Inverse impedance boundary problem via the conformal...

TAMTAM’09

Inverse impedance boundary problem via theconformal mapping method: the case of small

impedances

F. Ben Hassen1 , Y. Boukari1 , H. Haddar2

1ENIT-LAMSIN, Ecole Nationale des Ingénieurs de Tunis BP 37, 1002 Tunis, [email protected], [email protected] INRIA Saclay Ile de France & CMAP, Ecole Polytechnique, Route de Saclay, 91128 PalaiseauCedex, [email protected]

RÉSUMÉ. Haddar and Kress [9] ont étendu l’utilisation de la méthode des applications conformes[2, 8] pour construire le bord intérieur Γi d’un domaine doublement connexe à partir de données deCauchy sur le bord extérieur d’une fonction harmonique qui satisfait une condition homogène d’impé-dance sur Γi. Cependant, l’analyse de la méthode montre la non-convergence de l’algorithme pro-posé dans [9] dans le cas des faibles impédances. Dans ce travail, nous proposons des modificationsde cet algorithme afin d’obtenir un schéma d’inversion convergent et stable dans ce cas. La mé-thode est ensuite validée par des exemples numériques qui incluent également le cas d’impédancesvariables.

ABSTRACT. Haddar and Kress [9] extended the use of the conformal mapping approach [2, 8] toreconstruct the internal boundary curve Γi of a doubly connected domain from the Cauchy data on theexternal boundary of a harmonic function satisfying a homogeneous impedance boundary conditionon Γi. However, the analysis of this scheme indicates non convergence of the proposed algorithmfor small values of the impedance. In this paper, we modify the algorithm proposed in [9] in order toobtain a convergent and stable inversion process for small impedances. We illustrate the performanceof the method through some numerical examples that also include the cases of variable impedances.

MOTS-CLÉS : tomographie électrique, problèmes inverses, applications conformes.

KEYWORDS : electric impedance tomography, inverse problems, conformal mapping.

Reçu le 06/10/2009, révisé le 06/05/2010, accepté le 13/09/2010

Revue ARIMA, vol. 13 (2010), pp. 47-62

48 F. Ben Hassen, Y. Boukari and H. Haddar

1. Introduction

We are interested in an inverse problem for harmonic functions that arises in the ma-thematical modeling of electrostatic or thermal imaging methods. As examples of applica-tions we can mention the detection of faults in metal plates via applying electric currents[3, 4], the monitoring of lung patients [7] or also the evaluation of wood quality both viaelectrical impedance tomography in long cylindrical objects [17]. In these applications,an unknown inclusion within a conducting host mediumD with constant conductivity isassessed from overdetermined Cauchy data on the accessible part of the boundaryΓe ofthe medium. We shall consider the cases where the effect of the inclusion on imposedcurrents can be modeled by an impedance boundary condition that has be to satisfied onits boundaryΓi. This model can be viewed for instance as an approximation of the trans-mission problem in the asymptotic regimes : inclusion with a very high or a very smallconductivity.In a series of papers, a simple and fast numerical scheme was developed to reconstructa perfectly conducting inclusion [2, 14] or a nonconducting inclusion [8], governed by ahomogeneous Dirichlet or Neumann boundary condition on the boundaryΓi of the inclu-sion. The approach is based on a conformal mappingΨ : B → D that takes an annulusBbounded by two concentric circles ontoD. In the first step, in terms of the given Cauchydata onΓe, by successive approximations one has to solve a non-local and nonlinear or-dinary differential equation for the boundary valuesΨ|C1

of this mapping on the exteriorboundary circle ofB. Then in the second step, a Cauchy problem for the holomorphicfunctionΨ in B has to be solved via a regularized Laurent expansion to obtain the unk-nown boundaryΓi = Ψ(Cρ) as the image of the interior boundary circleCρ.The application of the conformal mapping method to the inverse impedance problem wasfirst analyzed in [9]. Since the homogeneous impedance condition onΓi transforms intoan impedance condition onCρ that contains the trace of the derivative of the conformalmapΨ onCρ, the method does not completely separate the two steps indicated above. Theanalysis of the proposed algorithm indicated the convergence for large impedances (i.e.when the impedance boundary condition is close to a Dirichlet boundary condition). Ho-wever, as expected from known results on the Neumann problem, (and as numerically ob-served) the algorithm does not converge in the opposite regime, i.e. for small impedances.This paper is devoted to addressing this issue using similar ideas as in the treatment ofthe Neumann problem, namely by deriving the nonlinear equation of step 1) associatedwith the conjugate harmonic function. The impedance condition is transformed into a ge-neralized impedance condition (where second order tangential derivative is involved) andtherefore the subsequent analysis requires a special care. The analysis of the algorithmindicates convergence for sufficiently small impedances. This is confirmed numericallyeven in the case of variable impedances. As we shall point out theoretically, the values ofthe impedances for which convergence occurs depends on the size of the inclusion. Thisis also true in the case of [9]. For other work using conformal mapping ideas in the studyof inverse problems for the Laplace equation we refer to [1, 5, 6, 12, 11]. Concerningthe recovery of obstacles with impedance boundary conditions we also refer to [15, 16]where a Newton like method based on the use of appropriate integral representation ofthe solution is used. Compared to our method, this technique is indeed more general andallows one to seek both the impedance and the geometry of the obstacle. However, noconvergence proof is available for this technique and as opposed to our algorithm, closeenough initialization is required to ensure convergence.


