the mfs for numerical boundary identification in two-dimensional harmonic problems

Engineering Analysis with Boundary Elements 35 (2011) 342–354

Contents lists available at ScienceDirect

Engineering Analysis with Boundary Elements

0955-79

doi:10.1

� Corr

E-m

andreas

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

The MFS for numerical boundary identification in two-dimensionalharmonic problems

Liviu Marin a,�, Andreas Karageorghis b, Daniel Lesnic c

a Institute of Solid Mechanics, Romanian Academy, 15 Constantin Mille, P.O. Box 1-863, 010141 Bucharest, Romaniab Department of Mathematics and Statistics, University of Cyprus=Panepist �Zmio K �uprou, P.O. Box 20537, 1678 Nicosia=LeuKos�ia, Cyprus=K �uproBc Department of Applied Mathematics, University of Leeds, Leeds LS2 9JT, UK

a r t i c l e i n f o

Article history:

Received 16 June 2010

Accepted 26 September 2010Available online 20 October 2010

Keywords:

Inverse problem

Method of fundamental solutions (MFS)

Regularization

97/$ - see front matter & 2010 Elsevier Ltd. A

016/j.enganabound.2010.09.014

esponding author.

ail addresses: [email protected], liviu@im

[email protected] (A. Karageorghis), amt5ld@math

a b s t r a c t

In this study, we briefly review the applications of the method of fundamental solutions to inverse

problems over the last decade. Subsequently, we consider the inverse geometric problem of identifying an

unknown part of the boundary of a domain in which the Laplace equation is satisfied. Additional Cauchy

data are provided on the known part of the boundary. The method of fundamental solutions is employed

in conjunction with regularization in order to obtain a stable solution. Numerical results are presented

and discussed.

& 2010 Elsevier Ltd. All rights reserved.

1. Introduction

The method of fundamental solutions (MFS) is a meshlessboundary collocation method which is applicable to boundaryvalue problems in which a fundamental solution of the operator inthe governing equation is known explicitly. Despite this restriction,it has, in recent years, become very popular primarily because ofthe ease with which it can be implemented, in particular forproblems in complex geometries. The basic ideas of the methodwere first introduced by Kupradze and Aleksidze in the early 1960s,see e.g. [31]. Since its introduction as a numerical method byMathon and Johnston [44], it has been successfully applied to alarge variety of physical problems, an account of which may befound in the survey papers by Fairweather and Karageorghis [13],Fairweather et al. [14] and Golberg and Chen [17].

The ease of implementation of the MFS for problems withcomplex boundaries makes it an ideal candidate for problems inwhich the boundary is of major importance or requires specialattention, such as free boundary problems. A different but relatedclass of problems to which the MFS is naturally suited is the class ofinverse problems. Inverse problems can be subdivided into fourmain categories, namely Cauchy problems, inverse geometricproblems, source identification problems and parameter identifi-cation problems. For these reasons, the MFS has been usedincreasingly over the last decade for the numerical solution ofthe above classes of problems.

ll rights reserved.

sar.bu.edu.ro (L. Marin),

s.leeds.ac.uk (D. Lesnic).

The aim of this paper is, after briefly surveying the applicationsof the MFS to inverse problems in recent years, to study theapplication of the method to inverse geometric problems for theLaplace equation subject to various boundary conditions, and topresent the various implementational issues related to this appli-cation. More specifically, we shall consider the inverse boundaryvalue problem given by the Laplace equation

Du¼ 0 in O, ð1aÞ

subject to the boundary conditions

u¼ f1 and@u

@m¼ g1 on @O1, ð1bÞ

u¼ f2 on @O2, ð1cÞ

or

@u

@m¼ g2 on @O2, ð1dÞ

where O�Rd is a bounded domain, d is the dimension of the spacewhere the problem is posed, usually dAf1,2,3g, f1, g1, f2 and g2 areknown functions and the boundary @O¼ @O1

S@O2, where @O1 is

the known part of the boundary and @O2 is the unknown part of theboundary to be identified. Also, @=@m denotes the partial derivativein the direction of the outward unit normal vector m to theboundary. In Eqs. (1b) and (1c), @O1 and @O2 are, in general, twosimple arcs having in common the endpoints only, and thisproblem occurs in several contexts such as corrosion detectionby electrostatic measurements of the voltage f1 and the current fluxg1, see Kaup and Santosa [30], or crack detection in non-ferrousmetals subject to electromagnetic measurements, see McIver [45].

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L. Marin et al. / Engineering Analysis with Boundary Elements 35 (2011) 342–354 343

Robin linear convective or nonlinear boundary conditions canalso be considered, see Isakov [25] and Vogelius and Xu [55],instead of the Dirichlet boundary condition (1c) or the Neumannboundary condition (1d). For a comprehensive review of the aboveinverse geometric corrosion problem, including the transientanalysis, see Vessella [54].

The paper is organized as follows: In Section 2 we briefly surveythe application of the MFS to the various types of inverse problems.Then, in the rest of the paper, we focus on the specific application ofthe MFS to the boundary identification in corrosion engineering.Detailed accounts of the boundary discretization and the MFSapproximation are given in Section 3. Section 4 presents themethod of minimization of the nonlinear functional associatedwith problem (1), as well as details regarding the numericalimplementation of the proposed method. In Section 5 severalnumerical examples are considered. Finally, some conclusions andideas about possible future work are provided in Section 6.

2. The MFS for inverse problems: A brief review

2.1. Cauchy problems

In these problems the boundary @O of the domain O of theproblem under consideration is known. On part of the boundary(@O1) conditions (1b) are over specified, while on the remaining@O2 no boundary conditions are given. More specifically, considerthe inverse boundary value problem (1a) and (1b), and the goal is todetermine both u and @u=@m on @O2. Although the (local) existenceand uniqueness of the solution are ensured by the Cauchy–Kowalevskaya and Holmgren unique continuation theorems, theCauchy problem (1a) and (1b) is still ill-posed since the solutiondoes not depend continuously on the input data (1b), i.e. it isunstable with respect to small perturbations in the boundary dataon @O1.

