Detecting corrosion using thermal measurements

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Detecting corrosion using thermal measurements

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2007 Inverse Problems 23 53

(http://iopscience.iop.org/0266-5611/23/1/003)

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INSTITUTE OF PHYSICS PUBLISHING INVERSE PROBLEMS

Inverse Problems 23 (2007) 53–72 doi:10.1088/0266-5611/23/1/003

Detecting corrosion using thermal measurements

T Hohage1, M-L Rapun2 and F-J Sayas3

1 Inst. fur Numerische und Angewandte Mathematik, Universitat Gottingen, Lotzestr. 16–18,37083 Gottingen, Germany2 Dep. Matematica Aplicada (Biomatematica), Universidad Complutense de Madrid, C/ JoseAntonio Novais 2, 28040 Madrid, Spain3 Dep. Matematica Aplicada, CPS, Universidad de Zaragoza, C/ Marıa de Luna 3,50018 Zaragoza, Spain

E-mail: [email protected], [email protected] and [email protected]

Received 11 July 2006, in final form 18 September 2006Published 4 December 2006Online at stacks.iop.org/IP/23/53

AbstractThis paper deals with the inverse problem of detecting the level of corrosionat the interface of an inclusion given thermal measurements at the accessibleboundary of the sample. This leads to a transmission problem for the heatequation with an unknown coefficient in a transmission condition. We considerboth time-harmonic and delta-pulse excitations. In both cases we proveuniqueness results for the inverse problem. To reconstruct the unknown levelof corrosion numerically, we study a non-iterative method and a regularizedNewton method and compare their performances in a number of numericalexperiments.

1. Introduction

Photothermal techniques are suitable means of inspecting composite materials with non-destructive tests. We are interested in a technique that consists in heating the accessible sideof the material by a defocused laser beam. The goal is to reconstruct internal properties of thematerial (to detect structural defects, reconstruct the size, depth, orientation of the inclusionsand/or physical properties of them) from measurements of the temperature at the side thathas been thermically excited. Some recent papers on physical experiments with this kind oftechniques are [12, 27, 33, 34].

In this work we study the detection of the level of corrosion at the interface separating theinternal inhomogeneities from the matrix material. We explore two different kinds of thermalexcitement, produced by time-harmonic incident heating fields and by delta-pulse excitations.In the first one we deal with a steady-state problem. In the corresponding forward problem onelooks for time-harmonic solutions of an elliptic conductive-transmission problem. This kindof solution is commonly referred to as thermal waves. For a detailed study of these waves andtheir uses we refer to [3, 22–24] and references therein. The numerical solution of the forward

0266-5611/07/010053+20$30.00 © 2007 IOP Publishing Ltd Printed in the UK 53

54 T Hohage et al

problem using boundary element methods was discussed in [15, 31] and of a closely relatedproblem in [29, 30]. In this paper we first prove a uniqueness result for the reconstruction ofthe corrosion function in the time-harmonic setting and propose two different approaches torecover this function: a non-iterative method and a regularized Newton-type method.

When delta-pulse excitations are used, we deal with an evolution equation. In a paper [15]devoted to the forward problem it has been shown that with the help of the Laplace transformwith respect to time and a boundary element formulation in space, we can profit from thetechniques in the time-harmonic case. We will use these results to study the time-dependentinverse problem of recovering the corrosion function from the observed temperature on thesurface of the sample over a time interval. On the one hand, we perform a simple adaptation ofthe uniqueness result for the time-harmonic case to show uniqueness for the transient problem.On the other hand, we show how to recycle the codes of the time-harmonic setting to developa regularized Newton-type method for the numerical reconstruction of the corrosion function.Our numerical experiments suggest that depending on the size of the obstacles, careful choicesof the time interval and of the observation points allow more accurate reconstructions for agiven noise level than simple time-harmonic excitations.

There has been some previous work on the use of thermal waves for inverse problems. In[26] tomographic techniques based on thermal waves were applied to obtain the shape of aninhomogeneity in a turbid medium. Furthermore, domain identification in a two-dimensionalsetting, where part of the domain is known and the other part has to be reconstructedfrom measurements of the temperature in a subset of the known part, was considered in[6, 7, 13]. From the huge literature on lateral overdetermination for the heat equation we onlycite [10, 17] where the interested reader can find further references.

A similar problem to the detection of corrosion is the recovery of a coating factor indielectric materials as studied in [8]. That work deals only with the time-harmonic situation,but it also includes the treatment of partial coating. We will comment on the relation betweenour uniqueness results and those in [8] in section 3, after our main results are stated and proved.We also remark that the goal in [8] is to determine the shape of the scatterer and the L∞ normof the coating function, but that the function itself is not recovered. Also, even for knownscatterers, only numerical experiments for constant coating functions are shown in that work.Related results by the authors of [8] are referenced and explained therein.

The forward problem is modelled by a heat diffusion problem in the half planeR

d− := (x1, . . . , xd) ∈ R

d | xd < 0 with d = 2 or 3. We consider an inclusion − ⊂ Rd−,

which is a bounded open domain whose boundary := ∂− is a C2-curve/surface. Thematerials occupying − and + := R

d−\− have different thermal properties, i.e., their

corresponding diffusivities κ−, κ+ > 0 are different. Then the temperature distribution

U(x, t) :=

U−(x, t), in − × (0,∞),

U+(x, t), in + × (0,∞),

satisfies the heat equation

∂tU∓ = κ∓U∓, in ∓ × (0,∞). (1)

In the exterior domain + the total temperature U+ + Uhom is a superposition of U+ and a givensource field Uhom which satisfies

∂tUhom = κ+Uhom, in Rd− × (0,∞).