Inverse impedance boundary problem 49

The paper is organized as follows. We introduce in Section 2 the inverse impedanceproblem and recall the general ideas of the conformal mapping method and the recons-truction process proposed in [9]. In section 3, we describe the modified algorithm forsmall values of the impedance. The convergence of the proposed algorithm is discussedin Section 4. Finally, we conclude in section 5 with some numerical implementations tovalid our approach both for a constant and variable impedances.

2. The inverse problem and the conformal mapping method

LetD be a doubly connected bounded domain inR2 with a smooth boundary∂D that

consists of two disjoint closed Jordan curvesΓi andΓd, that is∂D = Γi ∪ Γe withΓi ∩ Γe = ∅ such thatΓi is contained in the interior ofΓe. By ν we denote the outwardunit normal toΓe. For a given functionf ∈ H1/2(Γe) and a positive constantλ weconsider the impedance problem for the Laplace equation

∆u = 0 in D, [1]

with boundary conditions∂u

∂ν− λu = 0 on Γi, [2]

andu = f on Γe. [3]

Note that the positivity ofλ ensures existence and uniqueness of a solutionu ∈ H1(D).The inverse problem we are interested in, is to determine the shape of the interior

boundary curveΓi from one or many pairs of Cauchy data

(f, g) =

(

u,∂u

∂ν

)∣

∣

∣

∣

Γe

, [4]

assuming that the impedance is known a priori. We shall use for that purpose the confor-mal mapping method introduced in [2, 8]. We shall first describe the principle of thismethod and the setting used in [9] to solve the case of large values ofλ.

Principle of the conformal mapping method : Let B := {z ∈ C : ρ < |z| < 1}denotes the annulus bounded by the two circlesCρ := {z ∈ C : |z| = ρ} andC1 :={z ∈ C : |z| = 1}. SinceD is a doubly connected domain, then by using the conformalmapping theorem (see [11]), there exists a uniquely determined radiusρ and a bijectiveholomorphic functionΨ : B → D that conformally mapsB ontoD such that theboundariesCρ andC1 are mapped ontoΓi andΓe, respectively. Denoting byL1 thelength ofΓe, and letγ : [0, L1] → Γe be a parametrization ofΓe in terms of the arclength. The mappingΨ is uniquely characterized by prescribingΨ(1) = γ(0).The inversion algorithm is based on the determination ofρ andΨ. To this end, we intro-duce the two functions :χ : [0, 2π] → C andϕ : [0, 2π] → [0, L1] defined by

χ(t) := Ψ(ρeit) and ϕ(t) := γ−1(Ψ(eit)). [5]

The functionχ is injective and parametrizes the interior boundary curveΓi. The functionϕ is strictly monotonically increasing and bijective, and roughly speaking it describes howΨ maps arc length onC1 onto arc length onΓe. Obviously, determining the mapχ solves



the inverse problem. The main idea behind setting up the conformal mapping method isthe derivation of a non linear differential equation satisfied by the boundary mapϕ. Thisequation is then solved iteratively to determineϕ then obtainχ.Iterative operator for large impedances (see [9]) : The derivation of the non lineardifferential equation used in [9] for large impedances is done as follows. We associatethe impedance problem (1)-(3) inD with an impedance problem posed in the annulusB.More precisely, ifu solves the impedance problem (1)-(3) thenv := u ◦Ψ is a harmonicfunction

∆v = 0 in B [6]

satisfying the boundary conditions

∂v

∂r(ρeit)− µρ,Ψ(ρe

it)v(ρeit) = 0, 0 ≤ t ≤ 2π, [7]

andv = f ◦Ψ on C1, [8]

where

µρ,Ψ(ρeit) :=

λ

ρ

∣

∣

∣

∣

d

dtΨ(ρeit)

