 900MHz central frequency

41 HomogeneousModel Wefirst test a simple homogeneousmodel where permittivity 120576 = 56064 times 10

minus12 Fm perme-ability 120583 = 0 and relative conductivity 120590 = minus52983 BothdampingGauss-Newtonmethod and perturbation homotopymethod select the same relative initial value 120590 = minus48993Iterations of damping Gauss-Newton method are five timeselapsed time is 1230426 seconds homotopy algorithmrsquos outeriterations 120582

119894 119894 = 1 10 are ten times inner local

iterations (damping Gauss-Newton method) are three timesand elapsed time is 1405536 seconds Figure 2 gives resultsof true model damping Gauss-Newton and perturbationhomotopy method The relative error 120590inv minus 120590true120590trueof (a) and (b) in Figure 2 is 00516 and of (a) and (c) is00830 where 120590inv and 120590true are inversion solution and truemodel respectively From Figure 2 we can see that wheninitial value is close to original conductivity both methodshave satisfactory result but perturbation homotopy is notmore efficient

42 One Circle Cavity Model The relative conductivity ofcavity is minus32775 other parameters have the same value asin the above model Both methods select the same initialvalue 120590 = minus16094 Elapsed time of damping Gauss-Newtonmethod and perturbation homotopy method is 11120536seconds and 12578436 seconds respectively The relativeerror 120590inv minus 120590true120590true of (a) and (b) in Figure 3 is

8 Mathematical Problems in Engineeringy

(m)

x (m)05 10 15 20 25

2

25

3

35

4

45

5

2

25

3

35

4

45

5

2

25

3

35

4

45

5

2

25

3

35

4

45

5

2

25

3

35

4

45

5

2

25

3

35

4

45

5

2

25

3

35

4

45

5

2

25

3

35

4

45

5

2

25

3

35

4

45

5

0204060810121416182022

x (m)05 10 15 20 25

y(m

)

0204060810121416182022

x (m)05 10 15 20 25

y(m

)

0204060810121416182022

x (m)05 10 15 20 25

y(m

)

0204060810121416182022

x (m)05 10 15 20 25

y(m

)

0204060810121416182022

x (m)05 10 15 20 25

y(m

)

0204060810121416182022

x (m)05 10 15 20 25

y(m

)

0204060810121416182022

y(m

)

x (m)05 10 15 20 25

0204060810121416182022

y(m

)

x (m)05 10 15 20 25

0204060810121416182022

Figure 12 Inversion results with different SNR measured data (1 dB 3 dB 5 dB 8 dB 10 dB 12 dB 15 dB 20 dB and 30 dB)

2535 and of (a) and (c) is 961 Although elapsed timeof perturbation homotopy is ten times more than dampingGauss-Newton method when initial value is far away fromtrue value the latter is inefficient compared to the former

43 Two Cavities Model The conductivity of two cavities isequal other parameters have the same value as in Section 42Both methods select the same relative initial value 120590 =

minus08775 Elapsed time of damping Gauss-Newton methodand perturbation homotopy method is 11080546 secondsand 15458768 seconds respectively Figure 4 gives results oforiginal model damping Gauss-Newton and perturbationhomotopy method The relative error 120590inv minus 120590true120590true of(a) and (b) in Figure 4 is 4935 and of (a) and (c) is 1285When the initial value is further away from original valuedamping Gauss-Newton method is invalid

44 One Square Cavity 2D Model The vertex coordinate ofsquare cavity is 12m 04m 13m 04m 12m 05m 13m05m respectively The original model structure schemeis shown in Figure 5 Inversion results by Gauss-Newtonmethod are shown in Figure 7 Measured data is shownin Figure 6 With the influence of local minima Figure 7

roughly describes the information of location and scale ofsquare cavity but the quality information error is hugeBecause traditional Tikhonov regularization would smooththe inversion solution the solution always appears blurringphenomenon at boundary Inversion results by perturbationhomotopy method with different homotopy parameter areshown in Figure 8 the relative error between inversion andoriginal model is 455 We select homotopy parameter as150001 15001 12001 11501 11001 1501 151 16 23100105 10001005 and 100000100005 From Figure 9 wecan conclude that the rate of convergence is fast but whenhomotopy parameter is close to 1 numerical solution isdiverging so the optimal solution is not at 120582 = 1

45 Three Square Cavities 2D Model The original modelstructure scheme is shown in Figure 10 Measured data isshown in Figure 11 In order to demonstrate that the methodis stable with noise we add different Gauss white noise tomeasured data The SNR (signal-to-noise ratio) is 1 dB 3 dB5 dB 8 dB 10 dB 12 dB 15 dB 20 dB and 30 dB respectivelyFrom Figure 12 we can see that with noise decrease numer-ical solution is convergent to original model So our methodis stable

Mathematical Problems in Engineering 9

1 2 3 4 5

50

100

150

200

250

300

350

400

450

500

y(m

)

x (m)

01

02

03

04

05

06

07

08

09

10

Figure 13 Original model

10 20 30 40 50 60 70 80 90 100

3

6

9

12

15

0

500

1000

1500

t(n

s)

Trace

minus1000

minus1500

minus500

Figure 14 Practice measured data

46 Practice Data Inversion The practice measured data isprovided by Wuhan Changjiang Engineering GeophysicalExploration amp Testing Co LtdChina The testing object isa concrete member with three circle rebar in it The centre ofcircle is 14m 06m 22m 04m 34m 07m respectivelyradius is 01mThe original model structure scheme is shownin Figure 13 Practice measured data is shown in Figure 14Inversion result is shown in Figure 15 The relative error is1575 Before the inversion it is necessary to preprocess thedata because practice measured data has large noise

5 Conclusion

This paper has addressed the perturbation homotopymethodfor solving inverse electromagnetic problem Perturbationhomotopy coupled with damping Gauss-Newton methodshas been used in the experiments Results show that wheninitial value is far away from true model local convergentGauss-Newton method just roughly gives the informationof location and scale the quality has huge error with true

50

100

150

200

250

300

350

400

450

500

y(m

)

01

02

03

04

05

06

07

08

09

10

x (m)05 10 15 20 25

Figure 15 Inversion results

model Perturbation homotopy method is a widely conver-gent algorithm it could give amore accurate inversion resultseven if initial value is far away from the original value Whenadding Gauss white noise to measured data we can see thatwith noise decrease numerical solution is convergence tooriginal model so numerical results support that ourmethodis stable Traditional homotopy theory tells us that optimumsolution is determined at 120582 = 1 but numerical resultsshow that the solution will diverge when 120582 is close to 1because of regularization parameter impact In a word thismethod has clear advantages over damping Gauss-Newtonmethod it is an effective method especially on aspects ofwide convergence computational efficiency and precisionand has broad application prospects

Conflict of Interests

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

Acknowledgments

This paper is supported by Natural Science Foundation ofChina Project no 41304093 the Fundamental ResearchFunds for the Central Universities Project no DL12BB20and the Educational Commission of Heilongjiang Provinceof China no 12533013