Uhom plays the role of an incident field, and U+ the role of a scattered field. On the commoninterface , the temperature satisfies the transmission conditions

U− + f ∂νU− = U+ + Uhom, on × (0,∞), (2a)

Detecting corrosion using thermal measurements 55

α∂νU− = ∂νU+ + ∂νUhom, on × (0,∞), (2b)

where α > 0 is the ratio of the interior and the exterior conductivities. Condition (2a) issometimes called an Engquist–Nedelec condition. The function f (·) f0 > 0 models acorrosion factor, proportional to the width of the coating at each point of the interface andit will be the unknown of the inverse problem. The pair of transmission conditions (2a), (2b)is also called conductive transmission (see [4] and references therein). The model of thetime-dependent forward problem is completed by the adiabatic boundary condition

∂νU+ = 0, on × (0,∞), (3)

on the upper boundary := ∂Rd− and the initial conditions

U−(·, 0) = 0, U+(·, 0) = 0. (4)

The first part of this paper is devoted to the important special case of time-harmonicexcitations, i.e., incident fields of the form Uhom(x, t) = Re(uhom(x) exp(−iωt)) with agiven frequency ω > 0. In this case we obtain asymptotically time-harmonic solutionsU∓(x, t) = Re (u∓(x) exp(−iωt)) to problem (1)–(3), and the space-dependent componentu∓ is the solution of an elliptic transmission problem, which will play a crucial role in ournumerical method for the solution of the time-dependent problem (see [15]). To write thiselliptic transmission problem for u∓ in compact form, we introduce the space

H := (u−, u+) ∈ H 1

(−) × H 1(+) | ∂νu+ = 0 on

,

where H 1(O) := u ∈ H 1(O) | u ∈ L2(O), and the differential operator

Lu :=u− + λ2

−u−, in −,

u+ + λ2+u+, in +,

with λ∓ := (1 + i)√

ω/(2κ∓). Then, the complex amplitude u := (u−, u+) satisfies

u ∈ H, Lu = 0, (5a)

u− + f ∂νu− − u+ = uhom, on , (5b)

α∂νu− − ∂νu+ = ∂νuhom, on . (5c)

The inverse problem is to find f given the incident field and measurements of u+(x) forx ∈ obs ⊂ or measurements of U+(x, t) for x ∈ obs and t in some time interval.

The remainder of this paper is organized as follows: in the following section we review thetime-harmonic forward problem and its numerical solution and study the Frechet derivative ofthe solution with respect to f . A uniqueness result for the time-harmonic inverse problem isgiven in section 3. In the following sections we discuss both a non-iterative method (section 4)and an iterative method (section 5) for the numerical solution of the time-harmonic inverseproblem. Finally, section 6 is devoted to the time-dependent problem, and a short conclusioncompletes the paper.

2. Forward solution operator and its Frechet derivative

2.1. Boundary integral formulations of the forward problem

In this section we present two different systems of integral equations that are equivalentto (5a)–(5c). Let

λ(x, y) :=(i/4)H

(1)0 (λ|x − y|) for d = 2

exp(iλ|x − y|)/(4π |x − y|) for d = 3,

56 T Hohage et al

denote the fundamental solution to the Helmholtz equation u + λ2u = 0 where H(1)0 is the

Hankel function of the first kind and order zero (see e.g. [9]).The first method was proposed and analysed in [15]. It is based on an indirect formulation

of the problem using the single layer potential ansatz

u = (Sλ−ψ−, Sλ+ψ+), (6)

where

Sλψ− :=∫

λ(·, y)ψ−(y) dγy : − → C, (7a)

Sλψ+ :=∫

(λ(·, y) + λ(·, y))ψ+(y) dγy : + → C, (7b)

with densities ψ∓ in the Sobolev space H−1/2(). Here y denotes the reflected point to y withrespect to , i.e., y is obtained by changing the sign of the last coordinate of y. Note thatthe use of the symmetrized fundamental solution to the Helmholtz equation in the definitionof (7b) guarantees the condition on the top barrier. With the ansatz (6) the transmissionconditions (5b) and (5c) are equivalent to a system of integral equations

Wf (ψ−, ψ+) = (uhom, ∂νuhom), (8)

with the operator

Wf :=[f

(12I + J λ−

)+ V λ− −V λ+

α(

12I + J λ−

)12I − J λ−

],

containing the single layer operator V λ and its normal derivative J λ defined by

V λϕ :=∫

λ(·, y)ϕ(y) dγy : → C,

J λϕ :=∫

∂ν(·)λ(·, y)ϕ(y) dγy : → C.

The boundary integral operators V λ and J λ are defined likewise by taking λ(·, y) + λ(·, y)

instead of λ(·, y). If f ∈ C1(), then Wf : Hr () → Hr () is bounded for −1 r 1(Hr () := Hr() × Hr()), whereas if f ∈ C(),Wf : H0() → H0() is bounded.As shown in [15], equation (8) has a unique solution in H−1/2() for sufficiently smooth f .A numerical approximation of the solution can be obtained by Petrov–Galerkin (or pureGalerkin) methods as discussed in [15, 31]. In particular, the spectral method (withtrigonometric polynomials or spherical harmonics) enjoys superalgebraic convergence orderif the corrosion function f and the boundary are infinitely smooth. For finitely smooth f

and , spline approximations are preferred. When is a curve in R2, parametrized by a one-

periodic function, fully discrete approximations based on spectral methods can be obtained inthe same way as in [9, 25, 29].

Alternatively, one may represent only the exterior part u+ of the solution to (5a)–(5c) asa symmetrized single layer potential, and recover the solution in − from its Cauchy dataξ := u− and η := ∂νu− on by the representation formula

u = (Sλ−η − Dλ−ξ, Sλ+ψ).

Here Dλ is the double layer potential

Dλϕ :=∫

∂ν(y)λ(·, y)ϕ(y) dγy : − → C.

Detecting corrosion using thermal measurements 57

Now the unknowns of the problem are ψ, ξ, η, and we arrive at the following equivalent systemof integral equations: −V λ+ f I I

0 −V λ− 12I + Kλ−

12I − J λ+ αI 0

ψ

η

ξ

= uhom

0∂νuhom

. (9)

The operator Kλ defined by

Kλϕ :=∫

∂ν(y)λ(·, y)ϕ(y) dγy : → C

is the adjoint operator to J λ with respect to the bilinear duality H−1/2() × H 1/2(). Notethat the second equation is the well-known identity for the interior Cauchy data of any solutionto an interior Helmholtz problem. The remaining equations are the transmission conditionson written in terms of the new unknowns. This approach can be easily studied fromboth the continuous and the numerical points of view by suitable adaptations of the resultsin [30, 31].

2.2. Frechet derivative

The function

uhom(·, x0) := λ+(·, x0) (10)

describes a point source at x0 ∈ +. Physically, only point sources on the boundary ofthe sample are possible, and the set of all source points is denoted by src ⊂ . Given f

we denote by u(·, x0) := (u−(·, x0), u+(·, x0)) the associated solution to (5a)–(5c) for thepoint source (10). We assume we can observe u+(·, x0)|obs on some subset obs ⊂ for allx0 ∈ src ⊂ . Using this notation we can introduce the forward solution operator

F : D(F) ⊂ f ∈ C() | inf

f > 0 −→ L2(obs × src)

f −→ u+|obs×src .