The MFS was, apparently, used for the first time for the solutionof a Cauchy problem when it was applied to problem (1a) and (1b)in Hon and Wei [21]. In Marin and Lesnic [40], the authors used themethod to solve the Cauchy problem in two-dimensional elasticity,while the same authors applied the MFS for the solution of theCauchy problem associated with the two-dimensional Helmholtzand modified Helmholtz equations in Marin and Lesnic [41]. InMarin [33], the MFS was applied to the Cauchy problem for steady-state heat transfer in two-dimensional functionally gradedmaterials, while in Marin and Lesnic [42] it was applied tovarious Cauchy problems for the biharmonic equation. The MFSwas used for the first time for the solution of three-dimensionalCauchy problems in Marin [34,35], where Helmholtz and elasticityproblems are considered, respectively. In Wei et al. [59] the authorsused the method with various regularization techniques for thesolution of Cauchy problems associated with the Laplace,Helmholtz and modified Helmholtz equations in two and threedimensions. Further applications of the MFS to inverse HelmholtzCauchy problems can be found in Jin and Zheng [27]. The methodwas applied to Cauchy problems for steady-state heat conductionin anisotropic media in two and three dimensions in Jin et al. [28],while various Cauchy problems associated with the two-dimensional Stokes equations were investigated in Chen et al.[10]. The application of the MFS, in conjunction with the method ofparticular solutions, for the numerical solution of inverse boundaryvalue problems in steady-state heat conduction with sources wasinvestigated in Marin [37]. The method was adapted to solve, in astable manner, singular direct and Cauchy problems for the Laplaceequation with perturbed data in Marin [38]. An analysis of theconditioning of the MFS coefficient matrices for the Cauchyproblem in two dimensions was carried out in Young et al. [64],

and in Shigeta and Young [51] it was used to solve the Cauchyproblem for the two-dimensional Laplace equation in the presenceof singular points. An interesting application was reported in Wangand Rudy [56] where the authors applied the MFS to a doubly-connected Cauchy problem arising in electrocardiography. Inparticular, they computed potentials on the surface of the heartfrom measured body surface electrocardiographic data. Theapplication of the method to additional doubly-connectedCauchy problems for the Laplace equation in two and threedimensions may be found in Zhou and Wei [66]. Recently, inWei and Zhou [58], the convergence of the MFS approximation wasproved for the Cauchy problem (1a) and (1b) in the case O is anannulus and @O1 is either the interior or exterior boundary circle ofthe annulus. In particular, convergence was proved for noisy dataand the MFS was combined with the discrete Tikhonovregularization. This result generalizes the original convergenceresult of Ohe and Ohnaka [50] for exact Cauchy data.

The corresponding Cauchy problem for the one-dimensionalheat equation was solved with the MFS in Hon and Wei [20], whilethe same authors extended this approach to more general two- andthree-dimensional inverse heat conduction problems in Hon andWei [22,23]. The corresponding two-dimensional inverse heatconduction problem in an anisotropic medium was studied inDong et al. [12]. A related inverse problem, namely the so-calledbackward inverse problem in which the initial temperature needsto be retrieved from known measurements inside the solutiondomain at later stages, was solved in one and two dimensions usingthe MFS in Mera [46].

2.2. Inverse geometric problems

In these problems the size, location and shape of part of theboundary are unknown. On the known part of the boundary theboundary conditions are over specified, and the unknown part ofthe boundary is determined by the satisfaction of a specificboundary condition on it. More specifically, consider the inverseboundary value problem given by Eqs. (1a), (1b) and (1c) or (1d). Inthis case, to account for the possibility of inverse geometricproblems with internal inclusions (cavities or rigid voids), theboundary portions @O1 and @O2 may be disjoint. The uniqueness ofsolution and stability estimates of logarithmic type are provided inBeretta and Vessella [7] and Cheng et al. [11] in two and threedimensions, respectively. However, the problem is still ill-posedunder no a priori regularity assumption on @O2 since its solution isunstable with respect to small perturbations in the boundary data(1b) on @O1.

The first attempt to solve such a problem by a meshless methodwas carried out in Hon and Wu [24] where radial basis functions areused to approximate the solution. Apparently, the MFS was used forthe first time in the solution of an inverse boundary determinationproblem in Mera and Lesnic [47], where the authors solve thecorresponding inverse problem associated with the three-dimensional Laplace equation arising in potential corrosiondamage. In Zeb et al. [65], the authors used the method for thesolution of an inverse boundary determination problem associatedwith the two-dimensional biharmonic equation. In Hon and Li [19],the MFS was applied to one- and two-dimensional inverseboundary determination heat conduction problems. Problem (1)was solved as a Cauchy level-set problem using the MFS by Yanget al. [62,63]. The detection of cavities with the method, in variousproblems arising in electrical impedance tomography wasinvestigated in Borman et al. [9] and Karageorghis and Lesnic[29]. The similar problem of determining the location, size andshape of a body in the interior of a domain from boundarymeasurements, using the MFS, was studied, in the case of the

∂Ω2

∂Ω1z(k)

z(k+1) x(k)

z(1)z(N1 +1)

–r r x1

x2

z(N1+N2)

Fig. 1. Geometry and boundary discretization of the problem.

L. Marin et al. / Engineering Analysis with Boundary Elements 35 (2011) 342–354344

Laplace equation in Alves and Martins [3], in the case of the Stokesequations in Martins and Silvestre [43], and in the case of theCauchy–Navier equations of elasticity in Alves and Martins [4]. InWei and Li [57], the authors considered a one-dimensional heatconduction problem in a multi-layer domain with a movingboundary at the end of the last layer. The problem was solvedusing a regularized MFS.

2.3. Source identification problems

In these problems the boundary @O of the domain O of theproblem under consideration is known and on part of the boundary(@O1), conditions are over specified. The goal is to determine thesource function in the domain O. More specifically, suppose weconsider the inverse boundary value problem given by the Poissonequation

Du¼ F in O ð2Þ

subject to the boundary conditions (1b) and (1c), and the goal is todetermine the source function F in O.

The MFS was used, apparently for the first time for problems oftype (2) in Jin and Marin [26], under the assumption that F satisfieseither the Laplace or the Helmholtz equation. A thoroughinvestigation of the same problem was carried out in Alves et al.[2]. The solution of the corresponding source identificationproblem for the one-dimensional time-dependent heat equationvia the MFS was studied in Yan et al. [60]. While in Yan et al. [60] theheat source is taken to be a function of time only, the same problemin one, two and three dimensions in the case the heat source istaken to be a function of space only was considered in Yan et al.[61]. The case in which the heat source is taken to be a function ofspace only, in one dimension, was also studied in Nili Ahmadabadiet al. [49]. Recently, the inverse problem of identifying acousticsources from boundary data in two-dimensional domains wassolved in Alves et al. [6], where the MFS was used in conjunctionwith a Kansa-type MFS, see also Alves and Chen [1].

2.4. Parameter identification problems

In these problems the boundary of the domain of the problemunder consideration is known. On part of the boundary (@O1)conditions are over specified, while on the remaining part @O2 aheat transfer coefficient in a Robin convective boundary conditionis unknown. The goal is to determine this unknown heat transfercoefficient. More specifically, consider the inverse boundary valueproblem (1a) and (1b) and the Robin convective boundarycondition

uþa @u

@m¼ h2 on @O2, ð3Þ

and the goal is to determine the heat transfer coefficient a on @O2.This parameter identification problem is closely related to theinverse Cauchy problem of Section 2.1, which once solved in astable manner, through (3), it yields a¼ ðh2�uÞ=ð@u=@mÞ on @O2 (formore details see [15]).