References

[1] D Colton and R Kress Inverse Acoustic and ElectromagneticScattering Theory Springer New York NY USA 1998

[2] P A Eaton ldquo3-D electromagnetic inversion using integralequationsrdquo Geophysical Prospecting vol 37 pp 407ndash426 1989

[3] J Nocedal and S Wright Numerical Optimization SpringerNew York NY USA 1999

[4] G A Newman S Recher B Tezkan and F M Neubauerldquo3D inversion of a scalar radio magnetotelluric field data setrdquoGeophysics vol 68 no 3 pp 791ndash802 2003

10 Mathematical Problems in Engineering

[5] E Haber ldquoQuasi-Newton methods for large-scale electromag-netic inverse problemsrdquo Inverse Problems vol 21 no 1 pp 305ndash323 2005

[6] M S Zhdanov S Fang and G Hursan ldquoElectromagneticinversion using quasi-linear approximationrdquo Geophysics vol65 no 5 pp 1501ndash1513 2000

[7] C Torres-Verdın and T M Habashy ldquoRapid numerical sim-ulation of axisymmetric single-well induction data using theextended Born approximationrdquo Radio Science vol 36 no 6 pp1287ndash1306 2001

[8] H-W Tseng K H Lee and A Becker ldquo3D interpretation ofelectromagnetic data using a modified extended Born approxi-mationrdquo Geophysics vol 68 no 1 pp 127ndash137 2003

[9] Z Zhang ldquo3D resistivity mapping of airborne EM datardquoGeophysics vol 68 no 6 pp 1896ndash1905 2003

[10] M S Zhdanov Geophysical Inverse Theory and RegularizationProblems Elsevier Amsterdam The Netherlands 2002

[11] M S Zhdanov Geophysical Electromagnetic Theory and Meth-ods Elsevier Amsterdam The Netherlands 2009

[12] H B Han Bo K Z Kuang Zheng and L J-Q Liu Jia-Qi ldquoAmonotonous homotopy method for solving the resistivities ofthe earthrdquo Acta Geophysica Sinica vol 34 no 4 pp 517ndash5221991

[13] H S Fu BHan andGQGai ldquoAwaveletmultiscale-homotopymethod for the inverse problem of two-dimensional acousticwave equationrdquoAppliedMathematics andComputation vol 190no 1 pp 576ndash582 2007

[14] Y He and B Han ldquoA wavelet adaptive-homotopy method forinverse problem in the fluid-saturated porous mediardquo AppliedMathematics andComputation vol 208 no 1 pp 189ndash196 2009

[15] D W Vasco ldquoSingularity and branching a path-following for-malism for geophysical inverse problemsrdquo Geophysical JournalInternational vol 119 no 3 pp 809ndash830 1994

[16] D M Dunlavy D P OrsquoLeary D Klimov and D ThirumalaildquoHOPE a homotopy optimizationmethod for protein structurepredictionrdquo Journal of Computational Biology vol 12 no 10 pp1275ndash1288 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineeringy

(m)

y(m

)

x (m)05 10 15 20 25

x (m)05 10 15 20 25

2

25

3

35

4

45

5

2

25

3

35

4

45

5

2

25

3

35

4

45

5

2

25

3

35

4

45

5

2

25

3

35

4

45

5

2

25

3

35

4

45

5

2

25

3

35

4

45

5

2

25

3

35

4

45

5

2

25

3

35

4

45

5

2

25

3

35

4

45

5

2

25

3

35

4

45

5

2

25

3

35

4

45

5

y(m

)

x (m)05 10 15 20 25

y(m

)y

(m)

x (m)05 10 15 20 25

x (m)05 10 15 20 25

y(m

)

x (m)05 10 15 20 25

y(m

)

x (m)05 10 15 20 25

y(m

)

x (m)05 10 15 20 25

y(m

)

x (m)05 10 15 20 25

y(m

)

x (m)05 10 15 20 25

y(m

)

x (m)05 10 15 20 25

y(m

)

x (m)05 10 15 20 25

0204060810121416182022

0204060810121416182022

0204060810121416182022

0204060810121416182022

0204060810121416182022

0204060810121416182022

0204060810121416182022

0204060810121416182022

0204060810121416182022

0204060810121416182022

0204060810121416182022

0204060810121416182022

Figure 9 Inversion results with different homotopy parameters (150001 15001 12001 11501 11001 1501 151 16 23 100105 10001005and 100000100005)

(i) Input given minimum norm solution of 1206010 namely1205900= 120590ref 1205820 = 0 and Δ

119894gt 0 119894 = 1 119871

(ii) Update for 120582119894= 120582119894minus1

+ Δ119894minus1

we use damping Gauss-Newton iteration method compute an approximatesolution 120590

119894to minimize min

120590isin119883119867(120590 120582

119894) by initial

value 120590119894minus1

(iii) Output the iteration will be stopped if the solution 120590

119894

satisfied stopping criterion output 1205901 = 120590119898

The homotopy algorithmrsquos greatest advantage lies in increas-ing region of convergency But in the process of computationthe nonconvergent phenomenon often happens to homotopycurve for example standstill convolution and bifurcation

These difficulties can make the homotopy method quitecomplicated In order to make the homotopy more efficientsets of solutions are produced by perturbation for everyhomotopy parameter 120582 = 120582

119894 119894 = 0 119871 Although the

calculation increased it ensures the likelihood of obtainingaccurate solutionWith different homotopy parameter 120582

119894 the

minima is in the sequence and120590119894is solved by dampingGauss-

Newton method with the initial value 120590119894minus1

The next set oflocal minima (120582 = 120582

119894+1) is solved with the initial values in the

previous set (120582 = 120582119894) which has one or more perturbations of

each of those points in the set We denote the perturbationof solution per(120590) The perturbation stochastically disturbsone or more of the solutions 120590 in set We use a single initial

Mathematical Problems in Engineering 7

02

04

06

08

10

12

14

16

18

20

22

x (m)

y(m

)

05 10 15 20 252

25

3

35

4

45

5

Figure 10 Three square cavities model

20 40 60 80 100

0

500

1000

1500

2000

t(n

s)

72

144

216

288

36

Trace

minus1000

minus500

Figure 11 Measured data

value 1205900for 120582 = 120582

0 the number of solution solved at

each homotopy parameter120582 increases exponentiallyWithoutlimit on the size of the set the number of solutions in theset would be 2

119894 when homotopy iteration stops In order toconstraint on computational complexity it is necessary tolimit the size of the set We denote by 119873max the maximumnumber of solutions in a set If the number of local minimain the set at the current iteration is less than the maximumset size 119873max then every solution in the set is used inthe next iteration otherwise we will choose the preferablesolution An ideal measure of what constitutes the preferablesolutions is regularization functional value of (15) solutionwith the lower regularization functional values is consideredthe preferable The perturbation homotopy algorithm is asfollows