For the Newton method discussed in section 5 we need the Frechet derivative of F. This ischaracterized by the following proposition.

Proposition 2.1. The operator F is Frechet differentiable with respect to the supremum normon D(F). Moreover, for a given f , a perturbation h ∈ C() and x0 ∈ src, let u ∈ H denotethe solution to the forward problem (5a)–(5c) with uhom given by (10), and let u′ be the uniquesolution to the transmission problem

u′ ∈ H, Lu′ = 0, (11a)

u′− + f ∂νu

′− − u′

+ = −h∂νu−, on , (11b)

α∂νu′− − ∂νu

′+ = 0, on . (11c)

Then (F ′[f ]h)(·, x0) = u′+|obs .

Proof. By the results of the previous subsection the solution to the forward problem can bewritten as

u(·, x0)|obs = [0 V

λ+

]W−1

f u0(x0), x0 ∈ scr

where the mapping u0 : src → H0() is defined by

u0(x0) := (uhom(·, x0), ∂νuhom(·, x0)),

58 T Hohage et al

and the linear operator Vλ+ : H−1/2() → L2(obs) is given by

Vλ+ψ := (Sλ+ψ)|obs = 2

∫

λ+(·, y)ψ(y) dγy.

Because of the smoothness of the map u0 (with respect to x0) and by the decay propertiesof the layer potential on , it is simple to see that u ∈ L2(obs × scr) is well defined. Hencewe can succinctly write

F [f ] = [0 V

λ+

]W−1

f u0.

The map C() f → Wf ∈ L(H0()) is affine and continuous. Moreover, Wf is invertiblefor all f ∈ D(F). Hence, f → W−1

f is infinitely often differentiable from D(F) to L(H0())

and (W−1

f

)′h = −W−1

f (W ′f h)W−1

f , W ′f h = h

[ 12I + J λ− 0

0 0

].

Note that for x0 ∈ scr,

(W ′f h)W−1

f u0(x0) = (h(

12I + J λ−

)ψ−, 0

) = (h∂νu−, 0),

where (ψ−, ψ+) solves (8) and (u−, u+) solves (5a)–(5c). Hence,

(F ′[f ]h)(·, x0) = [0 V

λ+

]W−1

f (−h∂νu−, 0) = u′+|obs, (12)

where (u′−, u′

+) is the solution to (11a)–(11c).

Note that by proposition 2.1 the computation of F ′[f ]h requires the solution of a problemwith exactly the same structure as the direct problem (5a)–(5c), but with a new right-hand sideinvolving h and ∂νu−. By the results of section 2.1 we have

(F ′[f ]h)(·, x0) = Vλ+ϕ+,

where the exterior density ϕ+ can be computed by solving the system of integral equationsWf (ϕ−, ϕ+)

= (−h∂νu−, 0). The flux ∂νu− can be obtained as part of the solution of thesystem (9). This also yields a solution to the forward problem. Therefore, in the Newton-typemethod to be discussed in section 5 we use the second approach (9) to evaluate F and the firstapproach (8) to evaluate F ′[f ]h.

2.3. L2-adjoint of the Frechet derivative

Note that F ′[f ] has a unique extension to a bounded linear operator from the real Hilbertspace L2

R() to the complex Hilbert space L2

C(obs × src). To speak of an adjoint of F ′[f ]

we interpret L2C(obs ×src) as a real Hilbert space with inner product Re(〈g1, g2〉L2). Hence,

the L2-adjoint of F ′[f ] is the operator F ′[f ]∗L2 : L2

C(obs × src) → L2

R() defined by the

property∫

(F ′[f ]∗L2g)(y, x0)h(y) dγy = Re

(∫obs

∫src

g(x)(F ′[f ]h)(x, x0) dx dx0

)(13)

for all g ∈ L2C(obs × src) and h ∈ L2

R().

Lemma 2.2. If v := (v−, v+) solves

v ∈ H, Lv = 0, (14a)

v− + f ∂νv− − v+ = g1, on , (14b)

α∂νv− − ∂νv+ = g2, on , (14c)

Detecting corrosion using thermal measurements 59

then two of its Cauchy data satisfy the following uniquely solvable system of BIE

Wf

[−α∂νv−v+

]=

[−α(

12I + Kλ−

)g1

V λ+g2

],

where Wf denotes the transpose of Wf , i.e. the adjoint of Wf with respect to the bilinear

dual pairing on H0().

Proof. By Green’s third theorem we have12v+ = Kλ+v+ − V λ+∂νv+,

12v− = V λ−∂νv− − Kλ−v−.

Noting that

Wf =

[(12I + Kλ−

)f + V λ− α

(12I + Kλ−

)−V λ+ 1

2I − Kλ+

],

the result follows readily from substituting the transmission conditions (14b)–(14c) in thepreceding expressions. Proposition 2.3. Let V

λ+ : L2(obs) → H 0() be given by

Vλ+g := 2

∫obs

λ+(x, ·)g(x) dx.

Assume that src = x0. Then the operator F ′[f ]∗L2 defined by (13) is given by

F ′[f ]∗L2g = Re(α∂νu− ∂νv−), g ∈ L2(obs × x0)where u is the solution to (5a)–(5c) and v the solution to

v ∈ H, Lv = 0, (15a)

v− + f ∂νv− − v+ = 0, on , (15b)

α∂νv− − ∂νv+ = (V λ+)−1Vλ+g, on . (15c)

For general src we have F ′[f ]∗L2g = Re

(α

∫src

∂νu−(·, x0)∂νv−(·, x0) dx0).

Proof. First assume that src = x0. Let Qf : L2() → H0() be given by

Qf h := (W ′f h)W−1

f u0(x0) = (h∂νu−, 0),

where, as usual, (u−, u+) is the solution to (5a)–(5c) and u0 is defined as in the proof ofproposition 2.1. Qf is well defined on L2() since ∂νu− is continuous. To see this, note thatall the integral operators involved are compact in C(), so the Fredholm alternative can beapplied to Wf in C(). Then (12) allows us to write

F ′[f ] = − [0 V

λ+

]W−1

f Qf .

It follows from (13) that

F ′[f ]∗L2g = Re

(−Q∗

f

(W−1

f

)∗[

0(V

λ+

)∗]

g

)= Re

(−Q

f

(W−1

f

)[

0V

λ+

]g

).