Apparently, the only known applications of the MFS to inverseproblems of this type may be found in Alves and Martins [5]and Valle et al. [53], where the authors applied the MFS to theCauchy–Navier equations of elasticity to recover an unknownmatrix coefficient in a Robin boundary condition, and thereconstruction of the heat transfer coefficient in Helmholtz fin-type problems, respectively. The more general coefficientidentification problems in which the unknown parameters are inthe governing partial differential equation rather than in theboundary operator are yet to be investigated with the MFS.

In the next three sections we describe the MFS as applied to theinverse geometric problem (1) of Section 1. Further, we assume thatthe unknown boundary @O2 is the graph of an unknown Lipschitzfunction F : ½�r,r�-R, where the horizontal axis intersects theknown boundary @O1 at the points ð7r,0Þ.

3. Method of fundamental solutions

3.1. Boundary discretization

In order to discretize the boundary @O, we select the N1

boundary points ðzðiÞÞN1

i ¼ 1 on the boundary @O1 and N2 boundarypoints ðzðiÞÞN1þN2

i ¼ N1þ1 on the boundary @O2. In this way, the boundary@O may be approximated by (using the conventionzðN1þN2þ1Þ ¼ zð1Þ)

@O� @ ~O ¼[N1þN2

k ¼ 1

GðkÞ where GðkÞ ¼ ½zðkÞ,zðkþ1Þ�, k¼ 1, . . . ,N1þN2:

ð4Þ

Also, the boundary segments @O1 and @O2 are approximated by

@O1 � @ ~O1 ¼[N1

k ¼ 1

GðkÞ, @O2 � @ ~O2 ¼[N1þN2

k ¼ N1þ1

GðkÞ: ð5Þ

Now, we take the boundary collocation points in the MFS to be themidpoints ðxðkÞÞN1þN2

k ¼ 1 of each segment ðGðkÞÞN1þN2

k ¼ 1 , i.e.,

xðkÞ ¼ 12ðzðkÞ þzðkþ1ÞÞ, k¼ 1, . . . ,N1þN2: ð6Þ

In this way, the normal n to the approximate boundary @ ~O at theboundary collocation points is

nðxðkÞÞ ¼ðzðkþ1Þ

2 �zðkÞ2 ,�zðkþ1Þ1 þzðkÞ1 Þ

Jzðkþ1Þ�zðkÞJ, k¼ 1, . . . ,N1þN2, ð7Þ

where zðkÞ ¼ ðzðkÞ1 ,zðkÞ2 Þ.With respect to Fig. 1 we have

zðN1þN2þ1Þ ¼ zð1Þ ¼ ðr,0Þ, zðN1þ1Þ ¼ ð�r,0Þ,

zðiÞ1 ¼ r 1�2ði�1Þ

N1

� �, i¼ 1, . . . ,N1,


zðN1þ iÞ1 ¼�r 1�

2ði�1Þ

N2

� �, i¼ 1, . . . ,N2þ1: ð8Þ

3.2. MFS approximation

In the MFS the solution of problem (1) is approximated by

uMðc,n;xÞ ¼XMj ¼ 1

cjGðnðjÞ,xÞ, xAO, ð9Þ

where G is a fundamental solution of the Laplace equation in twodimensions, given by

Gðn,xÞ ¼ �1

2plogjn�xj: ð10Þ

Table 1Convergence of ezð0Þ, obtained using exact Dirichlet data on @O2, p1 ¼ p2 ¼ 0 and N1

¼ N2, for Example 1.

N1¼N2 5 10 20

ezð0Þ 3.66�10�1 1.37�10�2 8.89�10�3

–1 –0.5 0 0.5 1

–1

–0.5

0

0.5

x1

–1 –0.5 0 0.5 1x1

x 2

–1

–0.5

0

0.5

x 2

Dirichlet boundary data on ∂Ω2

ExactGuessp1=1%p1=5%p1=10%

Neumann boundary data on ∂Ω2


Fig. 2. Reconstructed curves for Dirichlet and Neumann data on @O2 in Example 1 with n

p1 ¼ 1,5 and 10% and (d) Case II: p2 ¼ 1,5 and 10%.

In (9), ðnðjÞÞMj ¼ 1 are pre-assigned singularities located outside O andc¼ ½c1,c2, . . . ,cM�

T are unknown coefficients to be determined fromthe satisfaction of the boundary conditions. Also, the normalderivative @u=@m is approximated by

@uM

@mðc,n;xÞ ¼

XMj ¼ 1

cj@G

@mðnðjÞ,xÞ, xA@O: ð11Þ

The satisfaction of the boundary conditions (1b)–(1d) yields

uMðc,n;xðiÞÞ ¼ f1ðxðiÞÞ, i¼ 1, . . . ,N1, ð12aÞ

@uM

@mðc,n;xðiÞÞ ¼ g1ðx

ðiÞÞ, i¼ 1, . . . ,N1, ð12bÞ

uMðc,n;xðN1þ iÞÞ ¼ f2ðxðN1þ iÞÞ, i¼ 1, . . . ,N2, ð12cÞ

or

@uM

@mðc,n;xðN1þ iÞÞ ¼ g2ðx

ðN1þ iÞÞ, i¼ 1, . . . ,N2:

Note: From Fig. 1 and

xðN1þ iÞ ¼ ðxðN1þ iÞ1 ,xðN1þ iÞ

2 Þ,

1–1 –0.5 0 0.5x1

–1

–0.5

0

0.5

x 2

–1

–0.5

0

0.5

x 2Dirichlet boundary data on ∂Ω2

–1 –0.5 0 0.5 1x1




o regularization. (a) Case I: p1 ¼ 1,5 and 10%, (b) Case I: p2 ¼ 1,5 and 10%, (c) Case II:


xðN1þ iÞ1 ¼

1

2ðzðN1þ iÞ

1 þzðN1þ iþ1Þ1 Þ ¼ �r 1�

2i�1

N2

� �,

xðN1þ iÞ2 ¼

1

2ðzðN1þ iÞ

2 þzðN1þ iþ1Þ2 Þ, i¼ 1, . . .N2, ð13Þ

it follows that the unknowns determining the unknown boundary@O2 are the heights ðzðN1þ2Þ

2 , . . . ,zðN1þN2Þ

2 ÞT¼ z.

In summary, Eqs. (12a)–(12c) provide 2N1+N2 equations for theM unknown coefficients c and the N2�1 unknown x2�coordinatesz. Thus, provided 2N1ZM�1 we have at least as many equations asunknowns.