(i) Input given minimum norm solution of 1206010 namely1205900= 120590ref 1205820 = 0119873max ge 1 and Δ

119894gt 0 119894 = 1 119871

(ii) Update for 120582119894

= 120582119894minus1

+ Δ119894minus1

we use damp-ing Gauss-Newton iteration method compute anapproximate solution 120590

119896

119894 119896 = 1 2

119894 to minimizemin120590isin119883

119867(120590 120582119894) by initial value 120590119895

119894minus1and (per(120590

119894minus1))119895

119895 = 1 2119894

(iii) Order the distinct solutions among 120590119896

119894 119896 = 1 2

119894from best to worst as 1205901

119894 1205902119894 and discard any worst

solution when the number of elements in 119883 largerthan119873max

(iv) Output the iteration will be stopped if the solution120590119898

119894 1 le 119898 le 2

119894 is the best solution in the set andsatisfied stopping criterion output 1205901 = 120590

119898

The sketch map of the algorithm is shown in Figure 1

4 Numerical Simulation

In this section we exhibit clear advantages of perturbationhomotopy method over traditional damping Gauss-Newtonmethod in the previous three experiments In the fourthexperiment we analysis the influence of homotopy parameteron inversion results In the fifth experiment we apply ouralgorithm to more complicated model and confirm that thisalgorithm is stable In the last experiment we solve a simplepractice data In the first three 3D experiments we selecthomotopy parameter 120582

119894= 11989410 119894 = 0 10 119873max = 2 and

internal Gauss-Newton iteration 5 with each homotopy stepin our numerical experiments The space domain is Ω Theinverse problem is considered on a uniform 32 times 32times16 gridThe transmitter source is 119904(119905) = 119905

2119890minus120572119905 sin(120596

0119905) where 120572 =

1205960radic3 120596

0= 2120587119891

0 and 119891

0= 100MHz is the central

frequency In the last three 2D experiments 119873max = 4 andinternal Gauss-Newton iteration is 10 The inversion spaceΩ = 25m times 25m space step Δ119909 = Δ119910 = 005m samplingtime 119905 = 36 ns time step 120591 = 18 ns and the transmitter sourceis Ricker wavelet with 900MHz central frequency

41 HomogeneousModel Wefirst test a simple homogeneousmodel where permittivity 120576 = 56064 times 10

minus12 Fm perme-ability 120583 = 0 and relative conductivity 120590 = minus52983 BothdampingGauss-Newtonmethod and perturbation homotopymethod select the same relative initial value 120590 = minus48993Iterations of damping Gauss-Newton method are five timeselapsed time is 1230426 seconds homotopy algorithmrsquos outeriterations 120582

119894 119894 = 1 10 are ten times inner local

iterations (damping Gauss-Newton method) are three timesand elapsed time is 1405536 seconds Figure 2 gives resultsof true model damping Gauss-Newton and perturbationhomotopy method The relative error 120590inv minus 120590true120590trueof (a) and (b) in Figure 2 is 00516 and of (a) and (c) is00830 where 120590inv and 120590true are inversion solution and truemodel respectively From Figure 2 we can see that wheninitial value is close to original conductivity both methodshave satisfactory result but perturbation homotopy is notmore efficient

42 One Circle Cavity Model The relative conductivity ofcavity is minus32775 other parameters have the same value asin the above model Both methods select the same initialvalue 120590 = minus16094 Elapsed time of damping Gauss-Newtonmethod and perturbation homotopy method is 11120536seconds and 12578436 seconds respectively The relativeerror 120590inv minus 120590true120590true of (a) and (b) in Figure 3 is

8 Mathematical Problems in Engineeringy

(m)

x (m)05 10 15 20 25

2

25

3

35

4

45

5

2

25

3

35

4

45

5

2

25

3

35

4

45

5

2

25

3

35

4

45

5

2

25

3

35

4

45

5

2

25

3

35

4

45

5

2

25

3

35

4

45

5

2

25

3

35

4

45

5

2

25

3

35

4

45

5

0204060810121416182022

x (m)05 10 15 20 25

y(m

)

0204060810121416182022

x (m)05 10 15 20 25

y(m

)

0204060810121416182022

x (m)05 10 15 20 25

y(m

)

0204060810121416182022

x (m)05 10 15 20 25

y(m

)

0204060810121416182022

x (m)05 10 15 20 25

y(m

)

0204060810121416182022

x (m)05 10 15 20 25

y(m

)

0204060810121416182022

y(m

)

x (m)05 10 15 20 25

0204060810121416182022

y(m

)

x (m)05 10 15 20 25

0204060810121416182022

Figure 12 Inversion results with different SNR measured data (1 dB 3 dB 5 dB 8 dB 10 dB 12 dB 15 dB 20 dB and 30 dB)

2535 and of (a) and (c) is 961 Although elapsed timeof perturbation homotopy is ten times more than dampingGauss-Newton method when initial value is far away fromtrue value the latter is inefficient compared to the former

43 Two Cavities Model The conductivity of two cavities isequal other parameters have the same value as in Section 42Both methods select the same relative initial value 120590 =

minus08775 Elapsed time of damping Gauss-Newton methodand perturbation homotopy method is 11080546 secondsand 15458768 seconds respectively Figure 4 gives results oforiginal model damping Gauss-Newton and perturbationhomotopy method The relative error 120590inv minus 120590true120590true of(a) and (b) in Figure 4 is 4935 and of (a) and (c) is 1285When the initial value is further away from original valuedamping Gauss-Newton method is invalid

44 One Square Cavity 2D Model The vertex coordinate ofsquare cavity is 12m 04m 13m 04m 12m 05m 13m05m respectively The original model structure schemeis shown in Figure 5 Inversion results by Gauss-Newtonmethod are shown in Figure 7 Measured data is shownin Figure 6 With the influence of local minima Figure 7

roughly describes the information of location and scale ofsquare cavity but the quality information error is hugeBecause traditional Tikhonov regularization would smooththe inversion solution the solution always appears blurringphenomenon at boundary Inversion results by perturbationhomotopy method with different homotopy parameter areshown in Figure 8 the relative error between inversion andoriginal model is 455 We select homotopy parameter as150001 15001 12001 11501 11001 1501 151 16 23100105 10001005 and 100000100005 From Figure 9 wecan conclude that the rate of convergence is fast but whenhomotopy parameter is close to 1 numerical solution isdiverging so the optimal solution is not at 120582 = 1

45 Three Square Cavities 2D Model The original modelstructure scheme is shown in Figure 10 Measured data isshown in Figure 11 In order to demonstrate that the methodis stable with noise we add different Gauss white noise tomeasured data The SNR (signal-to-noise ratio) is 1 dB 3 dB5 dB 8 dB 10 dB 12 dB 15 dB 20 dB and 30 dB respectivelyFrom Figure 12 we can see that with noise decrease numer-ical solution is convergent to original model So our methodis stable