Applying lemma 2.2 it follows that[ϕ−ϕ+

]= −(

W−1f

)[

0V

λ+g

]implies that ϕ− = α∂νv−, where (v−, v+) is the solution to (15a)–(15c). Finally,

F ′[f ]∗L2g = Re

(Q

f

[ϕ−ϕ+

])= Re(∂νu− ϕ−) = Re(α∂νu− ∂νv−).

The result for more general scr follows from a simple modification of this argument.

60 T Hohage et al

3. Uniqueness of the inverse problem

In this section we study the question if the data u+(·, x0)|obs for all x0 ∈ src are sufficientto determine f . There is an obvious potential obstruction for uniqueness: if there exists someopen subset ′ ⊂ such that the heat flux ∂νu−(x, x0) vanishes for all x ∈ ′ and all sourcepoints x0 ∈ src, then the solution u± is not affected by arbitrary changes of f on ′. Hence,f is not uniquely determined by the data. The following theorem asserts that up to thisobvious potential source of non-uniqueness f is uniquely determined by the data if obs hasnon-empty interior.

Theorem 3.1. Assume that int(obs) = ∅, let f (1), f (2) > 0 be two corrosion factors, anddenote the corresponding solutions to (5a)–(5c) by u

(1)± (·, x0) and u

(2)± (·, x0) for all x0 ∈ src.

Suppose that u(1)+ |obs×src ≡ u

(2)+ |obs×src . Then

f (1) = f (2) on := x ∈ : ∃x0 ∈ src ∂νu(1)− (x, x0) = 0. (16)

Proof. Let x0 ∈ src and write u(1,2)± = u

(1,2)± (·, x0) for simplicity. Since u

(1)+

∣∣obs

= u(2)+

∣∣obs

and ∂νu(1)+

∣∣obs

= 0 = ∂νu(2)+

∣∣obs

and since int(obs) = ∅, it follows from the Cauchy–

Kovalevskaya theorem that u(1)+ = u

(2)+ in +. From the transmission condition (5c) we obtain

∂νu(1)− = ∂νu

(2)− on . Since λ2

− ∈ iR is not a Neumann eigenvalue of the Laplace operator in

−, it follows that u(1)− = u

(2)− in −. Now the transmission condition (5b) implies that

f (1)∂νu(1)− = f (2)∂νu

(2)− on .

Finally, since also ∂νu(1)− (·, x0) = ∂νu

(2)− (·, x0) for all x0 ∈ src, we obtain the

conclusion (16).

The question that arises now is if it is possible that the heat flux ∂νu−(·, x0) vanishesidentically on some open subset of ′ for all x0 ∈ src. We have not been able to exclude thispossibility if src contains just one point, but theorem 3.4 below asserts that this cannot happenif src has non-empty interior. To prove this we use point source techniques as studied in[16, 19, 28]. As a preparation we need the following lemma.

Lemma 3.2. Assume that int(src) = ∅, and let D ⊂ Rd− be a smooth, bounded domain with

finite distance to . Then spanλ+(·, x0) | ∂D : x0 ∈ src is complete in H 1/2(∂D).

Proof. Let ϕ ∈ H−1/2(∂D) satisfy∫∂D

ϕ(y)λ+(y, x0) dγy = 0 for all x0 ∈ src.

Then the modified single layer potential

v(x) :=∫

∂D

(λ+(y, x) + λ+(y, x))ϕ(y) dγy, x ∈ Rd−\∂D

with y defined after (7b) satisfies v + λ2+v = 0 in R

d−\∂D, ∂νv|src = 0, and v|src = 0 since

λ+(y, x0) = λ+(y, x0) for x0 ∈ src. As int(src) = ∅ the Cauchy–Kovalevskaya theoremimplies that v = 0 in R

d−\D. Therefore, v−|∂D = v+|∂D = 0. Since λ2

+ is not a Dirichleteigenvalue of − in D, it follows that v = 0 in D. By the jump relations, we finally obtainϕ = ∂νv−|∂D − ∂νv+|∂D = 0.

In the following we will make free use of supremum (or essential supremum) norms on aset O, denoted as ‖·‖∞,O. For the norm of the Holder space Ck,β(O) we will write ‖·‖k,β,O.

Detecting corrosion using thermal measurements 61

Lemma 3.3. For x ∈ + we define the Green function for the exterior Dirichlet problem as

G(·, x) ∈ λ+(·, x) + H 2(+),(− − λ2+

)G(·, x) = δx,

G(·, x) = 0, on ,

∂νG(·, x) = 0, on .

If xn ⊂ + satisfies xn → x∗ ∈ , then for any ′ ⊂ surrounding x∗, it holds

‖∂νG(·, xn)‖∞,′ → ∞.

Proof. For g ∈ C(), the solution to

v + λ2+v = 0, in +,

v = g, on ,

∂νv = 0, on ,

can be written in integral form as

v(x) =∫

∂ν(y)G(y, x)g(y) dγy, x ∈ +. (17)

Assume that there exists a subsequence (also denoted xn), ′ as in the statement and C0 > 0such that for all n

‖∂νG(·, xn)‖∞,′ C0. (18)

We take g ∈ C() such that

g(x∗) = 1, supp g ⊂ ′,∫

supp g

dγy 1/(2C0). (19)

Since v is continuous, by (17)–(19) we deduce now that

1 = |g(x∗)| = limn

|v(xn)| 1/2.

Therefore, condition (18) cannot occur, and the statement of the lemma holds true.

Theorem 3.4. In addition to the assumptions of theorem 3.1 suppose that int(src) = ∅. Then = , i.e., f (1) = f (2) everywhere on .

Proof. By virtue of theorem 3.1, it suffices to show that the assumption that there exists anon-empty open subset ′ ⊂ such that ∂νu−(·, x0)|′ vanishes for all x0 ∈ src leads to acontradiction. This will be done in four steps.

Step 1. We first show that

∂νu−(·, x)|′ = 0 for all x ∈ +. (20)

To prove this for some fixed x ∈ +, we choose a smooth, bounded domain D ⊂ Rd−

such that − ⊂ D and x /∈ D. By lemma 3.2 there exists a sequence u(n)hom in

spanλ+(·, x0)|∂D : x0 ∈ src such that u(n)hom → uhom(·, x) in H 1/2(∂D) as n → ∞.

