THE GENERALIZED LINEAR SAMPLING AND FACTORIZATION METHODS ONLY DEPENDS ON THE SIGN OF CONTRAST ON THE BOUNDARY

. We extend the applicability of the Generalized Linear Sampling Method (GLSM) [2] and the Factorization Method (FM)[16] to the case of inhomogeneities where the contrast changes sign. Both methods give an exact characterization of the target shapes in terms of the farﬁeld operator (at a ﬁxed frequency) using the coercivity property of a special solution operator. We prove this property assuming that the contrast has a ﬁxed sign in a neighborhood of the inhomogeneities boundary. We treat both isotropic and anisotropic scatterers with possibly diﬀerent supports for the isotropic and anisotropic parts. We ﬁnally validate the methods through some numerical tests in two dimensions.

1. Introduction. We extend the applicability of the Generalized Linear Sampling Method (GLSM) [2] and the Factorization Method (FM) [16] to the case of inhomogeneities with sign changing contrast. More specifically we shall prove that these methods are applicable under the assumption that the contrast sign is kept constant only in a neighborhood of the inhomogeneities boundary. This type of assumption is compatible with the one used for proving discreteness of the Transmission Eigenfrequencies (frequencies for which the homogeneous Interior Transmission Problemhas non trivial solutions [6]) which is another additional ingredient for these sampling methods. Under these assumptions one obtains in particular an exact characterization of the target shapes in terms of the range of the farfield operator (at a fixed frequency). One of the key ingredient to prove this exact characterization is based on a factorization of the farfield operator. This factorization involves three operators that should satisfy some specific properties. The main result of this paper is to prove this coercivity property in the case of sign changing contrast both for isotropic and anisotropic scatters with possibly different supports for the isotropic and anisotropic parts.
This work is indeed closely related to the study of ITP for sign changing contrasts [6,17,9,4,8,15]. However, proving the above mentioned coercivity property requires to also deal with bounds on the solution outside the inhomogeneity. The main idea of our proof is to introduce an artificial contrast in order to isolate the terms in the region where the contrast changes sign. Using classical regularity results for pde we then are able to demonstrate that these terms are only related to compact perturbations of a case with constant sign contrast. This is then enough to prove the shape characterization using FM [16,14] and GLSM [2] with the additional assumption that ITP is well posed. Let us quote that a similar result for FM and isotropic scatterers has been obtained independently by [10] using different techniques.
Our methodology could be applied to other type of perturbations (such as soundsoft or sound-hard inclusions) inside the inhomogeneities to demonstrate coercivity property in those cases [18].
The article is organized as follows. In Section 2 we introduce the scalar wave equation for orthotropic media and demonstrate that the farfield operator can be factorized in a similar way as for the isotropic case (although with some additional technicalities with respect to [5]). The obtained factorization does not require any correlation between the supports of the isotropic parameters and the anisotropic ones (which then may be different). In Section 3 we demonstrate the coercivity of the middle operator (denoted T ) which is shown to hold true if the contrasts have fixed (and compatible) sign in a neighborhood of the boundary of D. In Section 4 we conclude that the Generalized Linear Sampling Method or the Factorization Method could be applied in these cases. Finally in Section 5, we give some numerical illustrations.
2. Model problem. The model problem we are interested in is the scattering of scalar waves by an orthoptic medium in the frequency domain. For a wave number k > 0, the total field solves the following scalar wave equation: div(A∇u) + k 2 nu = 0 in R d with d = 2 or 3 and with n ∈ L ∞ (R d ) denoting the refractive index such that the support of n − 1 is included into D n with D n a bounded domain with Lipschitz boundary and connected complement and such that (n) ≥ 0. We assume that A is at least in L ∞ (R d ) d×d (we denote by Id the identity matrix) and that the support of A − Id is included into D A with D A a bounded domain with Lipschitz boundary and connected complement and such that (Aζ ·ζ) ≤ 0 and (A)ζ ·ζ ≥ c|ζ| 2 for ζ ∈ C d and for some positive constant c. We introduce a domain D such that D n ∪ D A ⊂ D with D a bounded domain with Lipschitz boundary and connected complement. In the following we will assume that the simply connected components of D will either have boundary where A is not equal to Id or n is not equal to one if A is identically equal to Id. Therefore we will have D n ∪ D A = D.
