Sampling methods for recovering buried corroded boundaries from partial electrostatic Cauchy data
Isaac Harris Andreas Kleefeld Heejin Lee






Sampling methods for recovering buried corroded boundaries from partial electrostatic Cauchy data






Isaac Harris and Heejin Lee

Department of Mathematics, Purdue University, West Lafayette, IN 47907 

Email:  and   


Andreas Kleefeld

Forschungszentrum Jülich GmbH, Jülich Supercomputing Centre,  

Wilhelm-Johnen-Straße, 52425 Jülich, Germany

University of Applied Sciences Aachen, Faculty of Medical Engineering and  

Technomathematics, Heinrich-Mußmann-Str. 1, 52428 Jülich, Germany

Email: 






§ ABSTRACT 
We consider the inverse shape and parameter problem for detecting corrosion from partial boundary measurements. This problem models the non-destructive testing for a partially buried object from electrostatic measurements on the accessible part of the boundary. The main novelty is the extension of the linear sampling and factorization methods to an electrostatic problem with partial measurements. These methods so far have only mainly applied to recovering interior defects which is a simpler problem. Another important aspect of this paper is in our numerics, where we derive a system of boundary integral equations to recover the mixed Green's function which is needed for our inversion. With this, we are able to analytically and numerically solve the inverse shape problem. For the inverse parameter problem, we prove uniqueness and Lipschitz-stability (in a finite dimensional function space) assuming that one has the associated Neumann-to-Dirichlet operator on the accessible part of the boundary. 





§ INTRODUCTION

In this paper, we consider an inverse shape and parameter problem coming from electrical impedance tomography (EIT). The model we study is for a partial buried object that was degraded via corrosion. This problem is motivated by non-destructive testing where one wishes to detect/recover the corroded part of the boundary without removing the object. To this end, we will study the linear sampling and factorization methods for recovering the corroded boundary. These methods were first introduced in <cit.>, respectively. This is novel due to the fact that we have data only on the accessible part of the boundary and we wish to recover the rest of the boundary. Our inversion is done by embedding the defective region into a `healthy' region and comparing the gap in voltages. The linear sampling and factorization methods have been studied for similar problems in <cit.> where one wishes to recover interior defects from either full or partial boundary data. Again, this problem is different in the fact that we have partial boundary data and we wish to recover unaccessible part of the boundary. 

In order to solve the inverse shape problem we will consider two well known qualitative reconstruction methods i.e. the linear sampling and factorization methods. These methods have been greatly studied over the years for different inverse shape problems, see <cit.> as well as the manuscripts <cit.> and the references therein. Iterative methods for this problem were studied in <cit.> which extends the method presented in <cit.>. In the aforementioned papers, a non-linear system of boundary integral equations was used to solve the inverse shape and parameter problems. We also note that in <cit.> the authors used a similar iterative method to solve the problem with a generalized impedance condition. One of the main advantageous for using a qualitative method is the fact that one needs little a priori information about the region of interest.  On the other hand iterative methods will often converge to a local minima rather than the global minima if the initial guess is not sufficiently close to the target. Therefore, in many non-destructive testing applications it may be useful to use a qualitative method. 

We also consider the inverse parameter problem. Here, we will assume that the corroded part of the boundary is known/recovered. Then, we prove that the knowledge of the Neumann-to-Dirichlet operator on the accessible part of the boundary can uniquely recover the corrosion(i.e. Robin) parameter. Once we have proven uniqueness, we then turn our attention to stability. Due to the fact that inverse EIT problem are exponentially ill-posed there is no hope to obtain a Lipschitz–stability estimate on standard function spaces. Here, we appeal to the techniques in <cit.> to prove Lipschitz–stability assuming the parameter is in a finite dimensional function space. This is useful for numerical reconstructions of the parameter since one will often discretize the unknown function to be a linear combination of finite basis functions. Numerical reconstructions of the parameter are not studied here but the algorithm in <cit.> can also be applied to this problem. 


The rest of the paper is structured as follows. In Section <ref> we setup the direct and inverse problem under consideration. Then, in Section <ref> we consider the inverse shape problem where we give the theoretical justification of the linear sampling and factorization methods for our model. This will give a computationally simple yet analytically rigorous method for recovering the corroded part of the boundary. Next, in Section <ref> we consider the inverse parameter problem assuming that the corroded boundary is known/recovered, where we prove uniqueness and stability with respect to the Neumann-to-Dirichlet operator. Then, we provided numerical examples in Section <ref> for recovering the corroded boundary. Finally, a summary and conclusion is given.




§ THE DIRECT PROBLEM

In this section, we will discuss the direct problem associated with the inverse problems under consideration. Again, this problem comes from EIT where one applies a current on the accessible part of the boundary and measures the resulting voltage. To begin, we let the known region D ⊂ℝ^m for m=2, 3 be an open bounded and simply connected domain with the piecewise C^1 boundary ∂ D. The boundary ∂ D can be decomposed into 

    ∂ D =  Γ_N∪Γ_D  where  Γ_N ∩Γ_D = ∅

and are relatively open subsets of ∂ D. We assume that part of the region D has been buried such that Γ_N is the accessible part of the boundary with Γ_D being the part of the boundary that has been buried. Being buried has caused part of the region to be corroded away. The part of the region that has corroded away will be denoted Ω. To this end, we let Ω⊂ D be an open subset of D such that a part of the boundary ∂Ω is Γ_D and the other part of the boundary is C^1 and denoted by Γ_C. Therefore, we have that 

    ∂ (D ∖Ω) = Γ_N∪Γ_C  where  Γ_N ∩Γ_C = ∅

and are relatively open. In Figure <ref>, we have illustrated the aforementioned setup. 




In order to determine if there is a non-trivial corroded region Ω we assume that a current denoted g is applied to the accessible part of the boundary Γ_N. This will produce an electrostatic potential function u for the defective material .  This gives that the direct problem can be modeled by the mixed Neumann–Robin boundary value problem: given g ∈ L^2(Γ_N), determine u ∈ H^1() such that

    Δ u= 0   in  D∖Ω,   u = g   on Γ_N,   and  u +γ u  = 0   on Γ_C.

Here ν denotes the outward unit normal to  and the corrosion coefficient γ∈ L^∞(Γ_C). We will assume that there are two real–valued constants γ_max and γ_min such that the corrosion coefficient satisfies 

    0< γ_min≤γ (x) ≤γ_max  for a.e.   x ∈Γ_C.
 
Note that our notation is that Γ_N is the Neumann boundary and that Γ_C corresponds to the corroded/Robin boundary. 


Now, we wish to establish the well-posedness of the direct problem (<ref>). This can be done by considering the equivalent variational formulation of (<ref>). To this end, we can take φ∈ H^1() and using Green's first identity we have that 

    ∫_∇ u ·∇φ x + ∫_Γ_Cγ u φ s
    =
    ∫_Γ_N g φ s.

Note that (<ref>) is satisfied for any test function φ∈ H^1(). Clearly this implies that the variational formulation is given by 

    A(u,φ) = ℓ (φ)
 
where the sesquilinear form A(· , ·): H^1() × H^1() →ℝ is given by

    A(u, φ) = 
    ∫_∇ u ·∇φ x + ∫_Γ_Cγ u φ s

and
the conjugate linear functional ℓ: H^1() →ℝ is given by

    ℓ(φ) = ∫_Γ_N g φ s.

Here, the integrals over Γ_C and Γ_N are interpreted as the inner–product on L^2(Γ_C) and L^2(Γ_N), respectively. These integrals are well defined by the Trace Theorem. For the well-posedness, notice that 

    A(u, u) ≥ u ^2_L^2() +γ_minu^2_L^2(Γ_C)

which implies that A(· , ·) is coercive on H^1() by appealing to a standard Poincaré type argument (see for e.g. <cit.> p. 487). From the Trace Theorem, we have that 

    |ℓ(φ)| ≤ Cg_L^2(Γ_N)φ_H^1().

With this we have the following result. 

The mixed Neumann–Robin boundary value problem (<ref>) has a unique solution u ∈ H^1() that satisfies the estimate

    u_H^1()≤ C g_L^2(Γ_N)

with C independent of g ∈ L^2(Γ_N).


With the well-poseness of (<ref>) established, we now consider an auxiliary boundary value problem for the electrostatic potential in the healthy domain D: given g ∈ L^2(Γ_N), determine u_0 ∈ H^1(D) such that 

    Δ u_0= 0   in  D,   u_0 = g   on Γ_N,   and   u_0 = 0  on Γ_D.

Similarly, it can be shown that the above boundary value problem (<ref>) is well-posed with the estimate 

    u_0_H^1(D)≤ C g_L^2(Γ_N).
 
This can be done by again appealing to the variational formulation as well as the fact that u_0 satisfies the Poincaré estimate 

    u_0_H^1(D)≤ C  u_0 _L^2(D)

due to the zero trace on Γ_D (see <cit.> p. 486). Note that in our notation Γ_D is the part of the boundary where we impose the homogeneous Dirichlet condition. Since D is known a priori we have that u_0 can always be computed numerically. 