4. Regularization method

4.1. Tikhonov regularization functional

The inverse geometric problem investigated in this paper issolved, in a stable manner, by minimizing the following first-orderTikhonov regularization functional Tikhonov et al. [52]

F lð�,�Þ : RM�RN2�1

�!½0,1Þ,

–1 –0.5 0 0.5 1

–1

–0.5

0

0.5

–1

–0.5

0

0.5

x1

x 2

–1 –0.5 0 0.5 1x1

x 2





Fig. 3. Reconstructed curves for Dirichlet and Neumann data on @O2 in Example 1 with


F lðc,zÞ ¼1

2

XN1

i ¼ 1

½F1ðc,n;xðiÞÞ�2þ1

2

XN1

i ¼ 1

½F 2ðc,n;xðiÞÞ�2

þ1

2

XN1þN2

i ¼ N1þ1

½F3ðc,n;xðiÞÞ�2þlJzuJ2, ð14Þ

where

F1ðc,n;xðiÞÞ ¼ uMðc,n;xðiÞÞ�f1ðxðiÞÞ, i¼ 1, . . . ,N1, ð15aÞ

F2ðc,n;xðiÞÞ ¼@uM

@mðc,n;xðiÞÞ�g1ðx

ðiÞÞ, i¼ 1, . . . ,N1, ð15bÞ

F3ðc,n;xðiÞÞ ¼

uMðc,n;xðiÞÞ�f2ðxðiÞÞ,

or@uM

@mðc,n;xðiÞÞ�g2ðx

ðiÞÞ,

8>>><>>>:

i¼N1þ1, . . . ,N1þN2:

ð15cÞ

In (14), zu ¼ ðzðN1þ2Þ2 �zðN1þ1Þ

2 , . . . ,zðN1þN2þ1Þ2 �zðN1þN2Þ

2 Þ denotes an

approximation to the first-order derivative, keeping in mind that

z1(i + 1)�z1

(i), i¼N1,y, N1+N2, is constant, and l40 is a regularizationparameter to be prescribed.

–1 –0.5 0 0.5 1

–1

–0.5

0

0.5

x1

x 2

–1 –0.5 0 0.5 1

–1

–0.5

0

0.5

x1

x 2





regularization. (a) Case I: p1 ¼ 1,5 and 10%, (b) Case I: p2 ¼ 1,5 and 10%, (c) Case II:


In order to retrieve an accurate and physically correct numericalsolution of the inverse geometric problem investigated herein, thefirst-order Tikhonov functional (14) is minimized subject to thefollowing simple bounds imposed for the components of theunknown vector z:

�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2�ðzðiÞ1 Þ

2q

ozðiÞ2 o0, i¼N1þ2, . . . ,N1þN2: ð16Þ

Here we have assumed that the singularities nðjÞ, j¼ 1, . . . ,M, havebeen chosen on the boundary, @OS, of the disk of radius R andcentered at the origin, OS ¼ fðx1,x2Þ : x2

1þx22oR2g, that encloses the

solution domain, as well as its boundary, i.e. O �OS. Consequently,the simple bounds (16) require that the x2-coordinates of theunknown boundary @O2 are situated below the x1-axis, while, at thesame time, the singularities are located outside O.

It is important to mention that we may write the Tikhonovfunctional given by Eq. (14) as

F lð�,�Þ : RM�RN2�1

�!½0,1Þ, F lðc,zÞ ¼F LSðc,zÞþRlðzÞ, ð17Þ

where F LS ¼ ðF 21þF2

2þF 23Þ=2 is the least squares functional

associated with the inverse geometric problem considered inthis study and Rl ¼ lJzuJ2 is the first-order regularization term.It should be emphasized that the zeroth-order Tikhonovregularization procedure, which is based on penalizing the normof the solution, i.e. Rl ¼ lJzJ2, rather than its derivative, i.e.Rl ¼ lJzuJ2, in Eq. (14) did not produce satisfactorily accurateand stable results for the unknown boundary @O2. This observationis consistent with the results obtained by Lesnic et al. [32], Marinand Lesnic [39] and Marin [36] who have solved, using the

–1 –0.5 0 0.5 1

–0.6

–0.4

–0.2

0

0.2

0.4

x1

x 2

–1 –0.5 0 0.5 1

–0.6

–0.4

–0.2

0

0.2

0.4

x1

x 2





Fig. 4. Reconstructed curves for Dirichlet and Neumann data on @O2 in Example 2 with n


boundary element method the similar problem for the Laplaceequation, the Lame system and Helmholtz-type equations,respectively. Alternatively, instead of using the functional (14) or(17), one may parameterize @O2 with various approximatingfunctions and the problem reduces to finding the coefficients ofthis approximation, see Birginie et al. [8]. To summarize, theTikhonov regularization method solves a physically constrainedminimization problem using a smoothness norm in order toprovide a stable solution which fits the data and also has aminimum structure.

4.2. Implementational details

In order to minimize the functional (17), we shall use the NAGsubroutine E04UNF [48] which minimizes a sum of squares subjectto constraints. This may include simple bounds, linear constraintsand smooth nonlinear constraints. Each iteration of the subroutineE04UNF includes the following: (i) the solution of a quadraticprogramming subproblem; (ii) a line search with an augmentedLagrangian function; and (iii) a quasi-Newton update of theapproximate Hessian of the Lagrangian function.

4.2.1. Least squares functional

To implement a code using E04UNF, in (17), we write the leastsquares functional in the following form:

F LSðc,zÞ ¼1

2

X2N1þN2

k ¼ 1

½yk�Fkðc,zÞ�2, ð18Þ

–1 –0.5 0 0.5 1

–0.6

–0.4

–0.2

0

0.2

0.4

x1

x 2

–1 –0.5 0 0.5 1

–0.6

–0.4

–0.2

0

0.2

0.4

x1

x 2





o regularization. (a) Case I: p1 ¼ 1,5 and 10%, (b) Case I: p2 ¼ 1,5 and 10%, (c) Case II:


where

Fkðc,zÞ ¼XMj ¼ 1

cjGðnðjÞ,xðkÞÞ, yk ¼ f1ðx

ðkÞÞ, k¼ 1, . . . ,N1, ð19aÞ

FN1þkðc,zÞ ¼XMj ¼ 1

cj@G

@mðnðjÞ,xðkÞÞ, yN1þk ¼ g1ðx

ðkÞÞ, k¼ 1, . . . ,N1,

ð19bÞ

F2N1þkðc,zÞ ¼XMj ¼ 1

cj

GðnðjÞ,xðkÞÞor@G

@mðnðjÞ,xðkÞÞ,

8>>><>>>:

y2N1þk ¼

f2ðxðkÞÞ

or

g2ðxðkÞÞ,

8><>: k¼ 1, . . . ,N2: ð19cÞ

Further, we take the vector g¼ ðc,zÞARMþN2�1 to denote the set ofunknowns, with

Z‘ ¼ c‘ , ‘¼ 1, . . . ,M, ZMþ ‘ ¼ zðN1þ ‘þ1Þ2 , ‘¼ 1, . . . ,N2�1: ð20Þ

4.2.2. Regularization term

In order to obtain a stable solution to the inverse geometricproblem given by Eqs. (1a, 1b) and either (1c) or (1d) whenimplementing a code that uses E04UNF, in (17), we add one

–1 –0.5 0 0.5 1

–0.6

–0.4

–0.2

0

0.2

0.4

x1

x 2

–1 –0.5 0 0.5 1

–0.6

–0.4

–0.2

0

0.2

0.4

x1

x 2







element to the vector F defined in (18), by taking

RlðzÞ ¼ lJzuJ2¼ l

XN2þ1

j ¼ 2

ðzðN1þ jÞ2 �zðN1þ j�1Þ

2 Þ2

ð21aÞ

or

F2N1þN2 þ1ðc,zÞ ¼ffiffiffiffiffiffi2lp XN2þ1

j ¼ 2

ðzðN1þ jÞ2 �zðN1þ j�1Þ

2 Þ2

24

35

1=2

, y2N1þN2þ1 ¼ 0:

ð21bÞ

4.2.3. Gradient of the Tikhonov regularization functional

Finally, we define the components of the gradient,

Jðc,zÞ ¼rgF lðc,zÞARð2N1þN2þ1Þ�ðMþN2�1Þ. It should be mentioned

that providing as many exact values as possible to the NAGsubroutine E04UNF for the components of the gradient resultsnot only in an improvement in the accuracy of the numericalapproximation of the unknown boundary, but also in a markeddecrease in the computational time required to minimize theTikhonov regularization functional given by (14) subject to thesimple bounds (16).