We introduce the unit sphere S d−1 := {x ∈ R d s.t. |x| = 1},we are interested in the cases where the total field is generated by plane waves, u i (θ, x) := e ikx·θ with x ∈ R d and θ ∈ S d−1 and we denote by u s the scattered field defined by which is assumed to be satisfying the Sommerfeld radiation condition, Our data for the inverse problem will be formed by noisy measurements of so-called farfield pattern u ∞ (θ,x) defined by as |x| → ∞ for all (θ,x) ∈ S d−1 × S d−1 . The goal is to be able to reconstruct D from these measurements (without knowing n and A). From those measurements we introduce the farfield operator F : By linearity of the forward scattering problem, F g is nothing but the farfield pattern of w solution of (1) with we identify X and its adjoint. Finally we consider the norm on X(D) defined by Now consider the (compact) operator H : and the (compact) operator G : R(H) ⊂ X → L 2 (S d−1 ) defined by where w ∞ is the farfield of w ∈ H 1 loc (R d ) solution of (1) and where R(H) denotes the closure of the range of H in X. Then clearly One can also decompose G to get the second factorization of the farfield operator. More precisely, for the case under consideration, since the farfield pattern of w has the following expression [5]: x · ∇(ψ(y) + w(y))dy, one simply has G = H * T ψ, where H * : X → L 2 (S d−1 ) is the adjoint of H given by and T : X → X is defined by with w ∈ H 1 loc (R d ) being the solution of (1)(with ψ 1 = ψ and ψ 2 = ∇ψ). Finally we get (6) F = H * T H, Remark 1. We remark that T is independent of the type of incident waves (either plane waves, point sources ...). We presented our results for plane waves but the properties of T presented in this paper remain true for other type of incident waves and measurements such as the one considered in [3].
3. Key properties of the factorization operators. In the following we will review important properties of the operators involved in the factorization (4) and (6). First we will state the classical properties of H and G, in particular a range characterization of the obstacle D which is at the heart of both the GLSM and the FM. Then we will study the coercivity of T for sign changing contrasts.
3.1. Range characterization. First we assume that our obstacle D will be composed of several disjoint simply connected components. Those components will either have A = Id and n = 1 or A = Id in a neighborhood of their boundaries. The characterization of the obstacle D in terms of the range of G is based on the solvability of the interior transmission problem (for given regular boundary values f and h): is a space of solutions that will be specified later. We will assume that the following hypothesis holds true. Hypothesis 1. We assume that k 2 ∈ R + is such that problem (7) has a unique solution for all regular (to be specified later) functions f and h.
This hypothesis and the interior transmission problem stated above are incomplete in the sense that we did not specify Y(D). This space actually depends on the properties of A and n. For example if we assume that D n ⊂ D A = D, (7) can be studied for (u, v) ∈ Y(D) = H 1 (D) × H 1 (D). In this case we know from [4] that hypothesis 1 is for instance true if A − Id and n − 1 have the same sign and do not change sign in a neighborhood of ∂D. The case where . In this case Hypothesis 1 is true if n − 1 does not change sign in a neighborhood of ∂D. The case where n = 1 in a neighborhood of ∂D has been less studied in the literature and the only case where we know that hypothesis 1 is true is when A − Id does not change sign in all D and n = 1 in D. Finally when A = Id in a neighborhood of ∂D, but not in all D, and n − 1 does not change sign in a neighborhood of ∂D, there is no clearly stated result in the literature about this case. Let us mention however that surface integral method applied to (7) (as proposed in [9]) would be an appropriate tool to study this case. We also give the following lemmas from [5]: Inverse Problems and Imaging Volume 11, No. 6 (2017), 1107-1119 Lemma 3.2. G is compact and if hypothesis 1 holds true, its range is dense in

3.2.