In order to determine the corroded subregion Ω, we will assume that the u|_Γ_N can be measured and that u_0|_Γ_N can be computed for any current g∈ L^2(Γ_N).  Now we define the Neumann-to-Dirichlet (NtD) operators 

    Λ  and  Λ_0 : L^2(Γ_N) → L^2(Γ_N)   given by  Λ g = u|_Γ_N  and  Λ_0 g = u_0|_Γ_N,

where u and u_0 are the unique solutions to (<ref>) and (<ref>), respectively. From the well-posedness of the boundary value problems it is clear that the operators Λ and Λ_0 are well-defined bounded linear operators. The inverse problems that we are interested in are the inverse shape problem of determining the corroded region Ω from the knowledge of the difference (Λ-Λ_0) and the inverse impedance problem of recovering the corrosion coefficient γ provided that Γ_C is known. In the following section, we will study the linear sampling and factorization methods to recover Ω. Then, we will turn our attention to proving that Λ uniquely recovers the corrosion coefficient γ as well as provide a stability estimate. 






§ THE INVERSE SHAPE PROBLEM

In this section, we are interested in the inverse shape problem of recovering Ω from the knowledge of the NtD operators. In order to solve this problem we will consider the linear sampling and factorization methods associated with (Λ-Λ_0). Our analysis will show that the linear sampling method can give an approximate reconstruction of Ω where as the factorization method can be used under a stricter set of assumptions on the corrosion coefficient. The factorization method is mathematically more advantageous to use due to the fact that it gives an explicit characterization of the region of interest from the spectral decomposition of an operator defined by the difference of the NtD maps. In either case, we need to decompose the operator (Λ-Λ_0) to obtain a more explicit relationship with the unknown region Ω.  

To begin, we first derive an initial factorization of the operator (Λ-Λ_0). Therefore, we first notice that the difference of the electrostatic potentials satisfies 

    Δ (u-u_0)= 0   in ,   (u-u_0) = 0   on Γ_N,   and  (u- u_0)|_Γ_C∈ H^1/2(Γ_C).

This motivates us to consider the auxiliary boundary value problem: given φ∈ H^1/2(Γ_C), determine  w ∈ H^1() such that 

    Δ w= 0   in ,   w = 0   on  Γ_N,   and  w=φ  on  Γ_C.

Arguing similarly to the previous section, we have that (<ref>) is well-posed which implies that we can define

    G: H^1/2(Γ_C) → L^2(Γ_N)   given by    Gφ=w|_Γ_N,

where w is the solution to (<ref>) as a bounded linear operator by appealing to the Trace Theorem. By the well-posedness of (<ref>), we see that if 

    φ =  (u- u_0)|_Γ_C   we obtain that   w = (u- u_0)    in  .

Now, we further define the bounded linear operators 

    L   and   L_0: L^2(Γ_N) → H^1/2(Γ_C)   given by    Lg=u|_Γ_C  and   L_0 g=u_0|_Γ_C.

With this we have our initial factorization (Λ-Λ_0) = G(L-L_0). 

With our initial factorization in hand we will analyze the properties of the operators defined above. First, we notice that due to the compact embedding of H^1/2(Γ_N) into L^2(Γ_N) we have compactness of the operator G defined in (<ref>). We also notice that by Holmgren's theorem (see for e.g. <cit.>) if φ is in the null-space of G, this would imply that 

    w = w = 0   on  Γ_N    giving that    w = 0   in  .

By the Trace Theorem φ = 0 which gives injectivity of the operator G. With this we now present a result that gives the analytical properties of the source-to-trace operator G. 


The operator G: H^1/2(Γ_C) → L^2(Γ_N) as defined in (<ref>) is compact and injective.


With this, in order to further analyze the operator G we need to compute its adjoint. The adjoint operator G^* will be a mapping from L^2(Γ_N) into H^-1/2(Γ_C). Note that the adjoint is computed via the relationship

    (G φ , ψ)_L^2(Γ_N) = ⟨φ , G^*ψ⟩_Γ_C
 
where ⟨· , ·⟩_Γ_C is the sesquilinear dual–product between the 

    Hilbert Space H^±1/2(Γ_C) and its dual space H^∓ 1/2 (Γ_C )
 
where L^2  (Γ_C) is the associated Hilbert pivot space, see <cit.> p. 99 for details. The Sobolev space H^s (Γ_C ) is the closure of C^∞_0(Γ_C) with respect to the H^s (Γ_C )–norm for any s∈. Now, with this in mind we can give another result for the analytical properties of G. 

 
The adjoint G^*: L^2(Γ_N) →H^-1/2(Γ_C) is given by G^*ψ =  -  ∂_ν v |_Γ_C where v ∈ H^1 () satisfies 

    Δ v= 0   in ,   v = ψ  on  Γ_N,   and   v =0   on  Γ_C.
 
Moreover, the operator G has a dense range.


To prove the claim, we first note that (<ref>) is well-posed for any ψ∈ L^2(Γ_N). Now, in order to compute the adjoint operator we apply Green's second identity to obtain 

    0 =  ∫_Γ_Nv ∂_νw - w  ∂_νv ds + ∫_Γ_Cv ∂_νw - w  ∂_νv ds,

where we have used the fact that both w and v are harmonic in  as well as the fact that ∂ (D ∖Ω) = Γ_N∪Γ_C with Γ_N ∩Γ_C = ∅. Using the boundary conditions in (<ref>) and (<ref>) we have that 

    ∫_Γ_N w  ψ ds = -  ∫_Γ_Cφ ∂_νv ds.

Notice, that the left hand side of the above equality is a bounded linear functional of φ∈ H^1/2(Γ_C). Therefore, by definition we have that Gφ=w|_Γ_N which implies that 

    (G φ , ψ)_L^2(Γ_N) =∫_Γ_N w  ψ ds = -  ∫_Γ_Cφ ∂_νv ds = ⟨φ , G^*ψ⟩_Γ_C
 
proving that G^*ψ =  -  ∂_ν v |_Γ_C. 

Now, proving that the operator G has a dense range is equivalent to proving that the adjoint G^* is injective (see <cit.>, p. 46). So we assume that ψ is in the null-space of G^* which implies that 

    v = v = 0   on  Γ_C    giving that    v = 0   in

where we again appeal to Holmgren's Theorem proving the claim by the Trace Theorem. 



Now that we have analyzed the operator G we will turn our attention to studying (L-L_0). This is the other operator used in our initial factorization of difference of the NtD operators. Notice that the dependance on the unknown region Ω is more explicit for these operators since they map to the traces of function on Γ_N ⊊∂Ω. 



The operator (L-L_0):L^2(Γ_N) → H^1/2(Γ_C) as defined in (<ref>) is injective provided that γ_max is sufficiently small or γ_min is sufficiently large.


We begin by assuming g is in the null-space of (L-L_0) which implies that 

    Δ(u-u_0)= 0   in ,   (u-u_0) = 0   on  Γ_N,   and  (u- u_0) = 0   on  Γ_C.

It is clear that the above boundary value problem only admits the trivial solution. Therefore, we have that u=u_0 in  and hence ∂_νu_0+γ u_0=0 on Γ_C. Notice, that by (<ref>) we have that u_0∈ H^1(Ω) is the solution of the boundary value problem

    Δ u_0 = 0  in Ω  with   u_0 = 0   on  Γ_D,   u_0 + γ u_0 = 0  on Γ_C.
 Recall, that ν is the inward unit normal to Ω (see Figure <ref>). From Green's second identity applied to u_0 in Ω and the Trace Theorem, we have that

    0    = ∫_Ω |∇ u_0|^2  x - ∫_Γ_Cγ |u_0|^2  s 
       ≥∇ u_0^2_L^2(Ω) - γ_maxu_0^2_L^2(Γ_C)
       ≥∇ u_0^2_L^2(Ω) - γ_max C u_0^2_H^1(Ω),

since γ≤γ_max a.e. on Γ_C from our assumptions. Notice, that since u_0|_Γ_D=0 we have the  Poincaré estimate u_0_H^1(Ω)≤ C∇ u_0_L^2(Ω). Therefore,

    0 ≥ (1-γ_max C) ∇ u_0^2_L^2(Ω)

which implies that if γ_max is small enough, then |∇ u_0| = 0 in Ω and hence u_0 = 0 in Ω due to the zero trace on Γ_D. By the unique continuation principle (see for e.g. <cit.>, p. 276), we obtain that u_0 = 0 in D, which implies that g=0 by the Trace Theorem. The other case can be proven similarly by considering the opposite sign of the above equality.   


With this, we wish to prove that (L-L_0) is compact with a dense range just as we did for the operator G. Note that the compactness is not obvious as in the previous case and to prove the density of the range we need to compute the adjoint operator (L-L_0)^*. To this end, let us consider the solution p∈ H^1() to 

    Δ p = 0   in    with  p  = 0   on  Γ_N ,   p + γ p = ξ  on  Γ_C

and the solution q ∈ H^1(D) to 

    Δ q = 0  in  D∖Γ_C  with  q|_Γ_N = 0,    q|_Γ_D=0,  and   [[ q]] |_Γ_C =  ξ

for any ξ∈H^-1/2(Γ_C). Here, we define the notation 

    [[ q]]  |_Γ_C = (  ∂_νq^+ -  ∂_νq^-) |_Γ_C,
 
where + and - indicate the limit obtained by approaching the boundary Γ_C from  and Ω, respectively.  Note that since q∈ H^1(D) it has continuous trace across Γ_C. With this we can further analyze the operator (L-L_0 ). 


 
The adjoint (L-L_0)^*: H^-1/2(Γ_C) →  L^2(Γ_N) is given by 

    (L-L_0)^* ξ  = (p-q)|_Γ_N,

where p and q are the solution to (<ref>) and (<ref>), respectively. Moreover, the operator (L-L_0) is compact with a dense range provided that γ_max is sufficiently small or γ_min is sufficiently large.