∣

∣

∣

∣

=λ

ρ|χ′(t)|, 0 ≤ t ≤ 2π. [9]

Denote byAρ,µ : H1/2(C1) → H−1/2(C1) the Dirichlet-to-Neumann operator for theimpedance problem (6)-(8). It maps functionsf ◦Ψ ∈ H1/2(C1) onto the normal deriva-tive ∂v/∂ν onC1 of the solution of (6)-(8).Using the Cauchy-Riemann equations, we get the following non-local differential equa-tion

∂ϕ

∂t=Aρ,µ(f ◦ γ ◦ ϕ)

g ◦ γ ◦ ϕ[10]

for the boundary mappingϕ which has to be complemented by the boundary conditions

ϕ(0) = 0 and ϕ(2π) = L1. [11]

In order to rewrite (10)-(11) as a fixed point equation we introduce the affine operatorV :

(V ψ)(t) =L1

2πt+ ψ(t), t ∈ [0, 2π]. [12]

Then, for a pair of Cauchy data(f, g), after setting

Uρ,µψ :=Aρ,µ(f ◦ γ ◦ V ψ)

g ◦ γ ◦ V ψ,

we define an operatorTρ,µ : H10 [0, 2π] → H1

0 [0, 2π] by

(Tρ,µψ)(t) :=

∫ t

0

[

(Uρ,µψ)(τ)−1

2π

∫ 2π

0

(Uρ,µψ)(θ) dθ

]

dτ, t ∈ [0, 2π]. [13]

Finally, if ϕ ∈ H1[0, 2π] is a solution of (10)-(11) thenψ = V −1ϕ ∈ H10 [0, 2π] is a fixed

point ofTρ,µ.The scheme proposed in [9] is mainly based on solving the fix point equationTρ,µψ =

ψ using successive approximations. However, in order to completely describe the schemeand obtain a convergent one we omitted three steps 1) stabilization of the division byg in



(10) by combining the use of two Cauchy data 2) derivation of an equation for the radius3) regularization of the mappingϕ→ χ. One obtains a convergent algorithm for large im-pedances mainly because, asλ → ∞ the operatorAρ,µ ”converges” to the DtN operatorassociated with a homogeneous Dirichlet boundary condition on the interior circle. Onecan also show that the algorithm is non convergent for sufficiently small impedances sincein that caseAρ,µ is close to the DtN map associated with a homogeneous Neumann boun-dary condition on the interior circle. To obtain a convergent scheme in the latter cases, weshall modify the right hand side of (10) using a DtN map associated with the conjugateharmonic. Asλ→ 0 this operator has a similar behavior asAρ,µ whenλ→ ∞.

3. The new scheme for small impedances

If u is a harmonic function inD with Cauchy data(f, g) onΓe satisfying the impe-dance condition (3) then from the Cauchy-Riemann equations it follows that a conjugateharmonicu of u satisfies

∆u = 0 in D [14]

with the ”generalized” impedance condition

∂u

∂ν+

1

λ

∂2u

∂s2= 0 on Γi [15]

and the Cauchy data

∂u

∂s= g on Γe, [16]

∂u

∂ν= −

∂f

∂son Γe. [17]

We observe in particular, from integrating (16) overΓe, that the existence of single valuedconjugate harmonic functionu is subjected to the condition

∫

Γe

g ds = 0. [18]

Let us assume for the moment that this condition is verified and letu be the (unique)solution to (14)-(16). Thev := u ◦Ψ satisfies

∆v = 0 in B [19]

with the following generalized impedance boundary condition onCρ

∂v

∂r(ρeit) +

1

ρ2∂

∂t

(

µ(t)∂

∂tv(ρeit)

)

= 0 t ∈ [0, 2π] [20]

and the Dirichlet boundary condition onC1

v(eit) = F (eit) t ∈ [0, 2π], [21]

where we have set :

µ(t) :=ρ

λ |χ′(t)|and F (eit) := f ◦ ϕ(t), t ∈ [0, 2π], [22]



with f denoting the function defined with respect to the arc length onΓe by

f(s) =

∫ s

0

g ◦ γ(t) dt, s ∈ [0, L1]. [23]

Let Aρ,µ : H1/2(C1) → H−1/2(C1) be the Dirichlet-Neumann operator :F 7→ ∂v/∂r|C1,

associated to the impedance problem (19)-(21). Similarly to above, using the Cauchy-Riemann equations, one get the following non-local differential equation