Moreover, by [9, theorem 5.4] the sequenceu

(n)hom

and all its derivatives converge uniformly

to uhom(·, x) on compact subsets of D, in particular on . It follows from our assumptionthat ∂νu

(n)− |′ = 0 for all n. Hence, (20) follows from the well- posedness of the transmission

problem (5a)–(5c).

Step 2. Choose x∗ ∈ ′ and define xn := x∗ + 1nν(x∗) for n ∈ N. Then xn ∈ + for n

sufficiently large. Define u(n)± := u±(·, xn) (this is a different sequence than that in step 1!).

62 T Hohage et al

Using [19, lemma 4.2] and the system of integral equations (8) in spaces with the weightedsupremum norm of [19] it follows that

‖∂νu(n)− ‖∞, = ‖∂νu

(n)− ‖∞,\′ C

with a generic constant C independent of n. From the well-posedness of the interior Neumannproblem for + λ2

− with respect to the supremum norm it follows that

‖u(n)− ‖∞,− C. (21)

Step 3. Let ′′ ⊂ such that dist(′′, \′) > 0, and let ′− ⊂ − be a subdomain

with ∂′− ∩ = ′′. Then by classical Schauder estimates in Holder spaces Ck,β (see [1,

theorem 7.3]) we have

‖u(n)− ‖

k,β,′− C(‖( + λ2

−)u(n)− ‖k−2,β,− + ‖∂νu

(n)− ‖k−1,β,′′ + ‖u(n)

− ‖∞,−) C ′

for any k = 2, 3, . . . , and β ∈ (0, 1) using (5a), (20) and (21). In particular,

‖u(n)− ‖k,β,′′ C. (22)

Step 4. Let now vn ∈ H 2(+) be the solution to

vn + λ2+vn = 0, in +,

vn = λ+(·, xn) + λ+(·, xn), on ,

∂νvn = 0, on ,

i.e., vn = λ+(·, xn) + λ+(·, xn) − G(·, xn), where G is the Green function of lemma 3.3. Letfinally wn := vn + u

(n)+ . Since ∂νu

(n)− = 0 on ′, the transmission condition (5b) implies that

wn = u(n)+ + λ+(·, xn) + λ+(·, xn) = u

(n)− + λ+(·, xn), on ′.

Thus, by (22),

‖wn‖k,β,′′ C. (23)

On the other hand, by [19, lemma 4.2], we can prove that∥∥u

(n)+

∥∥∞,\′′ C and therefore

‖wn‖∞, C. (24)

Since

wn + λ2+wn = 0, in +, ∂νwn = 0, on ,

the well-posedness of the exterior Dirichlet problem and (24) prove that

‖wn‖∞,+ C.

This bound and (23) imply (using the same Schauder estimates as before, now in the exteriordomain) that for ′′′ ⊂ ′′ such that dist(′′′, \′′) > 0,

‖∂νwn‖∞,′′′ C. (25)

By the transmission condition and the definitions of vn and wn it follows that

∂νwn = α∂νu(n)− + ∂ν

λ+(·, xn) − ∂νG(·, xn).

Finally, since ∂νu(n)− = 0 on ′′′ ⊂ ′, bound (25) implies that ‖∂νG(·, xn)‖∞,′′′ C, which

is not possible by lemma 3.3.

Theorem 3.1 above corresponds to a similar result for the problem of coated dielectrics[8, theorem 4.2]. However, we go further to prove uniqueness in theorem 3.4 assuming moresource points are employed, and we will extend these results to the transient case in section 6.

Detecting corrosion using thermal measurements 63

4. A non-iterative method

Based on the proof of theorem 3.1 we propose a constructive method for the approximatesolution of the inverse problem. A related approach to a different problem has been discussedin [2]. We limit the exposition to an open set of source points. Obvious modifications applyfor the discrete case. In the following let uδ be the observed data u+|obs×src plus measurementerrors of size δ. Our method consists of three steps. We start by representing the scatteredfield in + as a symmetrized single layer potential u+ = Sλ+ψ+. Since the linear first kindintegral equation (Sλ+ψ+)|obs = uδ is severely ill-posed, we reconstruct ψ+ approximately byTikhonov regularization in the first step:

ψδ+ := argminψ+∈H 1()

(‖Sλ+ψ+ − uδ‖2L2(obs×src)

+ β‖ψ+‖2L2(src;H 1())

). (26)

Here the regularization parameter β > 0 is chosen by Morozov’s discrepancy principle, i.e.as the largest positive number such that ‖Sλ+ψ+ − uδ‖ τδ with τ = 1.1. It is well knownfrom regularization theory (see e.g. [11]) that the worst-case error of ψδ

+ = ψδ+[uδ] tends to 0

with the noise level:

sup‖uδ−u+‖δ

∥∥ψδ+[uδ] − ψ+

∥∥L2(src;H 1())

δ→0−→ 0. (27)

In two space dimensions this implies

sup‖uδ−u+‖δ

∥∥ψδ+[uδ] − ψ+

∥∥L2(src;L∞())

δ→0−→ 0 (28)

by Sobolev’s embedding theorem. For d = 3 we would get the same result by replacing H 1

by a higher Sobolev norm.In the next step, we represent the total field u− in − as a single layer potential

u− = Sλ−ψ− with density ψ− ∈ C(), which leads to the equation of the second kind

α(

12I + J λ−

)ψδ

− = (− 12I + J λ+

)ψδ

+ + ∂νuhom, (29)

by virtue of the transmission condition (5c). Since this is a well-posed problem with respectto the L∞-norm, it follows from (28) that

sup‖uδ−u+‖δ

‖ψδ− − ψ−‖L2(src;L∞())

δ→0−→ 0. (30)

Finally, the corrosion function can be computed from the approximate densities ψδ+ and ψδ

−by solving (5b) for f as a least-squares problem:

f δ = argminf ∈L2()

∫src

∫

|∂νuδ−(x, x0)f (x) − rδ(x, x0)|2 dγx dγx0 . (31)

Here ∂νuδ− := (

12I + J λ−

)ψδ

− and rδ := −V λ−ψδ− + V λ+ψδ

+ + uhom. The normal equation ofthis least-squares problem can be solved explicitly,

f δ(x) = Re∫src

∂νuδ−(x, x0)r

δ(x, x0) dγx0∫src

|∂νuδ−(x, x0)|2 dγx0

, x ∈ (32)

provided the denominator vanishes nowhere. Assume that

infx∈

∫src

|∂νu−(x, x0)|2 dγx0 > 0. (33)

Sufficient conditions are given in theorem 3.4. It follows from (28) and (30) that‖∂νu

δ− − ∂νu−‖L2(src;L∞()) and ‖rδ − r‖L2(src;L∞()) tend to 0 as δ → 0. Hence, we obtain

the following result.