To prove the claim, we first compute the adjoints L^* and L^*_0 separately. We begin with computing L^*. Just as in Theorem <ref> we use Green's second identity to obtain that 

    0 =  ∫_Γ_Np ∂_νu - u  ∂_νp ds + ∫_Γ_Cp ∂_νu - u  ∂_νp ds,

where we have used the fact that the functions are both harmonic. By the boundary conditions in (<ref>) and (<ref>) we have that 

    ( g , L^*ξ)_L^2(Γ_N) =∫_Γ_Npg ds =  ∫_Γ_Cu [ ∂_νp  +γp] ds = ∫_Γ_Cuξ s =⟨ L g , ξ⟩_Γ_C,
 
which gives L^*ξ  = p|_Γ_N. 

Now, for computing L^*_0 we proceed is a similar manner where we apply Green's second identity in Ω and  to obtain that

    0 = - ∫_Γ_Cu_0∂_νq^- - ∂_νu_0q  s   and   0= - ∫_Γ_N∂_νu_0 q  s + ∫_Γ_Cu_0∂_νq^+ -  ∂_νu_0 q  s,

where we have used that the functions are harmonic as well as ∂_ν q =0 on Γ_N and q = u_0=0 on Γ_D. By adding the above equations and using the boundary conditions in (<ref>) and (<ref>) we obtain that

    ⟨ξ, L_0 g ⟩_Γ_N= ∫_Γ_Cu_0ξ s = ∫_Γ_Ng q  s = (L_0^*ξ , g)_L^2(Γ_N),

which gives L_0^*ξ  = q|_Γ_N.

With this, it is clear that (L-L_0)^* is compact by the compact embedding of H^1/2(Γ_N) into L^2(Γ_N) which implies that (L-L_0) is compact. Now, let ξ be in the null-space of (L-L_0) which gives that 

    Δ(p-q)= 0   in ,    (p-q)=(p-p_0) = 0   on  Γ_N.

Therefore, by Holmgren's Theorem we have that p=q in . By the boundary conditions on Γ_C 

    ∂_νp  +γp = ξ = ∂_νq^+ -  ∂_νq^-  on  Γ_C

which implies that 

    Δ q = 0  in Ω  with   q = 0  on Γ_D,   q^- + γ q = 0  on Γ_C.

Here, we have used that  ∂_νp =∂_νq^+ and p=q on Γ_C.  Then, by arguing just as in Theorem <ref> we have that q=0 provided that γ_max is small enough or γ_min is sufficiently large, which gives that ξ=0. 






 §.§ The Linear Sampling Method

Now that we have the above results we can infer the analytical properties of the difference of the NtD operators (Λ-Λ_0 ). These properties of the operator are essential for applying the linear sampling method (LSM) for solving the inverse shape problem. This method has been used to solve many inverse shape problems (see for e.g. <cit.>). This method connects the unknown region to range of the data operator (Λ-Λ_0 ) via the solution to an ill-posed operator equation. To proceed, we will discuss the necessary analysis to show that the linear sampling method can be applied to this problem. From the analysis in the previous section, we have the following result for the difference of the NtD operators. 


The difference of the NtD operators (Λ-Λ_0 ): L^2(Γ_N)  →  L^2(Γ_N) given by (<ref>) has the factorization 

    (Λ-Λ_0) = G(L-L_0),
 
where G and (L-L_0) are defined in (<ref>) and (<ref>), respectively. 
Moreover, the operator (Λ-Λ_0 ) is compact and injective with a dense range provided that γ_max is sufficiently small or γ_min is sufficiently large. 

 
To proceed, we need to determine an associated function that depends on the sampling point z ∈ D to derive a `range test' to reconstruct the unknown subregion Ω. To this end, we define the mixed Green's function (also referred to as the Zaremba function <cit.>, p. 
B209): for any z ∈ D, let 𝔾(·, z) ∈ H_loc^1(D∖{z}) be the solution to

    -Δ𝔾(·, z) = δ(· -z)  in  D,    𝔾(·, z) = 0  on Γ_N,    and   𝔾(·, z) = 0  on  Γ_D.

The following result shows that the range of the operator G given by (<ref>) uniquely determines the region of interest Ω.

Let G: H^1/2(Γ_C) → L^2(Γ_N) as defined in (<ref>). Then,

    𝔾(·, z)|_Γ_N∈Range(G)   if and only if   z ∈Ω.



To prove the claim, we first start with the case when the sampling point z ∈Ω. With this we see that 𝔾(·, z) ∈ H^1() satisfies

    Δ𝔾(·, z) =0  in ,    𝔾(·, z) = 0  on Γ_N,    and   𝔾(·, z)|_Γ_C := φ_z ∈ H^1/2(Γ_C).

From this we obtain that Gφ_z = 𝔾(·, z)|_Γ_N proving this case.   

Now, we consider the case when z ∈ and we proceed by contradiction. To this end, we assume that there is a w_z ∈ H^1() such that 

    Δ w_z =0  in ,    w_z = 0  on Γ_N,    and    w_z = φ_z  on Γ_C

for some φ_z ∈ H^1/2(Γ_C) where w_z = 𝔾(·, z) on Γ_N. By appealing to Holmgren's Theorem we can obtain that w_z = 𝔾(·, z) in the set (  ) ∖{z}. Using interior elliptic regularity 
(see <cit.> p. 536) we have that  w_z is continuous at the sampling point z. By the singularity at z for the mixed Green's function 𝔾 ( · ,z) we have that  

    | w_z (x)| < ∞   whereas   | 𝔾 (x,z) | →∞  as   x → z.

This proves the claim by contradiction. 


With the result in Theorem <ref> we can prove that the linear sampling method can be used to recover Ω from the NtD mapping. This is useful in non-destructive testing since there is no initial guess needed for this algorithm. With this, we can now state the main result in this subsection. 

Let the difference of the NtD operators (Λ-Λ_0 ): L^2(Γ_N)  →  L^2(Γ_N) be given by (<ref>). Then for any sequence {g_z , }_ >0∈ L^2(Γ_N) for z ∈ D satisfying 

    (Λ - Λ_0) g_z ,  - 𝔾(·, z)|_Γ_N_ L^2(Γ_N)⟶ 0   as → 0

we have that g_z , _ L^2(Γ_N)⟶∞ as → 0 for all z ∉Ω provided that γ_max is sufficiently small or γ_min is sufficiently large.


To prove the claim, we first note that by Theorem <ref> we have that (Λ-Λ_0 ) has a dense range in L^2(Γ_N). Therefore, for all z ∈ D we have that there exists an approximating sequence {g_z , }_ >0 such that (Λ - Λ_0) g_z , converges in norm to 𝔾(·, z)|_Γ_N. For a contradiction, assume that there is such an approximating sequence such that g_z , _L^2(Γ_N) is bounded as → 0. Then we can assume that (up to a subsequence) it is weakly convergent such that g_z ,  g_z,0 as → 0. By the compactness of the operator (Λ - Λ_0) we have that as → 0

    (Λ - Λ_0) g_z , ⟶ (Λ - Λ_0) g_z,0  which implies that   (Λ - Λ_0) g_z , 0 = 𝔾(·, z)|_Γ_N.

By the factorization (Λ-Λ_0) = G(L-L_0) this would imply that 𝔾(·, z)|_Γ_N∈Range(G). This clearly contradicts Theorem <ref> proving the claim by contradiction.



Notice, we have shown that the linear sampling method can be used to recover Ω in Theorem <ref>. In order to use this result to recover the corroded part of the region we find an approximate solution to 

    (Λ - Λ_0) g_z = 𝔾(·, z)|_Γ_N.

Since, the operator (Λ - Λ_0) is compact this implies that the above equation is ill-posed. But the fact that (Λ - Λ_0) has a dense range means we can construct an approximate solution using a regularization strategy (see for e.g. <cit.>). Here, we can take >0 to be the regularization parameter then we can recover Ω by plotting the imaging functional 

    W_LSM(z) = 1/ g_z , _L^2(Γ_N )

 where g_z , is the regularized solution to (<ref>). Theorem <ref> implies that W_LSM(z) ≈ 0  for any z ∉Ω. Note that we can not infer that W_LSM(z) will be bounded below in z ∈Ω. Therefore, we will consider the factorization method in the proceeding section.




 §.§ The Factorization Method

In this section, we will consider using the factorization method (FM) to recover the corroded region Ω. Even though we have already studied the linear sampling method, we see that Theorem <ref> does not prove that the corresponding imaging functional is bounded below for z ∈Ω. With this in mind, we consider the factorization method since it gives an exact characterization of the region of interest Ω using the spectral decomposition of an operator associated to (Λ - Λ_0). 

To begin, we need to derive a `symmetric' factorization of the operator (Λ - Λ_0). Therefore, we recall that by Theorem <ref> we have that (Λ - Λ_0)=G(L - L_0). Now, we define the bounded linear operator 

    T: H^-1/2(Γ_C) → H^1/2(Γ_C)   given by   Tξ = (p-q)|_Γ_C.

where p and q satisfy (<ref>) and (<ref>), respectively. It is clear that the boundedness of T follows from the well-posedness of (<ref>) and (<ref>) along with the Trace Theorem. With this, we notice that 

    GTξ = G((p-q)|_Γ_C) = (p-q)|_Γ_N = (L-L_0)^*ξ

where we have used the fact that (p-q) ∈ H^1() satisfies

    Δ (p-q)= 0   in ,   (p-q) = 0   on  Γ_N,   and  (p-q)|_Γ_C∈ H^1/2(Γ_C)

along with the definition of G in (<ref>) and (L-L_0)^* given in Theorem <ref>. Since this is true for any ξ∈H^-1/2(Γ_C) we have that GT = (L-L_0)^*. We now have the desired factorization of (Λ - Λ_0)=G T^* G^* by the calculation that (L-L_0) = T^* G^*. 