∂ϕ

∂t=Aρ,µ(f ◦ ϕ)

g ◦ ϕ[24]

for the boundary mappingϕ, which are complemented by the boundary conditions

ϕ(0) = 0 and ϕ(2π) = L1. [25]

The functionf is given by (23) whereasg is defined in terms of the arc length onΓe by

g(s) = −(f ◦ γ(s))′, s ∈ [0, L1]. [26]

3.1. Combination of three Cauchy pairs

In order to get a stable reconstruction algorithm, the functiong should have no zeros.Moreover, the assumption (18) is not satisfied in general for some Dirichlet dataf . Toremedy to these problems, we use three different pairs of Cauchy data(fi, gi), 1 ≤ i ≤ 3.We assume that at least one of the three pairs does not satisfy (18), for example the pairnumber 3. We then build two pairs of Cauchy data(fi, gi), i = 1, 2, verifying (18) bysetting :

fi := fi − θi f3 andgi := gi − θi g3.

where

θi =

(∫

Γe

gi ds

)

/

(∫

Γe

g3 ds

)

.

We definefi and gi as in (23) and (26) wheref andg are respectively replaced withfiandgi. It is then clear from 24 that

∂ϕ

∂t=Aρ,µ(fi ◦ ϕ)

gi ◦ ϕ=

(gi ◦ ϕ)Aρ,µ(fi ◦ ϕ)

|gi ◦ ϕ|2[27]

for i = 1, 2. To shorten the notation, we introduce the complex-valued functions (whichhave therefore different meaning from those introduced above)

f = f1 + if2 and g = g1 + ig2

and observe that (27) implies that one also have

dϕ

dt=

ℜ{

(g ◦ ϕ) Aρ,µ(f ◦ ϕ)}

|g ◦ ϕ|2. [28]

In order to write (28) as a fixed point equation, we define the operatorTmodρ,µ : H1

0 [0, 2π] →

H10 [0, 2π] by

(Tmodρ,µ ψ)(t) :=

∫ t

0

[

(Umodρ,µ ψ)(τ)−

1

2π

∫ 2π

0

(Umodρ,µ ψ)(θ) dθ

]

dτ, t ∈ [0, 2π]. [29]



where

Umodρ,µ ψ :=

ℜ{

(gV ψ) Aρ,µ(fV ψ)}

|gV ψ|2.

Here, the operatorV is still given by (12). Obviously, ifϕ ∈ H1[0, 2π] is a solution of(28) satisfying (25) thenψ = V −1ϕ ∈ H1

0 [0, 2π] is a fixed point ofTmodρ,µ , that is

ψ = Tmodρ,µ (ψ) [30]

is satisfied. Note that ifTmodρ,µ is a contraction, then (28)-(25) and (30) are equivalent.

3.2. Equations for ρ

The fixed point equation (30) has to be coupled with an equation for the unknownradiusρ. We shall adopt a similar procedure as in [9]. Letw be a harmonic function inBsatisfying the generalized impedance boundary condition (21) onCρ, with µ as definedby (22). Moreover, we assume thatw = F1 onC1 for a givenF1 ∈ H1/2(C1). Then, weintroduce the functionHΨ : (0, 1) → R given by

HΨ(ρ) :=

∫

C1

f ◦ ϕ∂w

∂νds. [31]

Applying the Green’s theorem tov = u ◦ Ψ andw in the annulusB, and since the twofunctions satisfy the same generalized impedance condition on the interior boundaryCρ,

∫

C1

v∂w

∂νds =

∫

C1

∂v

∂νF1 ds =

∫ 2π

0

g(γ(ϕ(t)))ϕ′(t)F1(t) dt

Therefore, we have

HΨ(ρ) =

∫ 2π

0

g(γ(ϕ(t)))ϕ′(t)F1(t) dt. [32]

To compute an approximate solution of (32) via Newton iterations, we need the derivativeof HΨ. This in turn requires to calculate the derivativew′ of the functionw.w′ is a harmonic function inB vanishing onC1. To derive the boundary condition satisfiedbyw′ onCρ, we indicate the dependence ofw on the radiusρ by writingwρ and expand,

∂wρ+δ

∂r(ρ+ δ) +

1

(ρ+ δ)2

∂

∂t

(

µρ+δ∂wρ+δ

∂t

)

(ρ+ δ)

=∂wρ

∂r(ρ) +

1

ρ2∂

∂t

(

µρ∂wρ

∂t

)

(ρ)

