THE FACTORIZATION METHOD FOR A PARTIALLY COATED CAVITY IN INVERSE SCATTERING

. We consider the interior inverse scattering problem of recovering the shape of an impenetrable partially coated cavity. The scattered ﬁelds incited by point source waves are measured on a closed curve inside the cavity. We prove the validity of the factorization method for reconstructing the shape of the cavity. However, we are not able to apply the basic theorem introduced by Kirsch and Grinberg to treat the key operator directly, and some auxiliary operators have to be considered. In this paper, we provide theoretical validation of the factorization method to the problem, and some numerical results are presented to show the viability of our method.


1.
Introduction. The inverse scattering problem for acoustic or electromagnetic waves has drawn increased attention in recent years due its importance in various areas, such as medical imaging, ultrasound tomography, nondestructive testing, material science, radar. The typical inverse scattering problems are exterior problems where the measurements are taken outside of the objects (see [5,6]). but in recent years there has been extensive interest in the interior scattering problem for determining the scattering objects and the structures [7,13,21,23,25,26,29,30]. The scattering problem with both point sources (incident waves) and measurements (scattered waves) inside a cavity is called the interior scattering problem. In this paper, we consider an inverse scattering problem from an impenetrable partially coated cavity. The goal is to reconstruct the shape of the scattering object from the near field data measured on some curve C inside the cavity. These problems may occur in many industrial applications of non-destructive testing.
Comparing with other sampling methods [5,6], the factorization method gives an exact characterization of the boundary using the behaviour of the indicator function. The factorization method developed by authors in [11,12,14,15], Kirsch and Grinberg summarized their works in the monograph [16] in 2008. More related works can be found in [3,4,21] and the reference therein.
In this paper, we try to use this method to retrieve the shape and location of a partially coated cavity in R 2 . However, in the case of a partially coated cavity, we are not able to apply the basic theorem introduced by Kirsch and Grinberg to treat the key operator. Some auxiliary operators have to be considered. The main challenge is to factorize the auxiliary operators suitably, and show some key properties to the related operators. Then the basic Theorem 2.15 in [16] can be used to retrieve the shape and location of the cavity.
The inverse scattering problem we consider in this paper is to determine the shape of the partially coated cavity from the knowledge of the near field data measured on a known closed curve C inside the cavity.
This paper is organized as follows. In the next section, we will formulate the direct scattering problem, and give the well-posedness of the direct scattering problem by using the variational method. The mathematical basis of the factorization method applied to treat the inverse scattering problem is given in section 3. In section 4, some numerical results are presented to show the viability of our method.
2. The direct scattering problem. We begin with the formulations of the scattering problem. Let k = ω c > 0 be the wave number, where ω > 0 is the frequency of a time harmonic wave and c > 0 is the sound speed. Let D ⊂ R 2 be a bounded simply connected domain with Lipschitz boundary Γ = Γ D ∪ Π ∪ Γ I , where Γ D and Γ I are disjoint, relatively open subsets of Γ, having Π as their common boundary in Γ. Furthermore, the Dirichlet and impedance type of boundary conditions are specified on Γ D and Γ I , respectively.
To be precise, we consider the two-dimensional scattering problem of determining the total field U = u s + u i , from the mixed boundary value problem where ν denotes the unit outward normal vector defined almost everywhere on Γ D ∪ Γ I , and λ is a complex-valued impedance coefficient. Firstly we make some assumptions.
Assumptions. (1). Let C be a Lipschitz closed curve inside D, and D 0 be the interior domain enclosed by C.
(2). The incident wave u i is a point source in the form 0 (·) is the Hankel function of zero order of the first kind. (3). The impedance coefficient λ satisfies (λ) > 0.
We introduce some spaces will be used in the following. H 1 (D) and H 1 loc (R 2 \D) are the usual Sobolev spaces, H If Γ 0 is a partial boundary of Γ, i.e., Γ 0 ⊂ Γ, we define is equipped with the norm induced from H 1 (D). The direct scattering problem is to find scattering wave u s ∈ H 1 (D) by the known incident wave u i and the boundary Γ. Since the incident wave u i = Φ(x, y) satisfies the Helmholtz equation, then the problem (1) is a special case of the following general boundary value problem In the variational sense, the problem (2) is equivalent to find u ∈ H 1 (D) satisfying for all test function ϕ ∈ H 1 0 (D, Γ D ). Theorem 2.1. Assume that (λ) > 0, then the interior scattering problem (2) has a unique weak solution u ∈ H 1 (D) (satisfying (3)), and for some positive constant C. Proof.
The proof of this theorem is completed.
3. The factorization method. The inverse problem we are concerned with is to determine the shape and location of the cavity D from the knowledge of the near field data u s (x, y), x ∈ C due to point sources Φ(·, y), y ∈ C.
Through the scattered fields u s (x, y), x, y ∈ C, we define a near field operator N : L 2 (C) → L 2 (C) given by (6) ( Our approach for solving the inverse problem is the factorization method, that is, we try to derive a factorization decomposition to the near field operator N in the form with different forms of the operators H and M , and H * denotes the adjoint operator of H. The operator H is compact with dense range and M is an isomorphism. Then we try to check that the related operators satisfy the fundamental theorem introduced by Kirsch and Grinberg (see Theorem 2.15 [16]).