With this new factorization acquired we now prove that (Λ - Λ_0)  is self-adjoint. This would imply that (Λ - Λ_0)=G T G^* where G and T are defined in (<ref>) and (<ref>), respectively. To this end, we notice that  

    (g_1, (Λ - Λ_0)g_2)_L^2(Γ_N) = ∫_Γ_N g_1 [u^(2)-u^(2)_0] s =  ∫_Γ_Nu^(2)u^(1)  - u^(2)_0u^(1)_0 s,

where the superscript (j) corresponds to the solution of (<ref>) and (<ref>) for g_j ∈ L^2(Γ_N) with j=1,2. We now apply Green's first identity to u^(j) and u^(j)_0 in  and D, respectively to obtain

    (g_1, (Λ - Λ_0)g_2)_L^2(Γ_N)=   ∫_∇ u^(1)·∇ u^(2) x + ∫_Γ_Cγ u^(1) u^(2) s - ∫_D  u^(1)_0 · u^(2)_0  x,

where we have used the boundary conditions. This implies that (Λ-Λ_0)=G T G^* is self-adjoint. 



In order to apply the theory of the factorization method <cit.> we need to study the operator T defined in (<ref>). In particular, we wish to show that under some assumptions that ± T is coercive on the range of G^*. This can be achieved by showing that ± T is a coercive operator from H^-1/2(Γ_C) to H^1/2(Γ_C). With this in mind, notice that by the boundary conditions on the corroded boundary Γ_C we have that 

    ⟨ T ξ , ξ⟩_Γ_C = ∫_Γ_C(p-q)ξ s = ∫_Γ_C p [ ∂_νp  +γp] - q [[ q]]  s

and by appealing to Green's first identity 

    ⟨ T ξ , ξ⟩_Γ_C = ∫_ | p|^2  x + ∫_Γ_Cγ |p|^2  s - ∫_D | q|^2  x.

By (<ref>) we see that there is no way for ± T to be coercive without some extra assumptions because of the negative multiplying the L^2(D)–norm of the gradient of q. Therefore, to proceed we will consider two cases 0< γ <1  or 1<γ a.e. on Γ_C. 


For the first case when 0< γ <1 a.e. on Γ_C we have that 

    ⟨ T ξ , ξ⟩_Γ_C   ≥γ_min[  p^2_L^2() + p_L^2(Γ_C)^2 ] -  q_L^2(D)^2   since 0< γ <1
       ≥γ_min[  p_L^2()^2 + p_L^2(Γ_C)^2 ] - Cξ_H^-1/2(Γ_C)^2   by the well-posedness.

Now, notice that by the boundary condition in (<ref>) we can estimate 

    ξ_H^-1/2(Γ_C)   = ∂_νp  +γp_H^-1/2(Γ_C)
       ≤∂_νp_H^-1/2(Γ_C) + p_H^1/2(Γ_C)  since 0< γ <1
       ≤ C p_H^1()  by Trace Theorems
       ≤ C √( p^2_L^2() + p_L^2(Γ_C)^2)

where we have used that 

    p^2_H^1()  is equivalent to    p^2_L^2() + p_L^2(Γ_C)^2.

With this we see that ∃  C_j>0 independent of γ for j=1,2 where 

    ⟨ T ξ , ξ⟩_Γ_C≥ (C_1γ_min -C_2 )ξ_H^-1/2(Γ_C)^2.

This implies that for C_2/C_1 < γ <1 a.e. on Γ_C then we have that T is coercive. 

Now, for the case 1<γ a.e. on Γ_C we have that 

    ∫_ | p|^2  x + ∫_Γ_Cγ |p|^2  s =∫_Γ_Cξp s

by Green's first identity. With this, by the Trace Theorem we have the estimate 

    p^2_L^2() + p_L^2(Γ_C)^2 ≤ C ξ_H^-1/2(Γ_C)p_H^1() .

By the aforementioned norm equivalence, we further establish that

    √( p^2_L^2() + p_L^2(Γ_C)^2)≤ C ξ_H^-1/2(Γ_C).

Using the boundary condition in (<ref>) we have that

    ξ_H^-1/2(Γ_C)   = ∂_νq^+ -  ∂_νq^-_H^-1/2(Γ_C)
       ≤ C [   q _H^1() +   q _H^1(Ω)]   by Trace Theorem
       ≤ C q_L^2(D)  by the Poincaré estimate since q|_Γ_D = 0.

Therefore, by (<ref>) we have that ∃  C_j>0 independent of γ for j=3,4 where 

    - ⟨ T ξ , ξ⟩_Γ_C   ≥q^2_L^2(D)  - γ_max[  p^2_L^2() + p_L^2(Γ_C)^2 ]  
       ≥ (C_3 -C_4  γ_max)ξ_H^-1/2(Γ_C)^2.

This implies that for C_3/C_4 > γ >1 a.e. on Γ_C then we have that -T is coercive. 

Even though we have proven the coercivity it is unclear if the assumption that 

    C_2/C_1 < γ <1    or    C_3/C_4 > γ >1   a.e. on Γ_C
 
is satisfied. This is due to the fact that the constants C_j are unknown and depend on the geometry. In order to continue in our investigation, we make the assumption that there exists regions  and Ω such that the above assumptions are valid for some given γ. With this we have the main result of this subsection. 


Let the difference of the NtD operators (Λ-Λ_0 ): L^2(Γ_N)  →  L^2(Γ_N) be given by (<ref>).  Provided that either ± T defined by (<ref>) is coercive, then 

    𝔾(·, z)|_Γ_N∈Range( |Λ-Λ_0 |^1/2)   if and only if   z ∈Ω.



This is due to the fact that with the factorization (Λ - Λ_0)=G T G^* and provided that ± T is coercive we have that Range( |Λ-Λ_0 |^1/2) = Range(G) by the result in <cit.>. Here, we note that |Λ-Λ_0 |^1/2 is defined in the standard way by the spectral decomposition of a self-adjoint compact operator. Then by appealing to Theorem <ref> proves the claim. 


With this result, we have another way to recover the corroded region Ω. Notice that since (Λ - Λ_0) is a self-adjoint compact operator Theorem <ref> can be reformulated as 

    ∑_j=1^∞1/σ_j|( 𝔾(·, z)   ,   g_j )_L^2(Γ_N)|^2 < ∞  if and only if   z ∈Ω

by appealing to Picard's criteria (see for e.g. <cit.>) where (σ_j , g_j ) ∈_+× L^2(Γ_N) is the eigenvalue decomposition of the absolute value for the difference of the NtD operators. This result is stronger than Theorem <ref> since the result is an equivalence that implies that Λ uniquely determines the subregion Ω. Also, to numerically recover Ω we can use the imaging functional 

    W_FM(z) = [ ∑_j=1^∞1/σ_j|( 𝔾(·, z)   ,   g_j )_L^2(Γ_N)|^2 ]^-1

which is positive only when z ∈Ω. Since (Λ - Λ_0) is compact we have that the eigenvalues σ_j tend to zero rapidly which can cause instability in using the imaging functional W_FM(z). In <cit.> it has been shown that adding a regularizer to the sum can regain stability while still given the unique reconstruction of Ω. 








§ INVERSE IMPEDANCE PROBLEM

In this section, we consider the inverse impedance problem, i.e. determine the corrosion parameter γ on Γ_C from the knowledge NtD mapping Λ_γ. Here, we will assume that the corroded boundary Γ_C is known. This would be the case, if it was reconstructed as discussed in the previous section. We will prove that γ↦Λ_γ is injective as a mapping from L^∞(Γ_C) into ℒ(L^2(Γ_N)) i.e the set of bounded linear operator acting on L^2(Γ_N). Then we will prove a Lipschitz–stability estimate for the inverse impedance problem. Similar result have been proven in <cit.> just to name a few recent works. This will imply that one can reconstruct γ on Γ_C from the known Cauchy data g and Λ_γ g on Γ_N. In order to show the uniqueness, let us first consider the following density result associated with solutions to (<ref>).


Let 

    𝒰 = { u|_Γ_C∈ L^2(Γ_C) : u ∈ H^1()  solves (<ref>) for any   g ∈ L^2(Γ_N)}.

Then, 𝒰 is dense subspace in L^2(Γ_C).


It is enough to show that 𝒰^⊥ is trivial. To this end, notice that for any ϕ∈𝒰^⊥ there exists v ∈ H^1() that is the unique solution of

    Δ v= 0   in  D∖Ω,   v = 0   on Γ_N,   and  v +γ v  = ϕ  on Γ_C .

From the boundary conditions, we have that 

    0 = ∫_Γ_Cu ϕ s = ∫_Γ_C u(v + γ  v) s = ∫_Γ_C uv - vu s.