64 T Hohage et al

−0.5 0 0.5

−1

−0.5

0

Ω− Ω+

ΠΓ

Figure 1. Geometry of the problem. The observation points are uniformly distributed in theinterval between the stars. Either one point source at ‘+’ is used (figures 2 and 3 left) or tenequidistant point sources between the ‘×’ (figure 3 right).

Theorem 4.1. Suppose the exact flux ∂νu− satisfies the condition (33). Then theapproximations f δ defined by the three-step method (26), (29), (31) converge to the exactcorrosion function f as δ → 0:

sup‖uδ−u+‖δ

‖f δ − f ‖L∞()δ→0−→ 0.

As a test example we choose the boundary parametrized by

x(s) := (18 r(s) cos(2πs), 1

6 r(s) sin(2πs) − 25

), s ∈ [0, 1],

where r(s) := cos(2π(s + 1/6)) + sin(4π(s + 1/6)) + 3 (see figure 1). We use one point sourceof the form (10) at x0 = (0.07, 0) and n = 1000 observation points. Our data vector uδ isgiven by

uδj = u+(xj ) + εj , j = 1, . . . , n,

with xj ∈ obs, where εj describe measurement errors. We generated the ‘exact’ data u+(xj )

using the mixed approach with trigonometric polynomials described in section 2.1 and addedi.i.d. Gaussian random variables such that the relative discrete error was 0.1% or 0.5%.

The results shown in figure 2(a) and (b) illustrate the severe ill-posedness of the firstreconstruction step (26): even though the measured data are fitted very accurately, we havea large error in the reconstruction ψδ

+. The solution ψδ− computed in the second step (29)

is shown in figure 2(c). We represent |∂νu−| and the reconstruction |∂νuδ| in figure 2(d).

Since |∂νu−| is very small on the lower part of the boundary , large error amplificationoccurs in (32) as shown in figure 2(e). Although the last step is well-posed in the limit, itis very ill-conditioned in our examples. Therefore, we obtained considerably better resultsusing Tikhonov regularization with an H 1-penalty in the last step (see figure 2(f)). Note thatindependently of the noise level, the left part of the reconstruction (s ∈ [0, 1/2]) correspondingto the top part of the boundary is more accurate, i.e. the closer the obstacle is to the surface, themore information we can recover. Even so, the accuracy of the results is not quite satisfactorydespite the large number of observation points.

Detecting corrosion using thermal measurements 65

0 0.5 10

1

2

3

(b) step 1: |ψ+|

exactapprox

0 0.5 10

1

2

3

(c) step 2: |ψ− |

exactapprox

−0.5 0 0.50

0.02

0.04

0.06

0.08

0.1

(a) step 1: |u+|

exactapprox

0 0.5 10.4

0.6

0.8

1

1.2

(f) step 3: f using Tikhonov

exact0.1% noise0.5% noise

0 0.5 10

0.5

1

(d) step 2: |∂ u−|

exactapprox

0 0.5 10.4

0.6

0.8

1

1.2

(e) step 3: f using (32)

exact0.1% noise

Figure 2. Illustration of the non-iterative method for the reconstruction of the corrosionfunction f x. Noise level is 0.1% (except for the bottom right panel). f (x1, x2) =exp(−(x1 + 0.2)(x2 + 1)), κ+ = 2, κ− = 1, ω = 4 (hence λ− = 1 + i, λ+ = √

2(1 + i)).

5. An iterative method

In this section we consider the inverse problem as the nonlinear operator equation

F [f ] = uδ, (34)

where uδ denotes noisy data with known noise level δ:

‖uδ − u+‖L2(obs×src) δ.

Since we need a Hilbert space structure, we will consider D(F) as a subspace of Hs()

with s > (d − 1)/2. As C() is continuously embedded in Hs() for such s, it followsfrom proposition 2.1 that F is Frechet differentiable with respect to these norms. Moreover,the adjoint of F ′[f ] is given by F ′[f ]∗ = j ∗F ′[f ]∗

L2 , where F ′[f ]∗L2 is the L2-adjoint

characterized in proposition 2.3 and j : Hs() → L2() is the embedding operator.To solve equation (34), we use the iteratively regularized Gauß–Newton method (see

[5, 18] and references therein). It is described by the iteration formula

f δn+1 = f δ

n − (F ′[f δ

n

]∗F

[f δ

n

]+ αnI

)−1(F ′[f δ

n

]∗(F

[f δ

n

] − uδ)

+ αn

(f δ

n − f0))

for a given initial guess f0 ∈ Hs(). Equivalently, hn := f δn+1 − f δ

n is the unique minimumof the Tikhonov functional

h → ∥∥F ′[f δn

]h + F

[f δ

n

] − uδ∥∥2

L2(obs×src)+ αn

∥∥h + f δn − f0

∥∥2Hs()

, h ∈ Hs().

The regularization parameters were chosen of the form αn = 2−nα0 with α0 = 0.001.The iteration was stopped according to the discrepancy principle, i.e. at the first iterate

66 T Hohage et al

0 0.2 0.4 0.6 0.8 10.5

0.6

0.7

0.8

0.9

1

1.1

1.2

One incident wave

initial guess5% noise, 17 iter0.5% noise, 23 iter0% noise, 60 iterexact

0 0.2 0.4 0.6 0.8 10.5

0.6

0.7

0.8

0.9

1

1.1

1.2

Ten incident waves

initial guess5% noise, 15 iter0.5% noise, 21 iter0% noise, 41 iterexact

Figure 3. Reconstructions of f x with 100 equidistant observation points (see figure 1). Theparameters are the same as in figure 2.

N = N(uδ, δ) for which∥∥F[f δ

N

] − uδ∥∥ τδ

with τ = 1.1.Although there exists a convergence theory for the IRGNM (see [5, 18]), one either has to

impose conditions on the degree of nonlinearity of the operator F, which we could not verifyfor our problem, or one has to assume that f − f0 belongs to the range of F ′[f ]∗ which is anextremely restrictive condition for an exponentially ill-posed problem.