Alternatively, one may use one of the public domain MINPACK

routines lmdif or lmder, see Garbow et al. [16]. The formerminimizes the sum of the squares ofM nonlinear functions in Nvariables by a modification of the Levenberg–Marquardt algorithm,and in it the user must provide a subroutine which calculates thefunctions while the Jacobian is calculated by a forward-differenceapproximation. In the latter routine the user must also provide the

–1 –0.5 0 0.5 1

–0.6

–0.4

–0.2

0

0.2

0.4

x1

x 2

–1 –0.5 0 0.5 1

–0.6

–0.4

–0.2

0

0.2

0.4

x1

x 2





regularization. (a) Case I: p1 ¼ 1,5 and 10%, (b) Case I: p2 ¼ 1,5 and 10%, (c) Case II:


Jacobian leading to a substantial improvement in the performanceof the method.

5. Numerical results

In all four examples considered, the functions f1, g1, f2 and g2 in(1b)–(1d) correspond to the exact solution of the problem which istaken to be

uðx1,x2Þ ¼ x1�1

2

� �2

� x2�1

4

� �2

: ð22Þ

In each example we consider two cases:Case I: We solve (1a) subject to the boundary conditions (1b) and

(1c), i.e. u is given on the unknown boundary @O2. We first considernoise levels of p1 in the Dirichlet boundary data f1 on the knownboundary @O1 and then consider noise levels of p2 in the Neumannboundary data g1 on @O1. In the first two examples we considernoise levels of p1¼p2¼1, 5 and 10%, while in the last two exampleswe consider noise levels of p1¼p2¼1, 3 and 5%.

Case II: We solve (1a) subject to the boundary conditions (1b)and (1d), i.e. @u=@m is given on the unknown boundary @O2.

–1 –0.5 0 0.5 1

–1

–0.5

0

0.5

x1

x 2

–1 –0.5 0 0.5 1

–1

–0.5

0

0.5

x1

x 2







In order to analyse the accuracy of the numerical resultsobtained for various values of the regularization parameter, l,we introduce the root mean square (RMS) error:

ezðlÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

N2�1

XN1þN2

i ¼ N1þ2

ðzði;lÞ2 �zði;anÞ2 Þ

2

vuut , lZ0, ð23Þ

where zði;lÞ2 is the numerically retrieved value corresponding to theregularization parameter lZ0, with the mention that the value l¼ 0is associated with the least-squares solution (i.e. no regularization),and z2

(i;an) is the exact value for the x2-coordinate of the boundary @O2.

5.1. Example 1

In this example we consider the case where O in problem (1) isthe unit disk, and the portion of the boundary to be recovered, @O2,is the semicircle fðx1,x2Þ : x2

1þx22 ¼ 1, x2o0g. In all the numerical

results presented for this example, we took the pre-assignedsingularities ðnðjÞÞMj ¼ 1 in (9) to be uniformly distributed on acircle of radius R¼3 centered at the origin. Also, we took N1¼N2

and M¼N1+N2 (cf. (4) and (9), respectively).

–1 –0.5 0 0.5 1

–1

–0.5

0

0.5

x1

x 2

–1 –0.5 0 0.5 1

–1

–0.5

0

0.5

x1

x 2ExactGuessp2=1%p2=3%p2=5%




no regularization. (a) Case I: p1 ¼ 1,3 and 5%, (b) Case I: p2 ¼ 1,3 and 5%, (c) Case II:


First, in order to investigate the convergence of the MFS withrespect to increasing the number of degrees of freedom, for exactCauchy data on the known boundary @O1, i.e. p1¼p2¼0, and exactDirichlet data on the unknown boundary @O2, in Table 1 we presentthe numerical values of ezð0Þ for various values ofN1 ¼N2Af5,10,20g. From this table it can be seen that ezð0Þdecreases as N1¼N2 increases. Similar behaviour was observedfor the other examples considered in this study.

For the remaining results for this example we fixed N1¼N2¼10and investigated the stability of the MFS approximation withrespect to the amount of noise included in the input data. Fornoisy data, larger values of N1¼N2 would only increase the ill-conditioning and the instability of the MFS approximations, if noregularization is augmented in (14). On the other hand, ifregularization (with l40) is included in (14), similar stableresults are obtained for larger values of N1¼N2.

5.1.1. Case I

In Figs. 2(a) and (b), we present the reconstructed boundaries forthe two cases of noise considered when no regularization is used.As can be observed from these figures the approximations are not

–1 –0.5 0 0.5 1

–1

–0.5

0

0.5

x1

x 2

–1 –0.5 0 0.5 1

–1

–0.5

0

0.5

x1

x 2





Fig. 7. Reconstructed curves for Dirichlet and Neumann data on @O2 in Example 3 with reg

and 5% and (d) Case II: p2 ¼ 1,3 and 5%.

so accurate and become oscillatory, especially as the amount ofnoise increases. In Figs. 3(a) and (b), we present the correspondingreconstructed boundaries with regularization for the optimalvalues of l and observe that these are in consistent agreementwith the exact boundary, while, at the same time, being stable.Furthermore, the numerical solutions seem to converge to theexact solution, as the amount of noise p1 or p2 decreases.

5.1.2. Case II

In Figs. 2(c) and (d), we present the reconstructed boundarieswhen no regularization is used. Again, as can be observed fromthese figures the approximations are poor. In Figs. 3(c) and (d), wepresent the corresponding reconstructed boundaries withregularization for the optimal values of l and observe that theseare in consistent agreement with the exact boundary. However, theaccuracy deteriorates considerably as the level of noise increases.