Then, by appealing to Green's second identity in  we obtain 

    0   = - ∫_Γ_N uv - vu s = ∫_Γ_N g v s,   for any  g ∈ L^2(Γ_C)

where we have used that both u and v are harmonic in . Therefore, v|_Γ_N=0 and v|_Γ_N=0 from the boundary condition, so we conclude that v vanishes in  by Holmgren's theorem. Hence, ϕ=0 on Γ_C by the Trace Theorem.


Now, we will show that the NtD operator Λ uniquely determines the boundary coefficient γ on  Γ_C. To this end, consider the solutions u and u_0 to (<ref>) and (<ref>), respectively and let 𝔾(·, z) be the mixed Green's function defined in (<ref>). Then, the following lemma allows one to rewrite (u-u_0)(z) for any z ∈ in terms of a boundary integral operator. 


For any z∈,

    -(u-u_0)(z) = ∫_Γ_C u(x) [𝔾(x, z)  +  γ(x) 𝔾(x, z)]  s(x).



For any z ∈, from the boundary conditions and Green's second identity,

    -(u-u_0)(z)     =∫_ (u-u_0) Δ𝔾(· , z)  - 𝔾(· , z)  Δ(u-u_0)   x 
       = ∫_Γ_C  (u-u_0)𝔾(· , z)- 𝔾(· , z)(u-u_0) s 
       = ∫_Γ_C u 𝔾(· , z)- 𝔾(· , z)u s - ∫_Γ_C u_0 𝔾(· , z)-  𝔾(·, z)u_0 s.

Applying Green's second identity to u_0 and 𝔾(· , z) in Ω,

    - ∫_Γ_C u_0𝔾(· , z)   - 𝔾( · , z)u_0 s= ∫_Γ_D u_0𝔾(· , z)- 𝔾( · , z)u_0 s = 0,

where we have used the fact that  u_0 and 𝔾(· , z) have zero trace on Γ_D which completes the proof. 



The result in Lemma <ref> will now be used to prove that the NtD operator uniquely determines the corrosion coefficient γ. We would like to also note that the representation formula above can be used as an integral equation to solve for γ. Assuming that the Cauchy data for u is known on Γ_N, we can recover the Cauchy data on Γ_C numerically as in <cit.>. Therefore, by restricting the representation formula in Lemma <ref> onto Γ_C (or Γ_N) gives an integral equation for the unknown coefficient. We now prove our uniqueness result. 



Assume that γ∈ L^∞(Γ_C) and satisfies the inequality in Section <ref>. Then, the mapping γ↦Λ_γ from L^∞(Γ_C) →ℒ(L^2(Γ_N)) is injective.


To prove the claim, let γ_j for j=1,2 be the corrosion coefficient in (<ref>) such that Λ_γ_1 = Λ_γ_2 for any g ∈ L^2(Γ_N). Then the corresponding solutions 

    u_γ_1=u_γ_2  and  u_γ_1=u_γ_2  on   Γ_N,
 
which implies that u_γ_1=u_γ_2 in  for any g ∈ L^2(Γ_N) by Holmgren's Theorem. If we denote u_γ_1=u_γ_2 by u, then by subtracting (<ref>) we have that 

    0 = ∫_Γ_C (γ_1-γ_2)(x) u(x)  𝔾(x, z)  s(x)    for all z ∈ and for any g ∈ L^2(Γ_N).

From Lemma <ref>, we obtain that (γ_1-γ_2)(x) 𝔾(x, z) = 0 for a.e. x ∈Γ_C and for all z ∈. Notice that by interior elliptic regularity for any x ∈ D∖{z} the mixed Green's is continuous at x. 


Now, by way of contradiction assume that γ_1-γ_2 _L^∞(Γ_C)≠ 0. This would imply that there exists a subset Σ⊂Γ_C with positive boundary measure such that |γ_1-γ_2| >0 on Σ. Therefore, for some x^* ∈Σ we hat that 𝔾(x^* , z) = 0 for all z ∈. Then, we can take a sequence z_n ∈ such that z_n → x^* as n →∞. This gives a contradiction since 

    𝔾(x^* , z_n ) = 0    for all n ∈   and    | 𝔾 (x^* , z_n ) | →∞  as   n →∞.

This implies that γ_1-γ_2 _L^∞(Γ_C) = 0, proving the claim. 


Now that we have proven our uniqueness result we turn our attention to proving a stability estimate. We will prove a  Lipschitz–stability estimate using similar techniques in <cit.>. This will employ a monotonicity estimate for the NtD operator Λ_γ with respect to the corrosion parameter γ as well as our density result in Lemma <ref>. With this in mind, we now present the monotonicity estimate. 


Let the NtD operators Λ_γ_j: L^2(Γ_N)  →  L^2(Γ_N) be given by (<ref>) with corrosion parameter γ_j ∈ L^∞(Γ_C) for j=1,2 and satisfies the inequality in Section <ref>. Then, 

    ∫_Γ_C(γ_1 - γ_2) |u_γ_2|^2  s
    ≥∫_Γ_N g (Λ_γ_2- Λ_γ_1)g s.


 The proof is identical to what is done in <cit.> so we omit the proof to avoid repetition. 


























With this we are ready to prove our Lipschitz–stability estimate. We will show that the inverse of the nonlinear mapping  γ↦Λ_γ is Lipschitz continuous from the set of bounded linear operators ℒ(L^2(Γ_N)) to a finite dimensional subspace of L^∞(Γ_C). To this end, we let 𝒜 be a finite dimensional subspace of L^∞(Γ_C) and define the compact set 

    𝒜_[γ_min,γ_max] = {γ∈𝒜 : γ_min≤γ≤γ_max on Γ_C }.

This would imply that inverse impedance problem has a unique solution that depends continuously on the NtD mapping. This fits nicely with the results from the previous section that assuming the factorization method is valid the inverse shape problem has a unique solution that depends continuously on the NtD mapping. 


Let the NtD operators Λ_γ_j: L^2(Γ_N)  →  L^2(Γ_N) be given by (<ref>) with corrosion parameter γ_j ∈𝒜_[γ_min,γ_max] for j=1,2 and satisfies the inequality in Section <ref>. Then, 

    γ_1-γ_2_L^∞(Γ_C)≤ C Λ_γ_1 - Λ_γ_2_ℒ(L^2(Γ_N)),

where C>0 is independent of γ_j ∈𝒜_[γ_min,γ_max]. 


To prove the claim, notice that from Lemma <ref>, we have that 

    - ∫_Γ_N g (Λ_γ_2 g- Λ_γ_1 g) s ≥∫_Γ_C(γ_2 - γ_1) |u_γ_2|^2  s

and interchanging the roles of γ_1 and γ_2, we obtain

    ∫_Γ_N g (Λ_γ_2 g- Λ_γ_1 g) s ≥∫_Γ_C(γ_1 - γ_2) |u_γ_1|^2  s.

Therefore, we have that 

    Λ_γ_1 - Λ_γ_2_ℒ(L^2(Γ_C))
       = sup_g=1| ∫_Γ_N g (Λ_γ_2- Λ_γ_1)g s |
    = sup_g=1max{±∫_Γ_N g (Λ_γ_2- Λ_γ_1)g s }
       ≥sup_g=1max{∫_Γ_C(γ_1 - γ_2) |u_γ_1|^2  s,  ∫_Γ_C(γ_2 - γ_1) |u_γ_2|^2  s }.

Notice that we have used the fact that Λ_γ_j is self-adjoint. Here we let · denote the L^2(Γ_N)–norm. This implies that

    Λ_γ_1 - Λ_γ_2_ℒ(L^2(Γ_C))/γ_1- γ_2_L^∞(Γ_C)
       ≥sup_g=1max{∫_Γ_C(γ_1 - γ_2)/γ_1- γ_2_L^∞(Γ_C) |u_γ_1|^2  s,  ∫_Γ_C- (γ_1 - γ_2)/γ_1- γ_2_L^∞(Γ_C) |u_γ_2|^2  s }.
 
Provided that γ_1 ≠γ_2, we now let 

    ζ= (γ_1 - γ_2)/γ_1- γ_2_L^∞(Γ_C)

and define Ψ : L^2(Γ_N ) →  given by

    Ψ(g; ζ , γ_1 , γ_2) =  max{∫_Γ_Cζ |u_γ_1^(g)|^2  s,  ∫_Γ_C- ζ |u_γ_2^(g)|^2  s }.
 
Then, to complete the proof, it suffices to show that 

    inf_ζ∈𝒞,
    κ_1, κ_2 ∈𝒜_[a,b]sup_g=1Ψ(g; ζ , κ_1 , κ_2)>0
 
where 𝒞 = {ζ∈𝒜 : ζ_L^∞(Γ_C)=1}. Notice that since 𝒜_[γ_min,γ_max] and 𝒞 are finite dimensional, we have that they are compact sets. 

To this end, since we have that ζ_L^∞(Γ_C)=1, then there exists a subset Σ⊂Γ_C with positive boundary measure such that for a.e. x ∈Σ either ζ(x) ≥ 1/2 or -ζ(x) ≥ 1/2. Without loss of generality assume that ζ(x) ≥ 1/2  a.e. for x ∈Σ and the other case can be handled in a similar way. From Lemma <ref>, there exists a sequence {g_n}_n=1^∞∈ L^2(Γ_N) such that the corresponding solution u_γ_1^(g_n) of (<ref>) satisfies 

    u_γ_1^(g_n)→2 χ_Σ/√(∫_Σ s)  as   n →∞   in the  L^2(Γ_C)–norm.