In our numerical implementation we use the Sobolev index s = 1 and choose f x as atrigonometric polynomial of degree M of the form

f x ≈ a0 +M∑

m=1

am cos(2πmt) +M∑

m=1

bm sin(2πmt), (35)

with coefficients am, bm ∈ R. This has the advantage that the H 1-norm of f x can easily beexpressed in terms of the Fourier coefficients. M was chosen large enough so that increasingM had no visible effect for the given noise level (or the given number of Newton steps inthe case of noise free data). The Tikhonov minimization problem can either be solved by aQR-decomposition or with the help of the adjoint F ′[fn]∗ by the conjugate gradient method.A preconditioning technique for exponentially ill-posed problems has been discussed in [14].

Figure 3 shows the reconstructions with no noise, with 0.5% and 5% noise (generated asdescribed in section 4), giving the corresponding number of iterations of the IRGNM whenchoosing M = 20 in (35). Without noise the reconstructed corrosion function f can hardlybe distinguished from the exact corrosion function in both cases. The use of ten instead of oneincident wave improves the quality of the reconstructions only a bit.

The results show that even for 100 observation points and higher noise levels, this methodprovides more accurate results than the non-iterative method in section 4. Note that in theilluminated part of the domain, the results for 5% noise are comparable to those in the non-iterative method for 0.1% noise (see figure 2(f)), being considerably better in the shadowregion. The price to pay for the iterative method is a higher computation time.

Detecting corrosion using thermal measurements 67

6. Time-dependent problem

In this section we deal with the general time-dependent diffusion problem (1)–(4). As in thetime-harmonic case, our aim is to reconstruct the corrosion function from measurements ofthe temperature at the top barrier at one or several different times. Now, a point source locatedat x0 ∈ is given by

Uhom(x, t) = t−d/2 exp(−|x − x0|2/(4κ+t)), x ∈ Rd , t > 0. (36)

6.1. Numerical solution to the forward problem

We use the approach proposed and analysed in [15], which is based on the computation of theLaplace transform of the solution with respect to time

u(x, s) =∫ ∞

0e−stU(x, t) dt, ∀x ∈ R

d−, Re s > 0. (37)

Then, one can recover the time-dependent solution by the inversion formula

U(x, t) = 1

2π i

∫C

estu(x, s) ds, t > 0, (38)

C being any path connecting −i∞ and i∞. Moreover, it is shown that for s ∈ C\(−∞, 0) thefunction u := u(·, s) is the unique solution to

u ∈ H, Lsu = 0, (39a)

u− + f ∂νu− − u+ = uhom, on , (39b)

α∂νu− − ∂νu+ = ∂νuhom, on , (39c)

where uhom(x) := ∫ ∞0 e−stUhom(x, t) dt and

Lsu :=u− − (s/κ−)u−, in −,

u+ − (s/κ+)u+, in +.

Note that this problem is nothing but (5a)–(5c) for wave numbers λ2∓(s) = −s/κ∓. Both the

indirect and the mixed formulations given in section 2.1 as well as their discretizations can beapplied for s ∈ C\(−∞, 0].

To invert the Laplace transform (38) we consider the hyperbolic path proposed in [21]

γ (θ) = µ(1 − sin(β + iθ)), θ ∈ R, (40)

where µ > 0 and 0 < β < π/2 are parameters that can be chosen to obtain an optimalperformance of the method in the desired time interval. Then, U(x, t) is approximated by atruncated trapezoidal rule

U(x, t) ≈m∑

j=−m

ωj etsj u(x, sj ) (41)

with nodes and weights

sj := γ

(log(m)

mj

), ωj := log(m)

2π imγ ′

(log(m)

mj

).

68 T Hohage et al

−1 −0.5 0 0.5 1

−1

−0.5

0

Ω−

Ω+

Π

Γ x

0

Figure 4. Geometry of the problem. Observation points are uniformly distributed in the intervalbetween the stars. The point source is located at x0.

This methods leads to 2m + 1 stationary problems of the form (39a)–(39c) to computeu(x, sj ). Due to the symmetry with respect to the real axis only m + 1 problems have to besolved.

Using trigonometric polynomials for the BEM approximation, this method enjoyssuperalgebraic order of convergence both with respect to time and space discretizationparameters for infinitely smooth boundaries and corrosion functions.

After the time discretization (41) the computation of the Frechet derivative of the time-dependent forward solution operator and its adjoint reduces to Frechet derivatives of time-harmonic solution operators at the quadrature points as characterized in propositions 2.1and 2.3.

6.2. Uniqueness

We obtain the following corollary to theorems 3.1 and 3.4.

Corollary 6.1. Let obs,src ⊂ with int(obs) = ∅ and 0 < t0 < T . Let f (1), f (2) > 0 betwo corrosion factors, and denote the corresponding solutions to (1)–(4) by U

(1)± (x, x0, t)

and U(2)± (x, x0, t) for x0 ∈ src. Suppose that U

(1)+ (x, x0, t) = U

(2)+ (x, x0, t) for all

x ∈ obs, x0 ∈ src and t ∈ [t0, T ]. Then

f (1) = f (2) on := x ∈ : ∃(x0, t) ∈ src × [t0, T ], ∂νU(1)− (x, x0, t) = 0. (42)

If additionally int(src) = ∅, then = , i.e., f (1) ≡ f (2) on .

Proof. Let x ∈ obs and x0 ∈ src. It follows from (38) that U(1,2)+ (x, x0, ·) has a

holomorphic extension from [t0, T ] to t ∈ C : arg(t) < θ, θ < π/2. Therefore,U

(1)+ (x, x0, t) = U

(2)+ (x, x0, t) for all t > 0. Let u

(1,2)± be the Laplace transforms of U

(1,2)± as

defined in (37). Then u(1,2)± (·, x0, s) satisfy the conditions of theorem 3.1 for all s ∈ C with

Re s > 0, in particular u(1)± (x, x0, s) = u

(2)± (x, x0, s) for all x ∈ obs, Re s > 0 and x0 ∈ src.

Moreover, if for a given x0 ∈ src and x ∈ there exists t > 0 such that ∂νU(1)− (x, x0, t) = 0,

then by the injectivity of the Laplace transform ∂νu(1)− (x, x0, s) = 0 for some s ∈ C with

Re s > 0. Hence, theorem 3.1 yields (42). Analogously, the second assertion follows fromtheorem 3.4.