5.2. Example 2

In this example we consider the case where O in problem (1)is a complicated peanut-shaped domain whose boundary is

–1 –0.5 0 0.5 1

–1

–0.5

0

0.5

x1

x 2

–1 –0.5 0 0.5 1

–1

–0.5

0

0.5

x1

x 2





ularization. (a) Case I: p1 ¼ 1,3 and 5%, (b) Case I: p2 ¼ 1,3 and 5%, (c) Case II: p1 ¼ 1,3


defined by

@O¼ ðx1,x2Þ : x1 ¼ rðyÞcosy, x2 ¼ rðyÞsiny, rðyÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifficos2yþ

1

4sin2y

r, yA ½0,2pÞ

( ),

ð24Þ

and the boundary portion @O2 to be recovered is the lower part of(24) for yAðp,2pÞ. For this example, we took the singularities to beuniformly distributed on a circle of radius R¼2 centered at theorigin and N1¼N2¼12, M¼N1+N2.

In Figs. 4(a)–(d), we present the reconstructed boundaries forthe cases considered when no regularization is used. As can beobserved from these figures the MFS approximations are not onlypoor, but also highly oscillatory and, in some cases unbounded, i.e.unstable. In Figs. 5(a)–(d), we present the correspondingreconstructed boundaries with regularization for the optimalvalues of l and observe that these are in consistent agreementwith the exact boundary, and that the highly oscillatory andunbounded behaviour of the unregularized results hasdisappeared. With a high level of up to 10% noise, the MFS withregularization still manages to yield reasonable stable numericalsolutions.



–1 –0.5 0 0.5 1–1

–0.5

0

0.5

x1

x 2

–1 –0.5 0 0.5 1–1

–0.5

0

0.5

x1

x 2




Fig. 8. Reconstructed curves for Dirichlet and Neumann data on @O2 in Example 4 with reg

and 5% and (d) Case II: p2 ¼ 1,3 and 5%.

5.3. Example 3

In this example we consider the case where O in problem (1) isthe rotated (by p=4) square ð�

ffiffiffi2p

,ffiffiffi2pÞ � ð�

ffiffiffi2p

,ffiffiffi2pÞ and take the

unknown boundary @O2 to be retrieved the lower part for x2o0.For this example, we took the singularities to be uniformlydistributed on a circle of radius R¼2 centered at the origin andN1¼N2¼12, M¼N1+N2.

In Figs. 6(a)–(d), we present the reconstructed boundaries for thecases considered when no regularization is used and as can beobserved from these figures the approximations are poor, unboundedand highly oscillatory. In Figs. 7(a)–(d), we present the correspondingreconstructed boundaries with regularization for the optimal valuesof l and observe that these are in consistent agreement with the exactboundary, as well as stable with respect to decreasing the amount ofnoise added into the Dirichlet or Neumann data on @O1.

5.4. Example 4

In this example we consider the case where @O in problem (1) isthe regular hexagon inscribed in the unit circle and the boundary

–1 –0.5 0 0.5 1–1

–0.5

0

0.5

x1

x 2

–1 –0.5 0 0.5 1–1

–0.5

0

0.5

x1

x 2





ularization. (a) Case I: p1 ¼ 1,3 and 5%, (b) Case I: p2 ¼ 1,3 and 5%, (c) Case II: p1 ¼ 1,3


@O2 to be retrieved is the lower part x2o0. For this example, wetook the singularities to be uniformly distributed on a circle ofradius R¼2 centered at the origin and N1¼N2¼12, M¼N1+N2.Fig. 8 is analogous to Fig. 7, and the same conclusions as forExample 3 can be drawn.

5.5. Selection of the regularization parameter

In the first-order Tikhonov regularization functional (14), thechoice of the optimal regularization parameter l¼ lopt is essentialin order to achieve the stability of the numerical solution. In thispaper, we have employed the L-curve criterion, see Hansen andO’Leary [18], which plots on a log-log scale the residual norm, i.e.the square root of the least squares functional,

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiF LSðc,zÞ

p, versus

the norm of the derivative of the solution, JzuJ, for various values ofthe regularization parameter, l. In Figs. 9(a) and (c) we present theL-curves for Example 2, obtained for the inverse geometricproblems given by Case I with perturbed Dirichlet and Neumanndata on @O1, respectively. The optimal values, lopt, of theregularization parameter, l, are chosen at the corners of thesecurves in order to balance the under-smooth regions, i.e. l toosmall, and over-smooth regions, i.e. l too large. Figs. 9(a) and (c)illustrate clearly the L-shaped curves and, therefore, the L-curve

0.4 0.6 0.8 1 2

0.02

0.04

0.07

0.1

0.2

Residual norm

Sol

utio

n no

rm

p1=1%p1=5%p1=10%

0.4 0.6 0.8 1 2

0.01

0.02

0.04

0.07

0.1

0.2

Residual norm

Sol

utio

n no

rm

p2=1%p2=5%p2=10%

Fig. 9. The L-curves and RMS errors for Dirichlet and Neumann data on @O2 in Example 2 w


criterion is applicable. The corresponding values for the optimalregularization parameter, obtained according to the L-curvecriterion, are as follows:

(i)

0.

0.

0

0

RM

S e

rror

ez

0.

0.

0

0

RM

S e

rror

ez

ith r

lopt ¼ 5:0� 10�4, lopt ¼ 5:0� 10�3 and lopt ¼ 1:0� 10�2 forp1 ¼ 1%, p1 ¼ 5% and p1 ¼ 10%, respectively;

(ii)
lopt ¼ 1:0� 10�3, lopt ¼ 5:0� 10�3 and lopt ¼ 10�2 for p2¼1%,p2 ¼ 5% and p2¼10%, respectively.
Figs. 9(b) and (d) present on a log-log scale the RMS error ez

given by (23), as a function of the regularization parameter, l,obtained for the inverse geometric problems given by Case I withperturbed Dirichlet and Neumann data on @O1, respectively. Bycomparing Figs. 9(a) and (b), and (c) and (d), respectively, it can beseen that for the RMS error to attain its minimum requires theoptimal value, lopt, of the regularization parameter, l, to be chosenaccording to the L-curve criterion. Furthermore, from Figs. 9(b) and(d) it can be seen that the minimum value of the RMS error, ez,decreases as the level of noise added into the Dirichlet andNeumann data decreases, respectively. The convergencebehaviour with respect to the noise level illustrated in Figs. 9(b)and (d) can be more clearly observed from Table 2 where wepresent the values of the error ezðloptÞ for Case I of Example 2 for

04

07

.1

.2

04

07

.1

.2

p1=1%p1=5%p1=10%

10–10 10–8 10–6 10–4 10–2 100

Regularization parameter λ

10–10 10–8 10–6 10–4 10–2 100

Regularization parameter λ

p2=1%p2=5%p2=10%

egularization. (a) Case I: p1 ¼ 1,5 and 10%, (b) Case I: p2 ¼ 1,5 and 10%, (c) Case II:

Table 2

Values of ezðloptÞ with increasing noise for Case I of Example 2.

Noise p1¼1% p1¼5% p1¼10% p2¼1% p2¼5% p2¼10%

ezðloptÞ 3.34�10�2 4.42�10�2 4.82�10�2 2.63�10�2 4.01�10�2 5.02�10�2


noise levels p1,p2Af1,5,10g%. From this table we can see that thevalue of ezðloptÞ increases as the noise level increases.