Mathematical Problems in Engineering 9

1 2 3 4 5

50

100

150

200

250

300

350

400

450

500

y(m

)

x (m)

01

02

03

04

05

06

07

08

09

10

Figure 13 Original model

10 20 30 40 50 60 70 80 90 100

3

6

9

12

15

0

500

1000

1500

t(n

s)

Trace

minus1000

minus1500

minus500

Figure 14 Practice measured data

46 Practice Data Inversion The practice measured data isprovided by Wuhan Changjiang Engineering GeophysicalExploration amp Testing Co LtdChina The testing object isa concrete member with three circle rebar in it The centre ofcircle is 14m 06m 22m 04m 34m 07m respectivelyradius is 01mThe original model structure scheme is shownin Figure 13 Practice measured data is shown in Figure 14Inversion result is shown in Figure 15 The relative error is1575 Before the inversion it is necessary to preprocess thedata because practice measured data has large noise

5 Conclusion

This paper has addressed the perturbation homotopymethodfor solving inverse electromagnetic problem Perturbationhomotopy coupled with damping Gauss-Newton methodshas been used in the experiments Results show that wheninitial value is far away from true model local convergentGauss-Newton method just roughly gives the informationof location and scale the quality has huge error with true

50

100

150

200

250

300

350

400

450

500

y(m

)

01

02

03

04

05

06

07

08

09

10

x (m)05 10 15 20 25

Figure 15 Inversion results

model Perturbation homotopy method is a widely conver-gent algorithm it could give amore accurate inversion resultseven if initial value is far away from the original value Whenadding Gauss white noise to measured data we can see thatwith noise decrease numerical solution is convergence tooriginal model so numerical results support that ourmethodis stable Traditional homotopy theory tells us that optimumsolution is determined at 120582 = 1 but numerical resultsshow that the solution will diverge when 120582 is close to 1because of regularization parameter impact In a word thismethod has clear advantages over damping Gauss-Newtonmethod it is an effective method especially on aspects ofwide convergence computational efficiency and precisionand has broad application prospects

Conflict of Interests

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

Acknowledgments

This paper is supported by Natural Science Foundation ofChina Project no 41304093 the Fundamental ResearchFunds for the Central Universities Project no DL12BB20and the Educational Commission of Heilongjiang Provinceof China no 12533013

References

[1] D Colton and R Kress Inverse Acoustic and ElectromagneticScattering Theory Springer New York NY USA 1998

[2] P A Eaton ldquo3-D electromagnetic inversion using integralequationsrdquo Geophysical Prospecting vol 37 pp 407ndash426 1989

[3] J Nocedal and S Wright Numerical Optimization SpringerNew York NY USA 1999

[4] G A Newman S Recher B Tezkan and F M Neubauerldquo3D inversion of a scalar radio magnetotelluric field data setrdquoGeophysics vol 68 no 3 pp 791ndash802 2003

10 Mathematical Problems in Engineering

[5] E Haber ldquoQuasi-Newton methods for large-scale electromag-netic inverse problemsrdquo Inverse Problems vol 21 no 1 pp 305ndash323 2005

[6] M S Zhdanov S Fang and G Hursan ldquoElectromagneticinversion using quasi-linear approximationrdquo Geophysics vol65 no 5 pp 1501ndash1513 2000

[7] C Torres-Verdın and T M Habashy ldquoRapid numerical sim-ulation of axisymmetric single-well induction data using theextended Born approximationrdquo Radio Science vol 36 no 6 pp1287ndash1306 2001

[8] H-W Tseng K H Lee and A Becker ldquo3D interpretation ofelectromagnetic data using a modified extended Born approxi-mationrdquo Geophysics vol 68 no 1 pp 127ndash137 2003

[9] Z Zhang ldquo3D resistivity mapping of airborne EM datardquoGeophysics vol 68 no 6 pp 1896ndash1905 2003

[10] M S Zhdanov Geophysical Inverse Theory and RegularizationProblems Elsevier Amsterdam The Netherlands 2002

[11] M S Zhdanov Geophysical Electromagnetic Theory and Meth-ods Elsevier Amsterdam The Netherlands 2009

[12] H B Han Bo K Z Kuang Zheng and L J-Q Liu Jia-Qi ldquoAmonotonous homotopy method for solving the resistivities ofthe earthrdquo Acta Geophysica Sinica vol 34 no 4 pp 517ndash5221991

[13] H S Fu BHan andGQGai ldquoAwaveletmultiscale-homotopymethod for the inverse problem of two-dimensional acousticwave equationrdquoAppliedMathematics andComputation vol 190no 1 pp 576ndash582 2007

[14] Y He and B Han ldquoA wavelet adaptive-homotopy method forinverse problem in the fluid-saturated porous mediardquo AppliedMathematics andComputation vol 208 no 1 pp 189ndash196 2009

[15] D W Vasco ldquoSingularity and branching a path-following for-malism for geophysical inverse problemsrdquo Geophysical JournalInternational vol 119 no 3 pp 809ndash830 1994

[16] D M Dunlavy D P OrsquoLeary D Klimov and D ThirumalaildquoHOPE a homotopy optimizationmethod for protein structurepredictionrdquo Journal of Computational Biology vol 12 no 10 pp1275ndash1288 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

02

04

06

08

10

12

14

16

18

20

22

x (m)

y(m

)

05 10 15 20 252

25

3

35

4

45

5

Figure 10 Three square cavities model

20 40 60 80 100

0

500

1000

1500

2000

t(n

s)

72

144

216

288

36

Trace

minus1000

minus500

Figure 11 Measured data

value 1205900for 120582 = 120582

0 the number of solution solved at

each homotopy parameter120582 increases exponentiallyWithoutlimit on the size of the set the number of solutions in theset would be 2

119894 when homotopy iteration stops In order toconstraint on computational complexity it is necessary tolimit the size of the set We denote by 119873max the maximumnumber of solutions in a set If the number of local minimain the set at the current iteration is less than the maximumset size 119873max then every solution in the set is used inthe next iteration otherwise we will choose the preferablesolution An ideal measure of what constitutes the preferablesolutions is regularization functional value of (15) solutionwith the lower regularization functional values is consideredthe preferable The perturbation homotopy algorithm is asfollows

(i) Input given minimum norm solution of 1206010 namely1205900= 120590ref 1205820 = 0119873max ge 1 and Δ

119894gt 0 119894 = 1 119871

(ii) Update for 120582119894

= 120582119894minus1

+ Δ119894minus1

we use damp-ing Gauss-Newton iteration method compute anapproximate solution 120590