Remark 2.
In fact, in the following discussion, we find that we are not able to apply the basic theorem introduced by Kirsch and Grinberg to treat the key operator N directly, and some auxiliary operators have to be considered.
The factorization method is based on the following fundamental theorem ( Theorem 2.15 [16]).
Theorem 3.1. Let X ⊂ U ⊂ X * be a Gelfand triple with a Hilbert space U and a reflexive Banach space X such that the imbedding is dense. Furthermore, let Y be a second Hilbert space and let N : Y → Y , G : X → Y and T : X * → X be linear bounded operators such that N = GT G * .
We have the following assumptions: (1). G is compact with dense range.
(2). There exists t ∈ [0, 2π] such that [e it T ] has the form [e it T ] = C + K with some compact operator K and some self-adjoint and coercive operator C : In order to decompose the near field operator N , we define a data-to-data op- which maps the boundary data into the near fields u s | C , i.e., where u s is the scattered field of the problem (1). Through a single-layer potential To the original problem (1), from the definitions (6)-(10), we have a decomposition So the adjoint operator of H is given by We use ( α, β) to denote the zero extension of (α, β) to Γ, where ( α, β) ∈ H − 1 2 (Γ)× H 1 2 (Γ), and define a potential (14) W Let x approach the boundary Γ from inside of the domain D, and using the jump relationships of the single-and double-layer potentials across the boundary, we have Remark 3. For simplicity, in our discussion, we use "(·) ± " or "(·) ± " to denote the limit approaching the boundary from outside and inside of the corresponding domain, respectively.
Analogous to [9], S, K, K and T are boundary integral operators defined by We use S Γ D and S Γ I to denote the restriction of S on the partial boundary Γ D and Γ I , respectively, and K Γ D , K Γ D , T Γ D , K Γ I , K Γ I , T Γ I have the similar meaning.
Proof. (1). Firstly from the expression of W (in (14)), we have From (17) we have Taking the imaginary part in (23), we have where D R = B R \D, and B R is a sufficient large circle with radius R.
The operator M can be decomposed into This implies that the invertibility of M is equivalent to the invertibility of M , and Because M is injective, we know that M is injective too. In order to use the Fredholm theory, it is sufficient to show that M is a Fredholm operator with index zero. From those we can obtain that the operator M has bounded inverse, and the operator M also is bounded invertible. With the help of the following Lemma 3.3, we will prove M is a Fredholm operator with index zero. .