6. Conclusions

In this study, a brief survey of the applications of the MFS tovarious inverse problems has been presented. The application ofthe method to a specific type of inverse problems, namelygeometric boundary identification in two-dimensional harmonicproblems has been described and several numerical examples fordomains with both smooth and piecewise smooth boundaries havebeen presented. In order to obtain a stable solution, the first-orderTikhonov regularization method has been employed. Although allfour numerical examples presented happened to be geometricallysymmetric with respect to the y-axis, this was not exploited in thenumerical simulations as no constraints were posed on this, i.e. themethod performs equally well for non-symmetric geometries.

Comparing numerical methods for a class of problems is not aneasy task in the sense that there too many open questions toaddress. The MFS appears to be well-suited for the general class ofinverse problems, as it is easy to implement and is appropriate forproblems in which the boundary of the solution domain is of primeinterest. This explains the very rapid increase in the applications ofthe MFS for the solution of such problems as evidenced in Section 2.The obvious competitor to the MFS would be the boundary elementmethod (BEM) which is, however, considerably more complicated,especially in three dimensions, as it involves integration overboundary panels. A detailed comparison of the two methods couldbe the subject of future research.

Areas where the MFS has not yet been widely used and whichmay be viewed as areas of future research include time-dependentparabolic and hyperbolic inverse problems.

Acknowledgement

The authors would like to thank the University of Cyprus forsupporting this research.

References

[1] Alves CJS, Chen CS. A new method of fundamental solutions applied tononhomogeneous elliptic problems. Advances in Computational Mathematics2005;23:125–42.

[2] Alves CJS, Colac-o MJ, Leit~ao VMA, Martins NFM, Orlande HRB, Roberty NC.Recovering the source term in a linear diffusion problem by the method offundamental solutions. Inverse Problems in Science and Engineering 2005;16:1005–21.

[3] Alves CJS, Martins NFM. Reconstruction of inclusions or cavities in potentialproblems using the MFS. In: Chen CS, Karageorghis A, Smyrlis YS, editors. Themethod of fundamental solutions—a meshless method. Atlanta: DynamicPublishers Inc.; 2008. p. 51–73.

[4] Alves CJS, Martins NFM. The direct method of fundamental solutions and theinverse Kirsch–Kress method for the reconstruction of elastic inclusions orcavities. Journal of Integral Equations and Applications 2009;21:153–78.

[5] Alves CJS, Martins NFM. On the determination of a Robin boundary coefficientin an elastic cavity using the MFS. In: Fereira AJM, Kansa EJ, Fasshauer GE, Leit~aoVMA, editors. Progress on meshless methods. Computational methods inapplied sciences, vol. 11. Berlin: Springer; 2009. p. 125–39.

[6] Alves CJS, Martins NFM, Roberty NC. Full identification of acoustic sources withmultiple frequencies and boundary measurements. Inverse Problems andImaging 2009;3:275–94.

[7] Beretta E, Vessella S. Stable determination of boundaries from Cauchy data.SIAM Journal on Mathematical Analysis 1998;30:220–32.

[8] Birginie JM, Allard F, Kassab AJ. Application of trigonometric boundaryelements to heat and mass transfer problems. In: Brebbia CA, Martin JB,Aliabadi N, Haie N, editors. BEM 18. Southampton: WIT Press; 1996. p.65–74.

[9] Borman D, Ingham DB, Johansson BT, Lesnic D. The method of fundamentalsolutions for detection of cavities in EIT. Journal of Integral Equations andApplications 2009;21:381–404.

[10] Chen CW, Young DL, Tsai CC, Murugesan K. The method of fundamentalsolutions for inverse 2D Stokes problems. Computational Mechanics 2005;37:2–14.

[11] Cheng J, Hon YC, Yamamoto M. Conditional stability estimation for an inverseboundary problem with non-smooth boundary in R3. Transactions of theAmerican Mathematical Society 2001;353:4123–38.

[12] Dong CF, Sun FY, Meng BQ. A method of fundamental solutions for inverse heatconduction problems in an anisotropic medium. Engineering Analysis withBoundary Elements 2007;31:75–82.

[13] Fairweather G, Karageorghis A. The method of fundamental solutions forelliptic boundary value problems. Advances in Computational Mathematics1998;9:69–95.

[14] Fairweather G, Karageorghis A, Martin PA. The method of fundamentalsolutions for scattering and radiation problems. Engineering Analysis withBoundary Elements 2003;27:759–69.

[15] Fasino D, Inglese G. An inverse Robin problem for Laplace’s equation:theoretical results and numerical methods. Inverse Problems 1999;15:41–8.

[16] Garbow BS, Hillstrom KE, More JJ. MINPACK Project. Argonne NationalLaboratory; 1980.

[17] Golberg MA, Chen CS. The method of fundamental solutions for potentialHelmholtz and diffusion problems. In: Golberg MA, editor. Boundary integralmethods: numerical and mathematical aspects. Boston: WIT Press andComputational Mechanics Publications; 1999. p. 105–76.

[18] Hansen PC, Leary O’DP. The use of the L-curve in the regularization of discreteill-posed problems. SIAM Journal on Scientific Computing 1993;14:1487–503.

[19] Hon YC, Li M. A computational method for inverse free boundary determina-tion problem. International Journal for Numerical Methods in Engineering2008;73:1291–309.

[20] Hon YC, Wei T. A meshless computational method for solving inverse heatconduction problem. International Series on Advances in Boundary Elements2002;13:135–44.

[21] Hon YC, Wei T. A meshless scheme for solving inverse problems of Laplaceequation. In: Hon YC, Yamamoto M, Cheng J, Lee JL, editors. Recent develop-ments in theories and numerics. International conference on inverse problems.Hong Kong: World Scientific; 2003. p. 291–300.

[22] Hon YC, Wei T. A fundamental solution method for inverse heat conductionproblems. Engineering Analysis with Boundary Elements 2004;28:489–95.

[23] Hon YC, Wei T. The method of fundamental solutions for solving multi-dimensional heat conduction problems. CMES: Computer Modeling in Engi-neering & Sciences 2005;7:119–32.

[24] Hon YC, Wu Z. A numerical computation for inverse boundary determinationproblem. Engineering Analysis with Boundary Elements 2000;24:599–606.

[25] Isakov V. On uniqueness of obstacles and boundary conditions for restricteddynamical scattering data. Inverse Problems and Imaging 2008;2:151–64.

[26] Jin B, Marin L. The method of fundamental solutions for inverse sourceproblems associated with the steady state heat conduction. InternationalJournal for Numerical Methods in Engineering 2007;69:1572–89.

[27] Jin B, Zheng Y. A meshless method for some inverse problems associated withthe Helmholtz equation. Computer Methods in Applied Mechanics andEngineering 2006;195:2270–80.

[28] Jin B, Zheng Y, Marin L. The method of fundamental solutions for inverseboundary value problems associated with the steady state heat conduction inanisotropic media. International Journal for Numerical Methods in Engineering2006;65:1865–91.