119896

119894 119896 = 1 2

119894 to minimizemin120590isin119883

119867(120590 120582119894) by initial value 120590119895

119894minus1and (per(120590

119894minus1))119895

119895 = 1 2119894

(iii) Order the distinct solutions among 120590119896

119894 119896 = 1 2

119894from best to worst as 1205901

119894 1205902119894 and discard any worst

solution when the number of elements in 119883 largerthan119873max

(iv) Output the iteration will be stopped if the solution120590119898

119894 1 le 119898 le 2

119894 is the best solution in the set andsatisfied stopping criterion output 1205901 = 120590

119898

The sketch map of the algorithm is shown in Figure 1

4 Numerical Simulation

In this section we exhibit clear advantages of perturbationhomotopy method over traditional damping Gauss-Newtonmethod in the previous three experiments In the fourthexperiment we analysis the influence of homotopy parameteron inversion results In the fifth experiment we apply ouralgorithm to more complicated model and confirm that thisalgorithm is stable In the last experiment we solve a simplepractice data In the first three 3D experiments we selecthomotopy parameter 120582

119894= 11989410 119894 = 0 10 119873max = 2 and

internal Gauss-Newton iteration 5 with each homotopy stepin our numerical experiments The space domain is Ω Theinverse problem is considered on a uniform 32 times 32times16 gridThe transmitter source is 119904(119905) = 119905

2119890minus120572119905 sin(120596

0119905) where 120572 =

1205960radic3 120596

0= 2120587119891

0 and 119891

0= 100MHz is the central

frequency In the last three 2D experiments 119873max = 4 andinternal Gauss-Newton iteration is 10 The inversion spaceΩ = 25m times 25m space step Δ119909 = Δ119910 = 005m samplingtime 119905 = 36 ns time step 120591 = 18 ns and the transmitter sourceis Ricker wavelet with 900MHz central frequency

41 HomogeneousModel Wefirst test a simple homogeneousmodel where permittivity 120576 = 56064 times 10

minus12 Fm perme-ability 120583 = 0 and relative conductivity 120590 = minus52983 BothdampingGauss-Newtonmethod and perturbation homotopymethod select the same relative initial value 120590 = minus48993Iterations of damping Gauss-Newton method are five timeselapsed time is 1230426 seconds homotopy algorithmrsquos outeriterations 120582

119894 119894 = 1 10 are ten times inner local

iterations (damping Gauss-Newton method) are three timesand elapsed time is 1405536 seconds Figure 2 gives resultsof true model damping Gauss-Newton and perturbationhomotopy method The relative error 120590inv minus 120590true120590trueof (a) and (b) in Figure 2 is 00516 and of (a) and (c) is00830 where 120590inv and 120590true are inversion solution and truemodel respectively From Figure 2 we can see that wheninitial value is close to original conductivity both methodshave satisfactory result but perturbation homotopy is notmore efficient

42 One Circle Cavity Model The relative conductivity ofcavity is minus32775 other parameters have the same value asin the above model Both methods select the same initialvalue 120590 = minus16094 Elapsed time of damping Gauss-Newtonmethod and perturbation homotopy method is 11120536seconds and 12578436 seconds respectively The relativeerror 120590inv minus 120590true120590true of (a) and (b) in Figure 3 is

8 Mathematical Problems in Engineeringy

(m)

x (m)05 10 15 20 25

2

25

3

35

4

45

5

2

25

3

35

4

45

5

2

25

3

35

4

45

5

2

25

3

35

4

45

5

2

25

3

35

4

45

5

2

25

3

35

4

45

5

2

25

3

35

4

45

5

2

25

3

35

4

45

5

2

25

3

35

4

45

5

0204060810121416182022

x (m)05 10 15 20 25

y(m

)

0204060810121416182022

x (m)05 10 15 20 25

y(m

)

0204060810121416182022

x (m)05 10 15 20 25

y(m

)

0204060810121416182022

x (m)05 10 15 20 25

y(m

)

0204060810121416182022

x (m)05 10 15 20 25

y(m

)

0204060810121416182022

x (m)05 10 15 20 25

y(m

)

0204060810121416182022

y(m

)

x (m)05 10 15 20 25

0204060810121416182022

y(m

)

x (m)05 10 15 20 25

0204060810121416182022

Figure 12 Inversion results with different SNR measured data (1 dB 3 dB 5 dB 8 dB 10 dB 12 dB 15 dB 20 dB and 30 dB)

2535 and of (a) and (c) is 961 Although elapsed timeof perturbation homotopy is ten times more than dampingGauss-Newton method when initial value is far away fromtrue value the latter is inefficient compared to the former

43 Two Cavities Model The conductivity of two cavities isequal other parameters have the same value as in Section 42Both methods select the same relative initial value 120590 =

minus08775 Elapsed time of damping Gauss-Newton methodand perturbation homotopy method is 11080546 secondsand 15458768 seconds respectively Figure 4 gives results oforiginal model damping Gauss-Newton and perturbationhomotopy method The relative error 120590inv minus 120590true120590true of(a) and (b) in Figure 4 is 4935 and of (a) and (c) is 1285When the initial value is further away from original valuedamping Gauss-Newton method is invalid

44 One Square Cavity 2D Model The vertex coordinate ofsquare cavity is 12m 04m 13m 04m 12m 05m 13m05m respectively The original model structure schemeis shown in Figure 5 Inversion results by Gauss-Newtonmethod are shown in Figure 7 Measured data is shownin Figure 6 With the influence of local minima Figure 7

roughly describes the information of location and scale ofsquare cavity but the quality information error is hugeBecause traditional Tikhonov regularization would smooththe inversion solution the solution always appears blurringphenomenon at boundary Inversion results by perturbationhomotopy method with different homotopy parameter areshown in Figure 8 the relative error between inversion andoriginal model is 455 We select homotopy parameter as150001 15001 12001 11501 11001 1501 151 16 23100105 10001005 and 100000100005 From Figure 9 wecan conclude that the rate of convergence is fast but whenhomotopy parameter is close to 1 numerical solution isdiverging so the optimal solution is not at 120582 = 1

45 Three Square Cavities 2D Model The original modelstructure scheme is shown in Figure 10 Measured data isshown in Figure 11 In order to demonstrate that the methodis stable with noise we add different Gauss white noise tomeasured data The SNR (signal-to-noise ratio) is 1 dB 3 dB5 dB 8 dB 10 dB 12 dB 15 dB 20 dB and 30 dB respectivelyFrom Figure 12 we can see that with noise decrease numer-ical solution is convergent to original model So our methodis stable

Mathematical Problems in Engineering 9

1 2 3 4 5

50

100

150

200

250

300

350

400

450

500

y(m

)

x (m)

01

02

03

04

05

06

07

08

09

10

Figure 13 Original model

10 20 30 40 50 60 70 80 90 100

3

6

9

12

15

0

500

1000

1500

t(n

s)