Detecting corrosion using thermal measurements 69

0 0.2 0.4 0.6 0.8 10.3

0.4

0.5

0.6

0.7

0.8

0.9

1

500 observation points, t=0.1

initial guess5% noise, 14 iter0.5% noise, 18 iter0% noise, 33 iterexact

0 0.2 0.4 0.6 0.8 10.3

0.4

0.5

0.6

0.7

0.8

0.9

1

50 obs. points, 10 values of t in [0.05,0.15]

initial guess5% noise, 12 iter0.5% noise, 17 iter0% noise, 31 iterexact

Figure 5. Reconstructions of f x for t = 0.1 (left) and t ∈ [0.05, 0.15] (right). f (x, y) =log(3x2 + 3(y + 0.75)2 + 1), κ+ = 2, κ− = 1, α = 1/2.

We do not know if the data U+(x, x0, t) for one x ∈ , one x0 ∈ and all t > 0 aresufficient to determine f uniquely even on part of the boundary . Our numerical experiments(see figure 8) suggest that if this inverse problem is uniquely solvable, it is extremely unstable.

6.3. Reconstruction of the corrosion function

We illustrate the performance of the IRGNM for time-dependent data at a test problem withthe geometry shown in figure 4. The boundary is parametrized by

x(s) := (r(s) cos(2πs),−0.5 + r(s) sin(2πs)), s ∈ [0, 1],

with r(s) := 1/2 + sin(2πs)/(4 cos(2πs) − 5). For all our experiments we use one pointsource

Uhom(x, t) := t−1 exp(−|x − x0|2/(4κ+t))

located at x0 := (0.042, 0). For the space discretization we use the space of (complex)trigonometric polynomials of dimension 40 and take M = 20 for the approximation of f

in (35). Time discretization is carried out by taking m = 25 in (41). We choose β = π/4in (40), and the value of µ is adapted to the time interval in each example according to µ ≈ 5/t

as suggested in [15].Numerical reconstructions of f x for different noise levels are shown in figures 5

and 6 for different time intervals, also giving the number of iterations required by theIRGNM. Whereas for the time interval centred at t = 0.1 (figure 5) there is a significantdifference in the quality of the reconstructions between the illuminated and the shadowpart of the inclusion, we find almost no difference for the time interval centred at t = 1(figure 6). Moreover, the overall quality of the reconstructions is much better than in figure 5.To illustrate this behaviour we have plotted the incident field Uhom(·, t)| at t = 0.1 and t = 1in figure 7. Uhom(·, 0.1)| is localized over the obstacle, and the scattered field U+ seemsto contain very little information on the values of f on the bottom part of , whereas the

70 T Hohage et al

0 0.2 0.4 0.6 0.8 10.3

0.4

0.5

0.6

0.7

0.8

0.9

1

500 observation points, t=1

initial guess5% noise, 18 iter0.5% noise, 25 iter0% noise, 39 iterexact

0 0.2 0.4 0.6 0.8 10.3

0.4

0.5

0.6

0.7

0.8

0.9

1

50 obs. points, 10 values of t in [0.5,1.5]

initial guess5% noise, 15 iter0.5% noise, 20 iter0% noise, 30 iterexact

Figure 6. Reconstructions of f x for t = 1 (left) and t ∈ [0.5, 1.5] (right). Same parameters asin figure 5.

5 4 3 2 1 0 1 2 3 4 50

2

4

6

8

10 uhom

( x,0.1)u

hom( x,1)

−5 −4 −3 −2 −1 0 1 2 3 4 5−1.5

−1

−0.5

0

− − − − −

Figure 7. Incident field at t = 0.1 and t = 1.

width of the pulse Uhom(·, 1)| is much larger than the diameter of , and U+ seems to carryinformation on the values of f both on the top and bottom part of .

Note that the problem is formally overdetermined if we can measure U+(x, t) for bothx and t in some interval since our data are functions of two variables whereas the unknownf x is a function of one variable. We have also considered the case that either x or t isfixed. In both cases we do not have a uniqueness result for the inverse problem. The resultsin the left panels of figures 5 and 6 show that one can still obtain reasonable reconstructionsfor fixed t, but the quality is much worse than for the same total number of observations atdifferent times. On the other hand, numerical experiments show that from observations of thetemperature U+(x, ·) at one single point x for many different times the reconstructions of f

are very poor. In figure 8 we present a numerical illustration of this fact. We see that for thesame total number of observations with three or four carefully chosen observation points wecan obtain reasonable reconstructions.

Detecting corrosion using thermal measurements 71

0 0.2 0.4 0.6 0.8 10.3

0.4

0.5

0.6

0.7

0.8

0.9

1 initial guess(d) 20 iter(c) 19 iter(b) 19 iter(a) 21 iterexact

Figure 8. Reconstructions with few observation points (0.5% noise): (a) 100 values of t in[0.5 1.5], observation point at 0.2 (b) 50 values of t in [0.5 1.5], observation points at −0.2 and 0.2(c) 33 values of t in [0.5 1.5], observation points at −0.2, 0.2 and 0.4 (d) 25 values of t in [0.5 1.5],observation points at −0.4, −0.2, 0.2 and 0.4.

7. Conclusions

We studied the detection of the level of corrosion f at the interface of an inclusion fromtemperature measurements on the surface of the sample both for time-harmonic and fortime-dependent data. In both cases we proved uniqueness results. Moreover, we discussedthree methods for the numerical reconstruction of f , a non-iterative and an iterative methodfor time-harmonic data and an iterative method for time-dependent data. For the non-iterativemethod we derived a convergence result.

The non-iterative method is much faster, easier to implement, and theoretically moresatisfactory. However, the quality of the results was considerably worse than for the iterativemethod. The iterative method gave better results for 5% noise than the non-iterative methodfor 0.1% noise. Therefore, from a practical point of view the non-iterative method can onlybe recommended to provide an initial guess for the iterative method or if computation time isa limiting factor as in online computations.

The numerical results in section 6 show that much more accurate reconstructions ofthe corrosion function f can be obtained for a delta-pulse excitation than for time-harmonicexcitations if the temperature on the surface can be observed over an interval of time. However,it is crucial to choose this time interval carefully depending on the size of the inclusion.Furthermore, time-dependent temperature measurements at a single observation point for oneincident field are not sufficient to obtain accurate reconstructions of f .

Acknowledgments

We would like to thank Rainer Kress for helpful discussions. The authors are partiallysupported by MEC/FEDER Project MTM2004-01905, Gobierno de Aragon (GrupoConsolidado PDIE), Gobierno de Navarra Ref. 18/2005 and DAAD scholarship Ref.A/06/12974.

72 T Hohage et al

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