[29] Karageorghis A, Lesnic D. Detection of cavities using the method of fundamentalsolutions. Inverse Problems in Science and Engineering 2009;17:803–20.

[30] Kaup P, Santosa F. Nondestructive evaluation of corrosion damage usingelectrostatic boundary measurements. Journal of Nondestructive Evaluation1995;14:127–36.

[31] Kupradze VD, Aleksidze MA. The method of functional equations for theapproximate solution of certain boundary value problems. U.S.S.R. Computa-tional Mathematics and Mathematical Physics 1964;4:82–126.

[32] Lesnic D, Berger JR, Martin PA. A boundary element regularization method forthe boundary determination in potential corrosion damage. Inverse Problemsin Engineering 2002;10:163–82.

[33] Marin L. Numerical solutions of the Cauchy problem for steady-state heattransfer in two-dimensional functionally graded materials. InternationalJournal of Solids and Structures 2005;42:4338–51.


[34] Marin L. A meshless method for the numerical solution of the Cauchy problemassociated with three-dimensional Helmholtz-type equations. Applied Mathe-matics and Computation 2005;165:355–74.

[35] Marin L. A meshless method for solving the Cauchy problem in three-dimensional elastostatics. Computers and Mathematics with Applications2005;50:73–92.

[36] Marin L. Numerical boundary identification for Helmholtz-type equations.Computational Mechanics 2006;39:25–40.

[37] Marin L. The method of fundamental solutions for inverse problems associatedwith the steady-state heat conduction in the presence of sources. CMES:Computer Modeling in Engineering & Sciences 2008;30:99–122.

[38] Marin L. Stable MFS solution to singular direct and inverse problems associatedwith the Laplace equation subjected to noisy data. CMES: Computer Modelingin Engineering & Sciences 2008;37:203–42.

[39] Marin L, Lesnic D. BEM first-order regularisation method in linear elasticity forboundary identification. Computer Methods in Applied Mechanics and Engi-neering 2003;192:2059–71.

[40] Marin L, Lesnic D. The method of fundamental solutions for the Cauchyproblem in two-dimensional linear elasticity. International Journal of Solidsand Structures 2004;41:3425–38.

[41] Marin L, Lesnic D. The method of fundamental solutions for the Cauchyproblem associated with two-dimensional Helmholtz-type equations. Com-puters & Structures 2005;83:267–78.

[42] Marin L, Lesnic D. The method of fundamental solutions for inverse boundaryvalue problems associated with the two-dimensional biharmonic equation.Mathematical and Computer Modelling 2005;42:261–78.

[43] Martins NFM, Silvestre AL. An iterative MFS approach for the detection ofimmersed obstacles. Engineering Analysis with Boundary Elements 2008;32:517–24.

[44] Mathon R, Johnston RL. The approximate solution of elliptic boundary valueproblems by fundamental solutions. SIAM Journal on Numerical Analysis1977;14:638–50.

[45] McIver M. An inverse problem in electromagnetic crack detection. IMA Journalof Applied Mathematics 1991;47:127–45.

[46] Mera NS. The method of fundamental solutions for the backward heat conductionproblem. Inverse Problems in Science and Engineering 2005;13:79–98.

[47] Mera NS, Lesnic D. A three-dimensional boundary determination problem inpotential corrosion damage. Computational Mechanics 2005;36:129–38.

[48] Numerical Algorithms Group Library Mark 21. NAG(UK) Ltd, Wilkinson House,Jordan Hill Road, Oxford, UK, 2007.

[49] Nili Ahmadabadi M, Arab M, Maalek Ghaini FM. The method of fundamentalsolutions for the inverse space-dependent heat source problem. EngineeringAnalysis with Boundary Elements 2009;33:1231–5.

[50] Ohe T, Ohnaka K. Uniqueness and convergence of numerical solution of theCauchy problem for the Laplace equation by a charge simulation method.Japanese Journal of Industrial and Applied Mathematics 2004;21:339–59.

[51] Shigeta T, Young DL. Method of fundamental solutions with optimalregularization techniques for the Cauchy problem of the Laplace equationwith singular points. Journal of Computational Physics 2009;228:1903–15.

[52] Tikhonov AN, Leonov AS, Yagola AG. Nonlinear ill-posed problems. London:Chapman & Hall; 1998.

[53] Valle MF, Colac-o MJ, Neto FS. Estimation of the heat transfer coefficient bymeans of the method of fundamental solutions. Inverse Problems in Scienceand Engineering 2008;16:777–95.

[54] Vessella S. Quantitative estimates of unique continuation for parabolicequations, determination of unknown time-varying boundaries and optimalstability estimates. Inverse Problems 2008;24:023001. (81p.).

[55] Vogelius M, Xu J. A nonlinear elliptic boundary value problem related tocorrosion modeling. Quarterly of Applied Mathematics 1998;56:479–505.

[56] Wang Y, Rudy Y. Application of the method of fundamental solutions topotential-based inverse electrocardiography. Annals of Biomedical Engineer-ing 2006;34:1272–88.

[57] Wei T, Li YS. An inverse boundary problem for one-dimensional heat equationwith a multilayer domain. Engineering Analysis with Boundary Elements2009;33:225–32.

[58] Wei T, Zhou DY. Convergence analysis for the Cauchy problem of Laplace’sequation by a regularized method of fundamental solutions. Advances inComputational Mathematics 2010;33:491–510.

[59] Wei T, Hon YC, Ling L. Method of fundamental solutions with regularizationtechniques for Cauchy problems of elliptic operators. Engineering Analysiswith Boundary Elements 2007;31:373–85.

[60] Yan L, Fu CL, Yang FL. The method of fundamental solutions for the inverse heatsource problem. Engineering Analysis with Boundary Elements 2008;32:216–22.

[61] Yan L, Fu CL, Yang FL. A meshless method for solving an inverse spacewise-dependent heat source problem. Journal of Computational Physics 2009;228:123–36.

[62] Yang FL, Yan L, Wei T. Reconstruction of the corrosion boundary for the Laplaceequation by using a boundary collocation method. Mathematics and Compu-ters in Simulation 2009;79:2148–56.

[63] Yang FL, Yan L, Wei T. Reconstruction of part of a boundary for the Laplaceequation by using a regularized method of fundamental solutions. InverseProblems in Science and Engineering 2009;17:1113–28.

[64] Young DL, Tsai CC, Chen CW, Fan CM. The method of fundamental solutions andcondition number analysis for inverse problems of Laplace equation. Compu-ters and Mathematics with Applications 2008;55:1189–200.

[65] Zeb A, Ingham DB, Lesnic D. The method of fundamental solutions for abiharmonic boundary determination. Computational Mechanics 2008;42:371–9.

[66] Zhou D, Wei T. The method of fundamental solutions for solving a Cauchyproblem of Laplace’s equation in a multi-connected domain. Inverse Problemsin Science and Engineering 2008;16:389–411.

the mfs for numerical boundary identification in two-dimensional harmonic problems

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