Furthermore (30)
Now we obtain that M is the Fredholm operator with zero index, combining M is injective, the operators M and M are invertible.
The denseness of Range(H * ) is equivalent to the injectivity of its adjoint operator H.
Then from the continuation of v ϕ , we have v ϕ = 0 in R 2 \D 0 . Since v ϕ is a single layer potential, we have v ϕ | ± C = 0. Hence v ϕ = 0 in D 0 since k 2 is not the eigenvalue of − in D 0 .
So we obtain H is injective, this implies that H * has dense range.
The proof of this theorem has been completed.
But the real part of the operator M fails to be the sum of a coercive operator and a compact operator, which is in contrast to the second part of Theorem 3.1. In what follows, we will introduce a modified operator N D to replace N .
To this end, we select that Ω is a priori known open and bounded domain with C 2 boundary ∂Ω, such that D 0 ⊂ Ω and Ω ⊂ D, and the boundary ∂Ω of Ω is disjoint with C.
Substituting (42) into (41), we have Set (ρ 1 ) > 0, then M 10 and M 1c are coercive and compact, respectively, since R 2 and M c are compact. Similarly, we have Then we rewrite M 2 as let (ρ 2 ) > 0, then M 20 is coercive. By the compactness of R 1 and M c , we have that M 2c is compact.
Let (ρ 1 ) > 0, then we have If M 1 φ, φ = 0, this implies Mφ,φ = (ρ 1 ) φ 2 2 = 0, i.e., (3). The compactness of the operator H * can been seen immediately, since Lemma 3.5. Assume that k 2 is neither a Dirichlet eigenvalue of − in Ω, nor a Dirichlet eigenvalue of − in D 0 . For any piecewise smooth nonintersecting arc L without cups, defining On the contrary, if we have L ⊆ Γ D , but Φ 1 L ∈ Range(H * 1 , H * ). This implies that there existsφ = (φ 1 ,φ 2 ) ∈H − 1 2 (Γ D ) × L 2 (∂Ω), such that We select a point z ∈ L, but z / ∈ Γ D , then Φ 1 L (x) has a singular point at x = z, but the right term in (46) is analytic. This is a contradiction. From Theorems 3.1, 3.4 and Lemma 3.5, we have Theorem 3.6. Assume that k 2 is neither a Dirichlet eigenvalue of − in Ω, nor a Dirichlet eigenvalue of − in D 0 . Then (1) Here Analogously, we can reshape the boundary of Γ I from the following conclusions.
Lemma 3.7. Assume that k 2 is neither a Dirichlet eigenvalue of − in Ω, nor a Dirichlet eigenvalue of − in D 0 . For any piecewise smooth nonintersecting arc L without cups, defining Here (λ j , ψ j ) is an eigensystem of N I , recall N I = | (N I )| + | (N I )|.

4.
The numerical examples. In this section, we study the applicability of our method through some numerical examples in R 2 . In every example, for the forward numerical solution, we use the fully discrete collocation method on the boundary integral equation method as suggested in [19,20,24].
In the examples, the near field operator N is represented by a C 64×64 matrix, where each entry is the near field pattern u s (x j , y l ), j, l = 1, ..., 64, with x j ∈ C and y l ∈ C are the equidistant points on the curve C. Then the operator N D = N − ρ 1 H * H can be computed explicitly, and notice that ( H * Hψ)(x) = 2πR |ŷ|=1 J 0 (kR|x −ŷ|)ψ(ŷ)ds(ŷ), |x| = 1.
Remark 6. Refer to [16] on page 85, we have an example to calculate ( H * Hψ)(x) to the case that Ω is a disk with center 0 and radius R.
The real and imaginary part of the matrix N 1 is given by We define the absolute value of a matrix A ∈ C N ×N with a singular value decomposition A = U ΛV * as |A| = U |Λ|V * with |Λ| = diag|λ j |, j = 1, ..., N. In what follows, we consider two different geometries as the cavity, one is a kite and the other is a peanut. The wave number k = 1 and the noise is 3%. We plot the contour of ln g(z) to reconstruct the shapes of the cavities.
We choose the curve C to be a circle with radius 0.8 center at the origin, and set a small square with side length 2 between C and ∂D. We specify the Dirichlet boundary condition for z ∈ [0, π] and the impedance boundary condition for z ∈ [π, 2π] with λ = 50(1 + 3i).
We choose the curve C to be a circle with radius 1.2 center at the origin, and set a small rectangle [−1, 1] × [−0.5, 0.5] between C and ∂D. We specify the Dirichlet boundary condition for z ∈ [0, π] and the impedance boundary condition for z ∈ [π, 2π] with λ = 2 + 6i.