Trace

minus1000

minus1500

minus500

Figure 14 Practice measured data

46 Practice Data Inversion The practice measured data isprovided by Wuhan Changjiang Engineering GeophysicalExploration amp Testing Co LtdChina The testing object isa concrete member with three circle rebar in it The centre ofcircle is 14m 06m 22m 04m 34m 07m respectivelyradius is 01mThe original model structure scheme is shownin Figure 13 Practice measured data is shown in Figure 14Inversion result is shown in Figure 15 The relative error is1575 Before the inversion it is necessary to preprocess thedata because practice measured data has large noise

5 Conclusion

This paper has addressed the perturbation homotopymethodfor solving inverse electromagnetic problem Perturbationhomotopy coupled with damping Gauss-Newton methodshas been used in the experiments Results show that wheninitial value is far away from true model local convergentGauss-Newton method just roughly gives the informationof location and scale the quality has huge error with true

50

100

150

200

250

300

350

400

450

500

y(m

)

01

02

03

04

05

06

07

08

09

10

x (m)05 10 15 20 25

Figure 15 Inversion results

model Perturbation homotopy method is a widely conver-gent algorithm it could give amore accurate inversion resultseven if initial value is far away from the original value Whenadding Gauss white noise to measured data we can see thatwith noise decrease numerical solution is convergence tooriginal model so numerical results support that ourmethodis stable Traditional homotopy theory tells us that optimumsolution is determined at 120582 = 1 but numerical resultsshow that the solution will diverge when 120582 is close to 1because of regularization parameter impact In a word thismethod has clear advantages over damping Gauss-Newtonmethod it is an effective method especially on aspects ofwide convergence computational efficiency and precisionand has broad application prospects

Conflict of Interests

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

Acknowledgments

This paper is supported by Natural Science Foundation ofChina Project no 41304093 the Fundamental ResearchFunds for the Central Universities Project no DL12BB20and the Educational Commission of Heilongjiang Provinceof China no 12533013

References

[1] D Colton and R Kress Inverse Acoustic and ElectromagneticScattering Theory Springer New York NY USA 1998

[2] P A Eaton ldquo3-D electromagnetic inversion using integralequationsrdquo Geophysical Prospecting vol 37 pp 407ndash426 1989

[3] J Nocedal and S Wright Numerical Optimization SpringerNew York NY USA 1999

[4] G A Newman S Recher B Tezkan and F M Neubauerldquo3D inversion of a scalar radio magnetotelluric field data setrdquoGeophysics vol 68 no 3 pp 791ndash802 2003

10 Mathematical Problems in Engineering

[5] E Haber ldquoQuasi-Newton methods for large-scale electromag-netic inverse problemsrdquo Inverse Problems vol 21 no 1 pp 305ndash323 2005

[6] M S Zhdanov S Fang and G Hursan ldquoElectromagneticinversion using quasi-linear approximationrdquo Geophysics vol65 no 5 pp 1501ndash1513 2000

[7] C Torres-Verdın and T M Habashy ldquoRapid numerical sim-ulation of axisymmetric single-well induction data using theextended Born approximationrdquo Radio Science vol 36 no 6 pp1287ndash1306 2001

[8] H-W Tseng K H Lee and A Becker ldquo3D interpretation ofelectromagnetic data using a modified extended Born approxi-mationrdquo Geophysics vol 68 no 1 pp 127ndash137 2003

[9] Z Zhang ldquo3D resistivity mapping of airborne EM datardquoGeophysics vol 68 no 6 pp 1896ndash1905 2003

[10] M S Zhdanov Geophysical Inverse Theory and RegularizationProblems Elsevier Amsterdam The Netherlands 2002

[11] M S Zhdanov Geophysical Electromagnetic Theory and Meth-ods Elsevier Amsterdam The Netherlands 2009

[12] H B Han Bo K Z Kuang Zheng and L J-Q Liu Jia-Qi ldquoAmonotonous homotopy method for solving the resistivities ofthe earthrdquo Acta Geophysica Sinica vol 34 no 4 pp 517ndash5221991

[13] H S Fu BHan andGQGai ldquoAwaveletmultiscale-homotopymethod for the inverse problem of two-dimensional acousticwave equationrdquoAppliedMathematics andComputation vol 190no 1 pp 576ndash582 2007

[14] Y He and B Han ldquoA wavelet adaptive-homotopy method forinverse problem in the fluid-saturated porous mediardquo AppliedMathematics andComputation vol 208 no 1 pp 189ndash196 2009

[15] D W Vasco ldquoSingularity and branching a path-following for-malism for geophysical inverse problemsrdquo Geophysical JournalInternational vol 119 no 3 pp 809ndash830 1994

[16] D M Dunlavy D P OrsquoLeary D Klimov and D ThirumalaildquoHOPE a homotopy optimizationmethod for protein structurepredictionrdquo Journal of Computational Biology vol 12 no 10 pp1275ndash1288 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineeringy

(m)

x (m)05 10 15 20 25

2

25

3

35

4

45

5

2

25

3

35

4

45

5

2

25

3

35

4

45

5

2

25

3

35

4

45

5

2

25

3

35

4

45

5

2

25

3

35

4

45

5

2

25

3

35

4

45

5

2

25

3

35

4

45

5

2

25

3

35

4

45

5

0204060810121416182022

x (m)05 10 15 20 25

y(m

)

0204060810121416182022

x (m)05 10 15 20 25

y(m

)

0204060810121416182022

x (m)05 10 15 20 25

y(m

)

0204060810121416182022

x (m)05 10 15 20 25

y(m

)

0204060810121416182022

x (m)05 10 15 20 25

y(m

)

0204060810121416182022

x (m)05 10 15 20 25

y(m

)

0204060810121416182022

y(m

)

x (m)05 10 15 20 25

0204060810121416182022

y(m

)

x (m)05 10 15 20 25

0204060810121416182022

Figure 12 Inversion results with different SNR measured data (1 dB 3 dB 5 dB 8 dB 10 dB 12 dB 15 dB 20 dB and 30 dB)

2535 and of (a) and (c) is 961 Although elapsed timeof perturbation homotopy is ten times more than dampingGauss-Newton method when initial value is far away fromtrue value the latter is inefficient compared to the former

43 Two Cavities Model The conductivity of two cavities isequal other parameters have the same value as in Section 42Both methods select the same relative initial value 120590 =

minus08775 Elapsed time of damping Gauss-Newton methodand perturbation homotopy method is 11080546 secondsand 15458768 seconds respectively Figure 4 gives results oforiginal model damping Gauss-Newton and perturbationhomotopy method The relative error 120590inv minus 120590true120590true of(a) and (b) in Figure 4 is 4935 and of (a) and (c) is 1285When the initial value is further away from original valuedamping Gauss-Newton method is invalid