+1

ρ2

{

∂

∂t

(

µρ

∂w′ρ

∂t

)

(ρ) +∂

∂t

(

µ′ρ

∂wρ

∂t

)

(ρ) +∂

∂t

(

µρ∂

∂t

∂wρ

∂r

)

(ρ)

}

δ

+

{

∂w′ρ

∂r(ρ) +

∂2wρ

∂r2(ρ)−

2

ρ3∂

∂t

(

µρ∂wρ

∂t

)

(ρ)

}

δ +O(δ2)



In view of the boundary condition (21) forwρ+δ andwρ and after dropping the subscriptρ, we have

∂w′

∂r+

∂

∂s

(

µ∂w′

∂s

)

= −∂2w

∂r2+

2

ρ

∂

∂s

(

µ∂w

∂s

)

−∂

∂s

(

µ′ ∂w

∂s

)

−∂

∂s

(

µ∂

∂s

∂w

∂r

)

onCρ.

Therefore, using the Laplace operator in polar coordinates and once again the boundarycondition (21) forw, the derivativew′ satisfies onCρ

∂w′

∂r+

∂

∂s

(

µ∂w′

∂s

)

=1

ρ

∂w

∂r+∂2w

∂s2+

2

ρ

∂

∂s

(

µ∂w

∂s

)

[33]

−∂

∂s

(

µ′ ∂w

∂s

)

−∂

∂s

(

µ∂2

∂s2

(

µ∂w

∂s

))

Finally, the derivative ofHΨ is

H ′Ψ(ρ) =

∫

C1

v∂w′

∂νds [34]

wherew′ is a harmonic function inB which vanishes onC1 and satisfies the boundarycondition 33 onCρ.

3.3. Regularization of the mapping ϕ 7→ χ

The nonlinear equations forφ andρ (which use in particular the impedanceµ) as wellas the reconstruction ofΓi requires a (stable) determination ofχ in terms ofφ andρ. Moreprecisely, knowing the radiusρ andΨ(eit) = (γ ◦ V ψ)(t), we need to solve the Cauchyproblem for determining the holomorphic functionΨ from its values onC1. Expandingin a Fourier series

(γ ◦ V ψ)(t) = eit∞∑

k=−∞

akeikt, t ∈ [0, 2π],

we obtainΨ by the Laurent series

Ψ(z) = z

∞∑

k=−∞

akzk. [35]

The severe ill-posedness of the Cauchy problem inherited from the ill-posedness of ourinverse problem is clearly illustrated, since small errors in the Fourier coefficientsak,k < 0, will be amplified by the exponentially increasing factorsρk. Hence, we incorporatea Tikhonov-type regularization and replace (35) by

Ψα(z) = z∞∑

k=0

akzk + z

∞∑

k=1

a−k|z|2k

α+ |z|2kz−k, 0 < |z| < 1, [36]

whereα > 0 serves as a regularization parameter. We then getχ via χ = For its choicevia a discrepancy principle we refer to [8]. To simplify notation, in the sequel byLα wedenote the map that takesψ intoΨα as defined by (36).



3.4. The inversion scheme

Now, we can summarize our inversion process in the following iteration scheme. Gi-ven an initializationψ0 = 0 (see Remark 3.1) for the boundary map and an arbitrarychoice for the initial radius byρ0 (we usedρ0 = 0.3 in the numerical examples), theiteration algorithm is slitted into the following steps

1) Compute the mappingΨα = Lαψn and the correspondingχ(t) = Ψα(ρneit).

2) ComputeHΨαgiven by (31) and its derivative, by evaluatingw andw′ associa-

ted with a chosen functionF1 on the unit circle.

3) Update the radius using one Newton iteration via

ρn+1 = ρn −HΨα

(ρn)−∫ 2π

0g(γ(V ψn(t)))(V ψn)

′(t)F1(t) dt

H ′Ψα

(ρn).

4) Update the impedance functionµn+1 =ρn+1

λ|χ′(t)|.

5) Update the mappingϕn+1 = V ψn+1 by solving the differential equation asso-ciated with the iteration

ψn+1 = Tmodρn+1,µn+1

(ψn).

Remark 3.1 A natural initial guess forψ would be the one that corresponds withΨbeing the conformal mapping that transforms the unit disk onto the domain enclosed byΓ1. Since in our experiments we choose the to be the unit disk, this initial guess coincideswithψ0 = 0.