With the above convergence we have that

    lim_n →∞∫_Σ | u_γ_1^(g_n) |^2 s = 4    and   lim_n →∞∫_Γ_C ∖Σ |u_γ_1^(g_n)|^2  s = 0.

Then, there exists ĝ∈ L^2(Γ_N) such that

    ∫_Σ |u_γ_1^(ĝ)|^2  s ≥ 2    and   ∫_Γ_C ∖Σ |u_γ_1^(ĝ)|^2  s ≤ 1/2.

If ζ(x) ≥ 1/2 for x ∈Σ, then since ζ(x) ≥ -1 for x ∈Γ_C ∖Σ we have the estimate 

    Ψ(ĝ; ζ , γ_1 , γ_2)    = ∫_Γ_Cζ |u_γ_1^(ĝ)|^2  s ≥1/2∫_Σ |u_γ_1^(ĝ)|^2 s - ∫_Γ_C ∖Σ |u_γ_1^(ĝ)|^2  s ≥1/2 .

By the linearity of (<ref>) we have that 

    Ψ(ĝ/ĝ ; ζ , γ_1 , γ_2) =  Ψ(ĝ; ζ , γ_1 , γ_2) /ĝ^2 ≥1/2ĝ^2 >0

which implies that sup_g=1Ψ(g; ζ , γ_1 , γ_2)≥1/2. Now by the proof of Theorem <ref> we have that the mapping 

    (ζ , κ_1 , κ_2) ↦sup_g=1Ψ(g; ζ , κ_1 , κ_2)
 
is semi-lower continuous on the compact set 𝒞×𝒜_[γ_min,γ_max]×𝒜_[γ_min,γ_max]. This implies that it obtains its global minimum which is strictly positive by the above inequality, proving the claim.



With this result we have completed our analytical study of the inverse shape and inverse parameter problem. To reiterate, we have prove that the inverse shape and inverse parameter problem have uniquely solvable solutions given the full NtD mapping of Γ_N. 





§ NUMERICAL RESULTS

In this section, we provide numerical examples for the reconstruction of Γ_C using the NtD mapping. To this end, we first derive the corresponding integral equations to obtain the solution u_0 and u on Γ_N for (<ref>) and (<ref>), respectively. For more details on the definition of the integral operators and their jump relations we refer the reader to <cit.>. Next, we explain how to obtain 𝔾(,z) on Γ_N for a given set of points z (see equation (<ref>)). Then, we illustrate how to discretize the NtD operator Λ-Λ_0 using the Galerkin approximation in order to apply the LSM (see equation (<ref>)) or the FM (see Theorem <ref>). Finally, we provide some reconstructions using both FM and LSM, respectively.

In order to provide numerical evidence of the effectiveness of the sampling methods, we need the following definitions.
We define 
	
    Φ(x,y)=-log(|x-y|)/2π ,    x≠ y
 
to be the fundamental solution of the Laplace equation in ^2. Assume that A ⊂^2 is an arbitrary domain with boundary ∂ A.
The single-layer potential for the Laplace equation over a given boundary ∂ A is denoted by

    SL^∂ A[ϕ](x)   =   ∫_∂ AΦ(x, y)  ϕ(y)  ds(y) ,   x∈ A ,

where ϕ is some density function. Now, we let ∂ A=  Γ_α∪Γ_β with Γ_α∩Γ_β =∅ be the boundary of the domain A. 
The single- and double-layer boundary integral operators over the boundary Γ_i evaluated at a point of Γ_j are given as

    S^Γ_i→Γ_j[ϕ|_Γ_i](x)   =   ∫_Γ_iΦ(x,y)ϕ(y) ds(y) ,   x∈Γ_j ,
    
    	T^Γ_i→Γ_j[ϕ|_Γ_i](x)   =   ∫_Γ_i∂_ν_j(x)Φ(x,y)ϕ(y) ds(y) ,   x∈Γ_j ,

where i,j∈{α,β}. Here, ∂_ν_j(x) denotes the normal derivative, where ν_j(x) is the exterior normal at x∈Γ_j.






 §.§ Integral equation for computing u_0 on Γ_N

We first consider the uncorroded (healthy) object D, refer also to (<ref>). Now, we are in position to explain how to obtain u_0 at any point of Γ_N.

	Let D be the domain representing the uncorroded object with boundary Γ_N∪Γ_D satisfying Γ_N∩Γ_D=∅. Then, the solution u_0 to (<ref>) on Γ_N for the uncorroded object is given by
	
    u_0|_Γ_N(x)=S^Γ_N→Γ_N[ϕ_0|_Γ_N](x)+S^Γ_D→Γ_N[ϕ_0|_Γ_D](x) ,   x∈Γ_N ,

	where ϕ_0|_Γ_N and ϕ_0|_Γ_D are given by the solution of
	
    ([      S^Γ_N→Γ_D      S^Γ_D→Γ_D; 1/2I+T^Γ_N→Γ_N      T^Γ_D→Γ_N ])
    		([ ϕ_0|_Γ_N; ϕ_0|_Γ_D ])=
    		([ 0; g ]) .


 
We use a single-layer ansatz to represent the solution u_0 inside D as

    u_0(x)   =   SL^∂ D[ϕ_0 ](x) = SL^Γ_N[ϕ_0|_Γ_N](x)+SL^Γ_D[ϕ_0|_Γ_D](x) ,     x∈ D ,

where we used the fact that the given boundary is a disjoint union of Γ_N and Γ_D. Here, ϕ_0|_Γ_N and ϕ_0|_Γ_D are yet unknown functions.
 Letting D∋ x→ x∈Γ_D in (<ref>) together with the jump relation and the boundary condition u_0|_Γ_D=0 gives the first boundary integral equation

    0= S^Γ_N→Γ_D[ϕ_0|_Γ_N](x)+S^Γ_D→Γ_D[ϕ_0|_Γ_D](x) ,   x∈Γ_D .

Taking the normal derivative of (<ref>) and letting D∋ x→ x∈Γ_N along with the jump and the boundary condition ∂_νu_0|_Γ_N=g yields the second boundary integral equation

    g(x)=T^Γ_N→Γ_N[ϕ_0|_Γ_N](x)+1/2ϕ_0|_Γ_N(x)+T^Γ_D→Γ_N[ϕ_0|_Γ_D](x) ,   x∈Γ_N .

Equations (<ref>) and (<ref>) can be written together as the system (<ref>)
which have to be solved for ϕ_0|_Γ_N and ϕ_0|_Γ_D. Here, I denotes the identity operator. With this, we can use (<ref>) to obtain u_0 at any point within D. Letting D∋ x→ x∈Γ_N along with the jump relations yields (<ref>).




 §.§ Integral equation for computing u on Γ_N

Next, we consider the corroded object D\Ω, refer also to (<ref>) . Now, we are in position to explain how to obtain u at any point of Γ_N.

	Let D\Ω be the domain representing the corroded object with boundary Γ_N ∪Γ_C satisfying Γ_N∩Γ_C=∅. Then, the solution u to (<ref>) on Γ_N for the corroded object is given by
	
    u|_Γ_N(x)=S^Γ_N→Γ_N[ϕ|_Γ_N](x)+S^Γ_C→Γ_N[ϕ|_Γ_C](x) ,   x∈Γ_N ,

	 where ϕ|_Γ_N and ϕ|_Γ_C are given by the solution of
	 
    ([       T^Γ_N→Γ_C+γS^Γ_N→Γ_C 1/2I+T^Γ_C→Γ_C+ γS^Γ_C→Γ_C;             1/2I+T^Γ_N→Γ_N                  T^Γ_C→Γ_N ])
    	 	([ ϕ|_Γ_N; ϕ|_Γ_C ])=
    	 	([ 0; g ]) .




Using a single-layer ansatz 

    u(x)   =   SL^∂ (D\Ω)[ϕ](x) = SL^Γ_N[ϕ|_Γ_N](x)+SL^Γ_C[ϕ|_Γ_C](x) ,     x∈ D\Ω ,

where ϕ|_Γ_N and ϕ|_Γ_C are again unknown functions. As before, we obtain on Γ_C

    u|_Γ_C(x)   =   S^Γ_N→Γ_C[ϕ|_Γ_N](x)+S^Γ_C→Γ_C[ϕ|_Γ_C](x) ,     x∈Γ_C ,
    ∂_ν u|_Γ_C(x)   =   T^Γ_N→Γ_C[ϕ|_Γ_N](x)+T^Γ_C→Γ_C[ϕ|_Γ_C](x)+1/2ϕ|_Γ_C ,     x∈Γ_C ,

and hence using the boundary condition ∂_ν u+γ u=0 on Γ_C we obtain the first boundary integral equation

    0   =   T^Γ_N→Γ_C[ϕ|_Γ_N](x)+T^Γ_C→Γ_C[ϕ|_Γ_C](x)+1/2ϕ|_Γ_C
       +   γS^Γ_N→Γ_C[ϕ|_Γ_N](x)+γS^Γ_C→Γ_C[ϕ|_Γ_C](x) ,     x∈Γ_C .

Using the boundary condition ∂_ν u|_Γ_N=g yields the second boundary integral equation 

    g(x)=T^Γ_N→Γ_N[ϕ|_Γ_N](x)+1/2ϕ|_Γ_N(x)+T^Γ_C→Γ_N[ϕ|_Γ_C](x) ,   x∈Γ_N .