44 One Square Cavity 2D Model The vertex coordinate ofsquare cavity is 12m 04m 13m 04m 12m 05m 13m05m respectively The original model structure schemeis shown in Figure 5 Inversion results by Gauss-Newtonmethod are shown in Figure 7 Measured data is shownin Figure 6 With the influence of local minima Figure 7

roughly describes the information of location and scale ofsquare cavity but the quality information error is hugeBecause traditional Tikhonov regularization would smooththe inversion solution the solution always appears blurringphenomenon at boundary Inversion results by perturbationhomotopy method with different homotopy parameter areshown in Figure 8 the relative error between inversion andoriginal model is 455 We select homotopy parameter as150001 15001 12001 11501 11001 1501 151 16 23100105 10001005 and 100000100005 From Figure 9 wecan conclude that the rate of convergence is fast but whenhomotopy parameter is close to 1 numerical solution isdiverging so the optimal solution is not at 120582 = 1

45 Three Square Cavities 2D Model The original modelstructure scheme is shown in Figure 10 Measured data isshown in Figure 11 In order to demonstrate that the methodis stable with noise we add different Gauss white noise tomeasured data The SNR (signal-to-noise ratio) is 1 dB 3 dB5 dB 8 dB 10 dB 12 dB 15 dB 20 dB and 30 dB respectivelyFrom Figure 12 we can see that with noise decrease numer-ical solution is convergent to original model So our methodis stable

Mathematical Problems in Engineering 9

1 2 3 4 5

50

100

150

200

250

300

350

400

450

500

y(m

)

x (m)

01

02

03

04

05

06

07

08

09

10

Figure 13 Original model

10 20 30 40 50 60 70 80 90 100

3

6

9

12

15

0

500

1000

1500

t(n

s)

Trace

minus1000

minus1500

minus500

Figure 14 Practice measured data

46 Practice Data Inversion The practice measured data isprovided by Wuhan Changjiang Engineering GeophysicalExploration amp Testing Co LtdChina The testing object isa concrete member with three circle rebar in it The centre ofcircle is 14m 06m 22m 04m 34m 07m respectivelyradius is 01mThe original model structure scheme is shownin Figure 13 Practice measured data is shown in Figure 14Inversion result is shown in Figure 15 The relative error is1575 Before the inversion it is necessary to preprocess thedata because practice measured data has large noise

5 Conclusion

This paper has addressed the perturbation homotopymethodfor solving inverse electromagnetic problem Perturbationhomotopy coupled with damping Gauss-Newton methodshas been used in the experiments Results show that wheninitial value is far away from true model local convergentGauss-Newton method just roughly gives the informationof location and scale the quality has huge error with true

50

100

150

200

250

300

350

400

450

500

y(m

)

01

02

03

04

05

06

07

08

09

10

x (m)05 10 15 20 25

Figure 15 Inversion results

model Perturbation homotopy method is a widely conver-gent algorithm it could give amore accurate inversion resultseven if initial value is far away from the original value Whenadding Gauss white noise to measured data we can see thatwith noise decrease numerical solution is convergence tooriginal model so numerical results support that ourmethodis stable Traditional homotopy theory tells us that optimumsolution is determined at 120582 = 1 but numerical resultsshow that the solution will diverge when 120582 is close to 1because of regularization parameter impact In a word thismethod has clear advantages over damping Gauss-Newtonmethod it is an effective method especially on aspects ofwide convergence computational efficiency and precisionand has broad application prospects

Conflict of Interests

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

Acknowledgments

This paper is supported by Natural Science Foundation ofChina Project no 41304093 the Fundamental ResearchFunds for the Central Universities Project no DL12BB20and the Educational Commission of Heilongjiang Provinceof China no 12533013

References

[1] D Colton and R Kress Inverse Acoustic and ElectromagneticScattering Theory Springer New York NY USA 1998

[2] P A Eaton ldquo3-D electromagnetic inversion using integralequationsrdquo Geophysical Prospecting vol 37 pp 407ndash426 1989

[3] J Nocedal and S Wright Numerical Optimization SpringerNew York NY USA 1999

[4] G A Newman S Recher B Tezkan and F M Neubauerldquo3D inversion of a scalar radio magnetotelluric field data setrdquoGeophysics vol 68 no 3 pp 791ndash802 2003

10 Mathematical Problems in Engineering

[5] E Haber ldquoQuasi-Newton methods for large-scale electromag-netic inverse problemsrdquo Inverse Problems vol 21 no 1 pp 305ndash323 2005

[6] M S Zhdanov S Fang and G Hursan ldquoElectromagneticinversion using quasi-linear approximationrdquo Geophysics vol65 no 5 pp 1501ndash1513 2000

[7] C Torres-Verdın and T M Habashy ldquoRapid numerical sim-ulation of axisymmetric single-well induction data using theextended Born approximationrdquo Radio Science vol 36 no 6 pp1287ndash1306 2001

[8] H-W Tseng K H Lee and A Becker ldquo3D interpretation ofelectromagnetic data using a modified extended Born approxi-mationrdquo Geophysics vol 68 no 1 pp 127ndash137 2003

[9] Z Zhang ldquo3D resistivity mapping of airborne EM datardquoGeophysics vol 68 no 6 pp 1896ndash1905 2003

[10] M S Zhdanov Geophysical Inverse Theory and RegularizationProblems Elsevier Amsterdam The Netherlands 2002

[11] M S Zhdanov Geophysical Electromagnetic Theory and Meth-ods Elsevier Amsterdam The Netherlands 2009

[12] H B Han Bo K Z Kuang Zheng and L J-Q Liu Jia-Qi ldquoAmonotonous homotopy method for solving the resistivities ofthe earthrdquo Acta Geophysica Sinica vol 34 no 4 pp 517ndash5221991

[13] H S Fu BHan andGQGai ldquoAwaveletmultiscale-homotopymethod for the inverse problem of two-dimensional acousticwave equationrdquoAppliedMathematics andComputation vol 190no 1 pp 576ndash582 2007

[14] Y He and B Han ldquoA wavelet adaptive-homotopy method forinverse problem in the fluid-saturated porous mediardquo AppliedMathematics andComputation vol 208 no 1 pp 189ndash196 2009

[15] D W Vasco ldquoSingularity and branching a path-following for-malism for geophysical inverse problemsrdquo Geophysical JournalInternational vol 119 no 3 pp 809ndash830 1994

[16] D M Dunlavy D P OrsquoLeary D Klimov and D ThirumalaildquoHOPE a homotopy optimizationmethod for protein structurepredictionrdquo Journal of Computational Biology vol 12 no 10 pp1275ndash1288 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

1 2 3 4 5

50

100

150

200

250

300

350

400

450

500

y(m

)

x (m)

01

02

03

04

05

06

07

08

09

10

Figure 13 Original model

10 20 30 40 50 60 70 80 90 100

3

6

9

12

15

0

500

1000

1500

t(n

s)

Trace

minus1000

minus1500

minus500

Figure 14 Practice measured data

46 Practice Data Inversion The practice measured data isprovided by Wuhan Changjiang Engineering GeophysicalExploration amp Testing Co LtdChina The testing object isa concrete member with three circle rebar in it The centre ofcircle is 14m 06m 22m 04m 34m 07m respectivelyradius is 01mThe original model structure scheme is shownin Figure 13 Practice measured data is shown in Figure 14Inversion result is shown in Figure 15 The relative error is1575 Before the inversion it is necessary to preprocess thedata because practice measured data has large noise