4. Convergence Analysis

In order to check the convergence of our algorithm for small impedances we shallrestrict ourselves to the simple case where the geometry is a small perturbation of anannulus. Moreover we shall assume that the exact value for the radius is known (andtherefore, in particularµ = 1/λ). The analysis is then reduced to evaluating‖T ′(0)‖where

Tψ := Tmodρ,1/λ(ψ). [37]

Following [8], one has

d

dtT ′(0)h = U ′(0)h−

1

2π

∫ 2π

0

U ′(0)h dτ. [38]

where

U ′(0)h =ℜ[

g Aρ,1/λ(h f′)− h g ′ g

]

|g|2[39]

We shall further restrict the analysis by choosing three pairs of Cauchy data(fi, gi), 1 ≤i ≤ 3, associated to

f1(γ(t)) = cosmt, f2(γ(t)) = sinmt, f3(γ(t)) = 1, [40]

wheret ∈ [0, 2π] andm ≥ 1.



For abbreviation, bycn, sn anden we denote the trigonometric functions given by

cn(t) := cosnt, sn(t) := sinnt and en(t) := eint

for n = 0,±1,±2, . . . . Then one easily check that

Aρ,1/λen = γnen [41]

with

γn = |n|1− bn1 + bn

and bn =λρ− |n|

λρ+ |n|ρ2|n|,

while for n 6= 0,Aρ,λen = γnen

with

γn = |n|1 + bn1− bn

.

We therefore conclude that

g1(γ(t)) = g1(γ(t)) = γmcm(t), and g2(γ(t) = g2(γ(t)) = γmsm(t).

Consequently,

f1(t) =1

mγmsm(t), and f2(t) = −

1

mγmcm(t).

andg1(t) = msm(t), and g2(t) = −mcm(t).

After simple manipulations, and usingγm/m2 = 1/γm, the expression ofU ′(0) simpli-fies into

U ′(0)h =m

γmℑ(

emAρ,1/λ(hem))

. [42]

We are now in a situation similar to the convergence proof in [8, Theorem 3.1]. Weconclude that

U ′(0)sn =m

2γm[γn+m − γn−m]cn

and consequently, by integration,

T ′(0)sn =m[γn+m − γn−m]

2nγmsn

for n ∈ N. Analogously,U ′(0)(cn − 1) = −m

2γm[γn+m − γn−m]sn and

T ′(0)(cn − 1) =m[γn+m − γn−m]

2nγm(cn − 1)

for n ∈ N. We therefore deduce that the eigenvalues of the compact and self-adjointoperatorT ′(0) are

βn(λ, ρ) =m[γn+m − γn−m]

2nγm[43]



Forn > 0 andρ < 1, we notice that asλ→ 0

βn(λ, ρ) → βn(0, ρ) [44]

whereβn(0, ρ) are the eigenvalues of the iterative operator in the case of an inclusionwith a homogeneous Dirichlet boundary condition. Using the result of [8, Theorem 3.1]we conclude thatsupn |βn(0, ρ)| < 1 and,

limλ→0

‖T ′(0)‖ < 1. [45]

We therefore can state the following theorem :

Theorem 4.1 Assume thatD = B and that the pairs of data correspond to (40). Thenthe sequence(ψn) defined by the successive approximationsψn+1 = Tψn whereT isdefined by (37) converges to0, provided thatλ is sufficiently small

In practice it is desirable to a quantification of the value ofλ for which convergence occur.This question was not studied in [9]. We shall indicate in the following that the values ofλfor which convergence holds depends onρ. This means in particular that the convergencein (44) is not uniform with respect toρ for all n. To see this it is sufficient to consider thecase of

βm(λ, ρ) =γ2m(2γm)

=(1− b2m)(1 + bm)

(1− bm)(1 + b2m).

Forλ > 0, we easily see that

limρ→1

βm(λ, ρ) → βm(λ, 1) = 2.

This implies for instance that for any value of0 < λ < m there exists a radiusρ close to1 such that‖T ′(0)‖ > 1 and therefore the algorithm is not converging for that radius.

We conclude this section by indicating that the ”non convergence region” for small valuesof λ is confined in a neighborhood ofρ = 1 whose width exponentially goes to 0 asλ → 0. Computing the derivative ofγn with respect ton indicates thatn → γn isdecreasing for

n2 ≥ n20 := (λρ)2 − (λρ)/ ln(ρ) [46]

Therefore, following the similar procedure as in the proof of [8, Theorem 3.1], one canprove :

Theorem 4.2 Let βn(λ, ρ) be defined by (43) forλ > 0 and0 < ρ < 1 and letn0 bedefined by (46). Then

0 < βn(λ, ρ) < 1 for all {m ≥ 1 and|n−m| ≥ n0} or {n = m andn+m ≥ n0}.