Equations (<ref>) and (<ref>) can be written together as the system (<ref>)
which have to be solved for ϕ|_Γ_N and ϕ|_Γ_C. With this, we can use (<ref>) to obtain u at any point within D\Ω. Letting D\Ω∋ x→ x∈Γ_N along with the jump condition yields (<ref>).




 §.§ Integral equation for computing 𝔾(,z)

In order to solve the inverse shape problem, for fixed z∈ D, we need to compute 𝔾(,z) on Γ_N (refer also to (<ref>)). Recall, that 𝔾(,z) satisfies

    -Δ𝔾(·, z) = δ(· -z)  in  D,    𝔾(·, z) = 0  on Γ_N,    and   𝔾(·, z) = 0  on  Γ_D.

Just as in <cit.>, we assume that 

    𝔾(,z)=w(,z)+Φ(,z) ,
 
where Φ(,z) is again the fundamental solution of the Laplace equation in ℝ^2. Then, w(,z) obviously satisfies

    Δ w(,z)= 0    in   D,     ∂_νw(,z)=-∂_νΦ(,z)   on Γ_N ,      w(,z)=-Φ(,z)   on Γ_D .

Our task now is to compute w(,z) on Γ_N in order to approximate 𝔾(,z) on Γ_N.

	Let z∈ D be fixed. Then 𝔾(,z) on Γ_N is given by
	
    𝔾(,z)|_Γ_N=w(,z)|_Γ_N+Φ(,z)|_Γ_N .

	Here, w(,z)|_Γ_N is obtained through
	
    w(x,z)|_Γ_N=S^Γ_N→Γ_N[ϕ_z|_Γ_N](x)+S^Γ_D→Γ_N[ϕ_z|_Γ_D](x) ,   x∈Γ_N ,

	where ϕ_z|_Γ_N and ϕ_z|_Γ_D are given by the solution of
	 
    ([      S^Γ_N→Γ_D      S^Γ_D→Γ_D; 1/2I+T^Γ_N→Γ_N      T^Γ_D→Γ_N ])
    	 	([ ϕ_z |_Γ_N; ϕ_z |_Γ_D ])=
    	 	([    -Φ(,z) |_Γ_N; -∂_νΦ(,z) |_Γ_D ]) .



We make the ansatz

    w(x,z)   =   SL^∂ D[ϕ_z](x) =SL^Γ_N[ϕ_z|_Γ_N](x)+SL^Γ_D[ϕ_z|_Γ_D](x) ,   x∈ D .

Here, ϕ_z|_Γ_N and ϕ_z|_Γ_D are yet unknown functions.
Letting D∋ x→ x∈Γ_D in (<ref>) together with the jump conditions and the boundary condition w(,z)|_Γ_D=-Φ(,z)|_Γ_D gives the first boundary integral equation

    -Φ(x,z)= S^Γ_N→Γ_D[ϕ_z|_Γ_N](x)+S^Γ_D→Γ_D[ϕ_z|_Γ_D](x) ,   x∈Γ_D .

Taking the normal derivative of (<ref>) and letting D∋ x→ x∈Γ_N along with the jump condition and the boundary condition ∂_νw(,z)|_Γ_N=-∂_νΦ(,z)|_Γ_N yields the second boundary integral equation

    -∂_νΦ(x,z)=T^Γ_N→Γ_N[ϕ_z|_Γ_N](x)+1/2ϕ_z|_Γ_N(x)+T^Γ_D→Γ_N[ϕ_z|_Γ_D](x) ,   x∈Γ_N .

Equations (<ref>) and (<ref>) can be written together as the system (<ref>) which we have to solve for ϕ_z|_Γ_N and ϕ_z|_Γ_D. With this, we can use (<ref>) to obtain w(,z) at any point within D. Letting D∋ x→ x∈Γ_N along with the jump conditions yields (<ref>).




 §.§ Discretization of the DtN operator Λ-Λ_0

Now, we illustrate how to approximate the equation 

    (Λ-Λ_0)g_z=𝔾(,z)|_Γ_N

with the Galerkin method for a fixed z∈ D. Without loss of generality, we assume that the functions on Γ_N are parametrized by x(θ) such that θ∈ (θ_1,θ_2)⊆ [0,2π]. For a given set of `Fourier' basis functions φ_n(θ) and yet unknown `Fourier' coefficients g_n^(z) 

    g_z(θ)=∑_n=0^∞ g_n^(z)φ_n(θ)

which is approximated by a finite sum

    g_z(θ) ≈∑_n=0^N_B g_n^(z)φ_n(θ)

(i.e. N_B +1 denotes the number of basis functions). Now, we insert this into (<ref>) to obtain

    ∑_n=0^N_Bg_n^(z) (Λ-Λ_0)φ_n(θ)=𝔾(θ,z) .

Multiplying this equation with φ_m (θ) for m∈{0,1,…,N_B} and integrating over [θ_1,θ_2] yields the linear system of size (N_B+1)× (N_B+1)

    ∑_n=0^N_Bg_n^(z)∫_θ_1^θ_2φ_m(θ)  (Λ-Λ_0)φ_n(θ) ds(θ) =∫_θ_1^θ_2φ_m(θ)𝔾(θ,z) ds(θ)  ,

where the unknown `Fourier' coefficients g_n^(z) are to be determined. We write this linear system abstractly as 

    Bg^(z)=b^(z) .

To compute the matrix entries for each m and n numerically

    B_mn=∫_θ_1^θ_2φ_m(θ)  (Λ-Λ_0)φ_n(θ) ds(θ)

we subdivide the interval [θ_1,θ_2] into n_f equidistant panels and apply to each panel Gauß-Legendre quadrature using three quadrature nodes and three weights. 

 Note that the matrix B should become symmetric for increasing n_f, since the operator (Λ-Λ_0) is self-adjoint. In the same way, we approximate the right hand side for each m

    b^(z)_m=∫_θ_1^θ_2φ_m(θ)𝔾(θ,z) ds(θ)  .


To obtain (u-u_0 )|_Γ_N = (Λ-Λ_0)φ_n(θ) as well as 𝔾(θ,z) for fixed z∈ D (compare also (<ref>) and (<ref>) as well as (<ref>) for the corresponding integral equation), we use as discretization the boundary element collocation method as done in <cit.> using the wave number 0. We use α=(1-√(3/5))/2, then the collocation nodes on Γ_N are exactly the three Gauß-Legendre nodes on each panel that are needed in the approximation of (<ref>) and (<ref>), respectively. Hence, we are now in position to create the matrix B which approximates (Λ-Λ_0) and the right hand side b^(z) for different domains. That is, we can now create synthetic data which then can be used for the reconstruction algorithms LSM or FM.





 §.§ Reconstructions with FMreg and LSMreg

Now, we present some reconstructions using FM and LSM. For the details on the implementation of the FM, we refer the reader to <cit.>. We compute W_FM(z) (see section <ref> for the definition) for z ∈𝒢, where 𝒢 is a set of grid points covering the domain of interest. In the following, we plot W_FM^log(z)=log (W_FM(z)). With this definition, we have that for z∈Ω W_FM^log(z)≫ 0 and z∈ D\Ω W_FM^log(z)≪ 0. We will use regularization for the factorization method (FMreg) by only using the singular values that are greater than 10^-5 since the problem is severely ill-posed. We denote the regularized version of W_FM^log(z) by W_FMreg^log(z). The regularized version of the linear sampling method (LSMreg) which is denoted by W_LSMreg^log(z) (refer to section <ref> for details), where we solve 

    (B^∗ B+α I)g^(z)=B^∗ b^(z)  with  α=10^-5
 
the Tikhonov regularization of Bg^(z)=b^(z).




Example 1: 
Let the domain be a square D=[0,2π]× [-2π,0] completely buried and only the upper part of the domain is visible. Parts of the square are corroded as shown in Figure <ref>. Precisely, the corroded part is given by the polygon with vertices (2π,0), (0,0), (π/2,-3π/2), and (3π/2,-3π/2). 
 
To create the data, we use the 20 basis functions φ_n(θ)=cos(nθ) for n∈{0,1,…,19} and θ∈ [0,2π] and n_f=300 on Γ_N for the boundary element collocation method. Furthermore, for simplicity we use γ=1/2 and γ=2. For the FMreg and LSMreg, we use an equidistant grid of size 100× 100 of D. We choose the level–curve=3/2 as threshold value for recovering Γ_C by the FMreg imaging functional. For the LSMreg imaging functional we choose the level–curve=-1/2 as threshold value. In Figure <ref> and <ref>, we present the reconstruction results with the FMreg and the LSMreg.
	
	
		
We observe reasonable reconstructions using the FMreg and LSMreg although not perfect which is expected as the problem is severely ill-posed. 






Example 2: We now consider a wedge-shaped domain using the angle of π/2. A certain part of it is corroded as shown in Figure <ref>. Precisely, the corroded part is given by the triangle with vertices (1,0), (0,1), and (0,0).
 
We use as 20 basis functions φ_n(θ)=cos(4nθ) for n∈{0,1,…,19} with θ∈ [0,π/2] and n_f=300 on Γ_N for the boundary element collocation method. Further, we use γ=1/2 and γ=2. For the FMreg and LSMreg, we use an equidistant grid of size 100× 100 of [0,1]× [0,1]. We choose the level–curve=0.25  and =-1 as threshold value for the FMreg imaging functional when γ=1/2 and γ=2, respectively. For the LSMreg imaging functional we choose level–curve=-1 and =-1.5 as threshold value for γ=1/2 and γ=2. In Figure <ref> and <ref>, we present the reconstruction results with the FMreg and the LSMreg.
	