5 Conclusion

This paper has addressed the perturbation homotopymethodfor solving inverse electromagnetic problem Perturbationhomotopy coupled with damping Gauss-Newton methodshas been used in the experiments Results show that wheninitial value is far away from true model local convergentGauss-Newton method just roughly gives the informationof location and scale the quality has huge error with true

50

100

150

200

250

300

350

400

450

500

y(m

)

01

02

03

04

05

06

07

08

09

10

x (m)05 10 15 20 25

Figure 15 Inversion results

model Perturbation homotopy method is a widely conver-gent algorithm it could give amore accurate inversion resultseven if initial value is far away from the original value Whenadding Gauss white noise to measured data we can see thatwith noise decrease numerical solution is convergence tooriginal model so numerical results support that ourmethodis stable Traditional homotopy theory tells us that optimumsolution is determined at 120582 = 1 but numerical resultsshow that the solution will diverge when 120582 is close to 1because of regularization parameter impact In a word thismethod has clear advantages over damping Gauss-Newtonmethod it is an effective method especially on aspects ofwide convergence computational efficiency and precisionand has broad application prospects

Conflict of Interests

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

Acknowledgments

This paper is supported by Natural Science Foundation ofChina Project no 41304093 the Fundamental ResearchFunds for the Central Universities Project no DL12BB20and the Educational Commission of Heilongjiang Provinceof China no 12533013

References

[1] D Colton and R Kress Inverse Acoustic and ElectromagneticScattering Theory Springer New York NY USA 1998

[2] P A Eaton ldquo3-D electromagnetic inversion using integralequationsrdquo Geophysical Prospecting vol 37 pp 407ndash426 1989

[3] J Nocedal and S Wright Numerical Optimization SpringerNew York NY USA 1999

[4] G A Newman S Recher B Tezkan and F M Neubauerldquo3D inversion of a scalar radio magnetotelluric field data setrdquoGeophysics vol 68 no 3 pp 791ndash802 2003

10 Mathematical Problems in Engineering

[5] E Haber ldquoQuasi-Newton methods for large-scale electromag-netic inverse problemsrdquo Inverse Problems vol 21 no 1 pp 305ndash323 2005

[6] M S Zhdanov S Fang and G Hursan ldquoElectromagneticinversion using quasi-linear approximationrdquo Geophysics vol65 no 5 pp 1501ndash1513 2000

[7] C Torres-Verdın and T M Habashy ldquoRapid numerical sim-ulation of axisymmetric single-well induction data using theextended Born approximationrdquo Radio Science vol 36 no 6 pp1287ndash1306 2001

[8] H-W Tseng K H Lee and A Becker ldquo3D interpretation ofelectromagnetic data using a modified extended Born approxi-mationrdquo Geophysics vol 68 no 1 pp 127ndash137 2003

[9] Z Zhang ldquo3D resistivity mapping of airborne EM datardquoGeophysics vol 68 no 6 pp 1896ndash1905 2003

[10] M S Zhdanov Geophysical Inverse Theory and RegularizationProblems Elsevier Amsterdam The Netherlands 2002

[11] M S Zhdanov Geophysical Electromagnetic Theory and Meth-ods Elsevier Amsterdam The Netherlands 2009

[12] H B Han Bo K Z Kuang Zheng and L J-Q Liu Jia-Qi ldquoAmonotonous homotopy method for solving the resistivities ofthe earthrdquo Acta Geophysica Sinica vol 34 no 4 pp 517ndash5221991

[13] H S Fu BHan andGQGai ldquoAwaveletmultiscale-homotopymethod for the inverse problem of two-dimensional acousticwave equationrdquoAppliedMathematics andComputation vol 190no 1 pp 576ndash582 2007

[14] Y He and B Han ldquoA wavelet adaptive-homotopy method forinverse problem in the fluid-saturated porous mediardquo AppliedMathematics andComputation vol 208 no 1 pp 189ndash196 2009

[15] D W Vasco ldquoSingularity and branching a path-following for-malism for geophysical inverse problemsrdquo Geophysical JournalInternational vol 119 no 3 pp 809ndash830 1994

[16] D M Dunlavy D P OrsquoLeary D Klimov and D ThirumalaildquoHOPE a homotopy optimizationmethod for protein structurepredictionrdquo Journal of Computational Biology vol 12 no 10 pp1275ndash1288 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

[5] E Haber ldquoQuasi-Newton methods for large-scale electromag-netic inverse problemsrdquo Inverse Problems vol 21 no 1 pp 305ndash323 2005

[6] M S Zhdanov S Fang and G Hursan ldquoElectromagneticinversion using quasi-linear approximationrdquo Geophysics vol65 no 5 pp 1501ndash1513 2000

[7] C Torres-Verdın and T M Habashy ldquoRapid numerical sim-ulation of axisymmetric single-well induction data using theextended Born approximationrdquo Radio Science vol 36 no 6 pp1287ndash1306 2001

[8] H-W Tseng K H Lee and A Becker ldquo3D interpretation ofelectromagnetic data using a modified extended Born approxi-mationrdquo Geophysics vol 68 no 1 pp 127ndash137 2003

[9] Z Zhang ldquo3D resistivity mapping of airborne EM datardquoGeophysics vol 68 no 6 pp 1896ndash1905 2003

[10] M S Zhdanov Geophysical Inverse Theory and RegularizationProblems Elsevier Amsterdam The Netherlands 2002

[11] M S Zhdanov Geophysical Electromagnetic Theory and Meth-ods Elsevier Amsterdam The Netherlands 2009

[12] H B Han Bo K Z Kuang Zheng and L J-Q Liu Jia-Qi ldquoAmonotonous homotopy method for solving the resistivities ofthe earthrdquo Acta Geophysica Sinica vol 34 no 4 pp 517ndash5221991

[13] H S Fu BHan andGQGai ldquoAwaveletmultiscale-homotopymethod for the inverse problem of two-dimensional acousticwave equationrdquoAppliedMathematics andComputation vol 190no 1 pp 576ndash582 2007

[14] Y He and B Han ldquoA wavelet adaptive-homotopy method forinverse problem in the fluid-saturated porous mediardquo AppliedMathematics andComputation vol 208 no 1 pp 189ndash196 2009

[15] D W Vasco ldquoSingularity and branching a path-following for-malism for geophysical inverse problemsrdquo Geophysical JournalInternational vol 119 no 3 pp 809ndash830 1994

[16] D M Dunlavy D P OrsquoLeary D Klimov and D ThirumalaildquoHOPE a homotopy optimizationmethod for protein structurepredictionrdquo Journal of Computational Biology vol 12 no 10 pp1275ndash1288 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of