In particular we see that ifn0 ≤ 1 then,

0 < βn(λ, ρ) < 1 for all n ≥ 1.

For sufficiently smallλ, n0 ≤ 1 for all 0 < ρ ≤ ρ0(λ) whereρ0 satisfies

ρ0 = exp(−(λρ0)/(1− (λρ0)2)).



One observes thatρ0 → 1 asλ→ 0 exponentially fast.We also remark that the ordern0 can be used to quantify the term “small impedance”

for a given value of the radiusρ : these small values should correspond withn0 ≤ 1. Forexample forρ = 0.3, n0 is smaller than1 if the value of the impedance is smaller than2.25.

We finally notice that this non uniform behavior with respect toρ is not restrictedto the case of small impedances. We expect that similar results hold in the case of highimpedances.

5. Numerical Results

In this section, we provide some numerical examples to illustrate the convergence ofour algorithm. In these examples, we consider the casem = 1 (for the data (40)), the unitcircle as the exterior boundary and different parametrization of the interior boundary. Thesimulation of the data and the evaluation of the DtN map in the iteration procedure is doneafter reformulating the boundary value problem into an integral equation on the boundaryof the domain. The corresponding integral equation is discretized using a Nyström method[13]. For the case of a generalized impedance boundary condition (which is needed toevaluateT ), the second order tangential derivative is discretized using a standard finitedifference method.

Figure 1. Exact (red) and reconstructed (green) geometries after 2 iterations (left) and 6iterations (right). λ = 0.01 and the Cauchy data to f1 = sin t , f2 = cos t and f3 = 1. Noadded random noise.



Figure 2. Exact (red) and reconstructed (green) geometries after 10 iterations (left) and15 iterations (right). λ = 0.01 and the Cauchy data to f1 = sin t , f2 = cos t and f3 = 1. Noadded random noise.

Figure 1 and Figure 2 respectively correspond to the two following parameterizations ofthe inetrior boundaryΓi :

x1(t) = −0.2 + 0.4 cos t, x2(t) = 0.4 sin t+ 0.2 sin t, t ∈ [0, 2π], and

x1(t) = 0.4 cos t+ 0.2 cos 2t, x2(t) = 0.8 sin t+ 0.2 sin 2t, t ∈ [0, 2π].

As shown in Figure 1 and Figure 2, we find forλ = 0.01 and after only 2 iterations ageometry close to the exact one.

Figure 3. Exact (red) and reconstructed (green) geometries after 10 iterations for λ = 0.1and three Cauchy data corresponding to f1 = sin t , f2 = cos t and f3 = 1. Reconstructiondone for different levels of added relative random noise : 1% (left), 10% (right).



Figure 4. Exact (red) and reconstructed (green) geometries after 10 iterations for λ =0.001 and three Cauchy data corresponding to f1 = sin t , f2 = cos t and f3 = 1. Recons-truction done for different levels of added relative random noise : 1% (left), 10% (right).

As shown in Figure 3 and Figure 4, even with the presence of noise, we find a goodreconstruction of the interior geometry.

The case of variable impedances In previous examples we considered the case of an in-clusion with constant impedances. In some cases, this impedance would model the rough-ness of the inclusion boundary which implies that the impedance may be variable. Eventhough not supported by the convergence analysis, we shall numerically test our algorithmin the case of variable impedances. The only needed modification in our algorithm is tochange the expression of the impedanceµ into

µ(t) =1

λ(arg(χ(t))

ρ

|χ′(t)|

wherearg denotes the argument of the complex number associated with a given vector.We have tested the algorithm in the caseλ(t) = 0.001(sin(t) − 1.001) for dif-

ferent geometries and obtain convergent and stable reconstructions as attested by Fi-gures 5 and 6.

Figure 5. Exact (red) and reconstructed (green) geometries after 9 iterations (left) and15 iterations (right) for λ = 0.001(sin(t) − 1.001) and three Cauchy data f1(e

it) =cos t, f2(e

it) = sin t and f3(eit) = 1.



Figure 6. Exact (red) and reconstructed (green) geometries after 9 iterations (left) and15 iterations (right) for λ = 0.001(sin(t) − 1.001) and three Cauchy data f1(e

it) =cos t, f2(e

it) = sin t and f3(eit) = 1.