	
		
We observe reasonable reconstructions using the FMreg and LSMreg which is much better than the one for the previous example. The FMreg performs much better. 






Example 3: Next, we use an ellipse with half-axis a=1.1 and b=1 that is half buried. Hence, Γ_N and Γ_D are given in parameter form as x_1 (θ)=1.1cos(θ) and x_2 (θ)=sin(θ) with θ∈ [0,π] and θ∈ (π,2π), respectively. The boundary Γ_C is given by x_1 (θ)=1.1cos(θ) and x_2(θ) =0.5sin(θ) with θ∈ (π,2π). The situation is depicted in Figure <ref>. 
 
We use as 20 basis functions φ_n(θ)=cos(2nθ) for n∈{0,1,…,19} with θ∈ [0,π/2] and n_f=300 on Γ_N for the boundary element collocation method. For this example, we use an equidistant grid of size 100× 100 of [-1.1,1.1]× [-1.1,1.1]. To reconstruct Γ_C we choose the level–curve=2.5  and =1.5 as threshold value for the FMreg imaging functional when γ=1/2 and γ=2, respectively. In Figure <ref>, we present the reconstruction results with the FMreg.
	
We observe good reconstructions using the FMreg which is much better than the one for the previous two examples.  






§ SUMMARY AND CONCLUSION

We have extended the applicability of the LSM and FM for recovering an unknown corroded boundary from partial Cauchy data. This main idea that we employed was to embed the `defective' region into a `healthy' region. This allows use to consider the problem of finding the corroded region. We also want to note that the analysis presented here also implies that the generalized linear sampling method <cit.> is a valid reconstruction method. In the numerical results, we have seen that the threshold value seems to depend on γ. To obtain the correct dependence, a further investigation is needed. Moreover, the FMreg seems to provide better reconstructions than the LSMreg which might be due to the choice of the regularization parameter. A thorough investigation is needed to find out if the correct choice of the regularization parameter on both methods can give similar reconstruction results. In sum, we are able to obtain reasonable reconstructions with both the FMreg and the LSMreg giving us a good idea of how much of the buried obstacle is corroded although the problem at hand is severely ill-posed. 



Acknowledgments: The research of I. Harris and H. Lee is partially supported by the NSF DMS Grant 2107891. 




99

Zaremba 
H. Ammari, O. Bruno, K. Imeri and N. Nigam, 
Wave enhancement through optimization of boundary conditions, 
SIAM J. Sci. Comput. 42(1), B207–B224 (2019).


atkinson1997 
K. E. Atkinson, 
“The Numerical Solution of Integral Equations of the Second Kind”, 
Cambridge University Press 1997.


GLSM 
L. Audibert and H. Haddar, 
 A generalized formulation of the linear sampling method with exact characterization of targets in terms of far-field measurements, 
Inverse Problems 30,  035011 (2014).


 
fm-waveguide
L. Borcea and S. Meng, 
Factorization method versus migration imaging in a waveguide, 
Inverse Problems 35, 124006 (2019).



Data-completion 
Y. Boukari and H. Haddar, 
A convergent data completion algorithm using surface integral equations. 
Inverse Problems 31, 035011 (2015).



Brezis
H. Brezis, 
“Functional Analysis, Sobolev Spaces and Partial Differential Equations”. 
Springer 2011.



FM-wave
F. Cakoni,  H. Haddar and A. Lechleiter, 
On the factorization method for a far field inverse scattering problem in the time domain, 
SIAM J. Math. Anal. 51(2), 854–872 (2019).



CK-gibc
F. Cakoni , Y. Hu and R. Kress, 
Simultaneous reconstruction of shape and generalized impedance functions in electrostatic imaging, 
Inverse Problems 30, 105009 (2014).



CK-ibc
F. Cakoni and R. Kress, 
Integral equations for inverse problems in corrosion detection from partial Cauchy data, 
Inverse Problems and Imaging 1(2),  229–245 (2007).



CK-ibc2
F. Cakoni, R. Kress and C. Schuft,
Simultaneous reconstruction of shape and impedance in corrosion detection, 
J. Methods and Applications of Analysis 17(4),  357–378 (2010).



 
EIT-InnerStability
S. Chaabane and M. Jaoua,
Identification of Robin coefficients by the means of boundary measurements,
Inverse Problems 15, 1425 (1999).


fm-gbc 
M. Chamaillard, N. Chaulet, and H. Haddar, 
Analysis of the factorization method for a general class of boundary conditions, 
J. Inverse Ill-Posed Probl. 22(5), 643–670 (2014).

CK 
D. Colton and A. Kirsch, 
A simple method for solving inverse scattering problems in the resonance region, 
Inverse Problems 12, 383–393 (1996).

Colto2013
D. Colton and R. Kress.
Inverse Acoustic and Electromagnetic Scattering Theory.
Springer, 3rd edition, 2013.






embry
M.R. Embry,
Factorization of operators on Banach space
Proc. Amer. Math. Soc. 38,  587–590 (1973).



EIT-granados1 
G. Granados and I. Harris,
Reconstruction of small and extended regions in EIT with a Robin transmission condition.
Inverse Problems 38, 105009 (2022).



EIT-granados2 
G. Granados, I. Harris and H. Lee, 
Reconstruction of extended regions in EIT with a generalized Robin transmission condition.
Comm. on Analysis & Computation 1(4), 347–368 (2023).



fm-GR
R. Griesmaier and H.-G. Raumer, 
The factorization method and Capon’s method for random source identification in experimental aeroacoustics. 
Inverse Problems 38, 115004 (2022).



 
GLSM-elastic
B. Guzina and T.-P. Nguyen, 
Generalized Linear Sampling Method for the inverse elastic scattering of fractures in finite solid bodies, 
Inverse Problems 35, 104002 (2019).


EIT-finiteElectrode 
B. Harrach,
Uniqueness and Lipschitz stability in electrical impedance tomography with finitely many electrodes, 
Inverse Problems 35, 024005 (2019).


eit-transmission2 
B. Harrach,
Uniqueness, stability and global convergence for a discrete inverse elliptic Robin transmission problem, 
Numer. Math. 147, 29–70 (2021).



eit-transmission1 
B. Harrach and H. Meftahi,
Global Uniqueness and Lipschitz-Stability for the Inverse Robin Transmission Problem, 
SIAM J. App. Math. 79(2), 525–550 (2019).



eit-harris
I. Harris, 
A direct method for reconstructing inclusions and boundary conditions from electrostatic data, 
Applicable Analysis  102(5), 1511–1529 (2023).


regfm1
I. Harris, 
Regularization of the Factorization Method applied to diffuse optical tomography, 
Inverse Problems 37, 125010 (2021).


regfm2
I. Harris, 
Regularized factorization method for a perturbed positive compact operator applied to inverse scattering,
Inverse Problems 39, 115007 (2023).
 
 
 
holmgren
H.  Hedenmalm, 
On the uniqueness theorem of Holmgren,
Math. Z.  281, 357–378 (2015). 



firstFM
A. Kirsch, 
Characterization of the shape of the scattering obstacle by the spectral data of the far field operator,
Inverse Problems 14, 1489–512 (1998). 



FMiso
A. Kirsch, 
The MUSIC-algorithm and the factorization method in inverse scattering theory for inhomogeneous media,
Inverse Problems 18, 1025 (2002). 

 
 
kirschipbook
A. Kirsch, 
“An Introduction to the Mathematical Theory of Inverse Problems”,
2^nd edition, Springer 2011.



kirschbook
A. Kirsch and N.  Grinberg, 
“The Factorization Method for Inverse Problems”,
Oxford University Press, 2008.



kleefeldhotspot 
A. Kleefeld, 
The hot spots conjecture can be false: some numerical examples, 
Advances in Computational Mathematics 47, 85 (2021).

 
 
JIIP 
A. Laurain and H. Meftahi,
Shape and parameter reconstruction for the Robin transmission inverse problem.
J. Inverse Ill-Posed Probl. 24(6), 643–662 (2016).



glsm-stokes 
M. Liu and J. Yang, 
The Sampling Method for Inverse Exterior Stokes Problems, 
SIAM J. Sci. Computing 44(3), B429–B456 (2022).


McLean
W. McLean, “Strongly elliptic systems and boundary integral equations”, Cambridge University Press 2000.


invrobin-Meftahi 
H. Meftahi, 
Stability analysis in the inverse Robin transmission problem, 
Math. Methods Appl. Sci. 40,  2505–2521 (2017).



 
lsm-heat 
G. Nakamura and H. Wang, 
Linear sampling method for the heat equation with inclusions, 
Inverse Problems 29, 104015 (2013).



 
applied-lsm 
F. Pourahmadian and H. Yue, 
Laboratory application of sampling approaches to inverse scattering,
Inverse Problems 37, 055012 (2021).


 
Rundell 
W. Rundell. 
Recovering an obstacle and its impedance from Cauchy data. 
Inverse Problems 21, 045003 (2008).

salsa 
S. Salsa,
“Partial Differential Equations in Action From Modelling to Theory”,
Springer Italia, 2nd edition (2015).
 
 
 
LSM-periodic 
J. Yang, B. Zhang and R. Zhang,
A sampling method for the inverse transmission problem for periodic media,
Inverse Problems 28, 035004 (2012).