Conclusion

We have studied the problem of determining the shape of a simply connected obs-tacle inside a simply connected domain, from the knowledge of Cauchy data associatedwith an electric potential, using the conformal mapping method. The potential satisfiesthe Laplace equation with an impedance boundary condition on the boundary of the obs-tacle. Our investigations focus on the case of small impedances as being an extensionof the work proposed in [9]. We explained how a modification of the algorithm in [9]using conjugate harmonic functions provides a convergent iterative scheme. The conver-gence proof of the proposed algorithm is done in the simple case where the domain is asmall perturbation of an annulus. Extending our approach to general geometries wouldbe desirable, but is not obvious, since it is based on evaluating the spectral radius of thederivative of the iterative operator. The expression of this operator is very complex andhardly exploitable in the general case. We also pointed out, in the convergence analysis,that for a given value ofλ, the algorithm is not convergent for all possible values of the ra-diusρ. We gave a precise quantification of this behavior for the case of an annulus. Thesetheoretical considerations have been validated by means of some numerical tests. Thosetests have been conducted for different configurations (that are not small perturbationsof an annulus) and small impedance values, demonstrating the efficiency of the proposedalgorithm. A perspective of the present work would the extension of this approach to thecase of inclusions with small conductivities, which would be complementary to the workinitiated in [10].

6. Bibliographie

[1] G. ALESSANDRINI, A. MORASSI, E. ROSSET, « Detecting cavities by electrostatic boundarymeasurements. »,Inverse Problems, vol. 18 , no 5 , 1333-1353 , 2002.

[2] I. A KDUMAN , R. KRESS, « Electrostatic imaging via conformal mapping. »,Inverse Pro-blems, vol. 18 , no 6 , 2002.



[3] N. D. APARICIO , M. K. PIDCOCK, « The boundary inverse problem for the Laplace equationin two dimensions. »,Inverse Problems, vol. 12, 565-77, 1996.

[4] H. T. BANKS, F. KOJIMA, « Boundary shape identification in two-dimensional electrostaticproblems using SQUIDs. »,J. Inv. Ill-Posed Problemsvol. 8, 467-504, 2000.

[5] L. BARATCHART, A. BEN ABDA, F. BEN HASSEN, J. LEBLOND , « Recovery of pointwisesources or small inclusions in 2D domains and rational approximation. »,Inverse Problemsvol. 21, no 1, 51-74, 2005.

[6] S. CHAABANE , C. ELHECHMI, M. JAOUA, « A stable recovery method for the Robin inverseproblem. », Math. Comput. Simulationvol. 66, no 4-5, 367-383, 2004.

[7] I. FRERICHS, J. HINZ, P. HERRMAN , G. WEISSER, G. HAHN, M. QUINTEL , G. HELLIGE.« Regional lung perfusion as determined by electrical impedance tomography in comparisonwith electron beam CT imaging.IEEE Trans. Med Imaging», vol. 21, 646-52, 2002.

[8] H. HADDAR, R. KRESS, « Electrostatic imaging via conformal mapping. »,Inverse Pro-blems, vol. 21 , no 3 , 2005.

[9] H. HADDAR, R. KRESS, « Conformal mapping and an inverse impedance boundary valueproblem. »,J. Inv. Ill-Posed Problems, vol. 14 , no 1 , 2006.

[10] H. HADDAR, R. KRESS, « Conformal mapping and impedance tomography. »,InverseProblems, vol. 26 , no 7 , 2010.

[11] P. HENRICI « Applied and computational complex analysis », vol. 1,Wiley Classics Library,New York,1988.

[12] M. JAOUA, J. LEBLOND, M. MAHJOUB, J. R. ,PARTINGTON, « Robust numerical algorithmsbased on analytic approximation for the solution of inverse problems in annular domains. »,IMA J. Appl. Math, vol. 74 , no 2 , 481-506, 2009.

[13] R. KRESS. Linear integral equations, volume 82 ofApplied Mathematical Sciences. Springer-Verlag, New York, second edition, 1999.

[14] R. KRESS. « Inverse Dirichlet problem and conformal mapping »,Math. and Computer inSim.vol. 66, 255-65, 2004.

[15] W. RUNDELL « Recovering an obstacle and its impedance from Cauchy data. »,InverseProblemsvol. 24, no 4, 2008.

[16] W. RUNDELL « Recovering an obstacle and a nonlinear conductivity from Cauchy data. »,Inverse Problemsvol. 24, no 5, 2008.

[17] U. WEIHS„ F. Krummheuer, V. Dubbel « Zerstorungsfreie Baumdiagnose mittels elektri-scher Widerstandstomographie »,Jahrbuch der Baumpege, 505- 58, 2001,(Braunschweig :Thalacker Medien).


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