Coercivity of the middle operator T . Our hypothesis on D implies that we can split the simply connected component into two categories. The first one is such that A − Id does not equal zero on a neighborhood of the boundary and the second one is such that A−Id = 0 and n−1 does not change sign in a neighborhood of the boundary. We will give a coercivity result for each of those two configurations and then merge them into a combined condition on n and A under which we have the coercivity of T defined in (5). For both cases we will need the following equality.
Lemma 3.4. We have the following identity, for ψ = (ψ 1 , ψ 2 ) ∈ X(D) and T defined in (5) : Dn (1 − n)(ψ + w)w Using (1) and integrating by parts over a ball B R such that D ⊂ B R we have: (8), taking the imaginary part and letting R to +∞ prove the lemma.
3.2.1. The case where D A D n . We first consider the case where D A D n which can be seen as an extension of the case D A = ∅. The Herglotz wave operator reduces to Theorem 3.5. IfD A D n = D and hypothesis 2 and 1 holds true , then there exists µ > 0 such that the operator T defined in (5) verifies Proof. We will proceed by a contradiction argument, therefore we assume: ψ X(D) = 1 ∆ψ + k 2 ψ = 0 in D and |(T ψ , ψ )| → 0 as → ∞.
Up to changing the initial sequence, one can assume that ψ weakly converges to ψ in L 2 (D). One easily see that ψ satisfies ∆ψ + k 2 ψ = 0 in D.
We denoted w ∈ H 2 loc (R d ) the solution of (9) w converges weakly in H 2 loc (R d ) and strongly in H 1 (D) to some w ∈ H 2 loc (R d ) that satifies with ψ equation (9). Lemma 3.4 implies that w ∞ → 0 in L 2 (S d−1 ) and therefore w ∞ = 0. The Rellich theorem and the unique continuation principle imply that w = 0 outside D. Thus we have that u = ψ+w and v = ψ solve the interior transmission eigenvalue problem (7) with f = g = 0. Hypothesis 1 implies that that ψ = w = 0.
Our hypothesis on n allows us to introduce n 0 such that n 0 = n in some domain V ⊂ D and there exist α ≥ 0 and c > 0 such that either (n 0 − 1) + α (n 0 ) ≥ c or (1 − n 0 ) + α (n 0 ) ≥ c in D. We introduce Ω = supp(n 0 − n) ∪ D A . By assumption we have that Ω D and we can choose V such that V ∩ Ω = ∅. We introduce the intermediate scattered field u s 0, ∈ H 2 loc (R d ) that satisfies : We denoted by u 0, = u s 0, + ψ the associated total field. We also introduced the scattered field u s that satisfies : Using the same argument as for w we get that u s 0, converges strongly to zero in H 1 (D). Since Ω is strictly included inside D, we have that u 0, is bounded in H 2 (Ω) (by interior elliptic regularity). Therefore u 0, ∈ H 2 (Ω) converges strongly to zero in H 1 (Ω) together with the continuity of the forward scattering problem for u s , we deduce that u s converges strongly to zero in H 1 loc (R d ). Finally the interior elliptic regularity implies that ψ strongly converges to zero in H 1 (Ω). Applying those strong convergence results to we deduce that the last four terms go to zero. The first two terms on the right hand side can be bounded from below : where the last term on the right hand side goes to zero (because of the strong convergence results), and using the assumption on n 0 we conclude that lim →0 T ψ , ψ ≥ k 2 c/2 > 0, which is a contradiction.

3.2.2.
The case D n ⊂ D A . Without loss of generality we will consider the case where D n = ∅ and D = D A rather than the case D n ⊂ D A = D in order to lighten the notation. This is possible because thanks to compact embedding from H 1 to L 2 , terms that come from contrast in n will go to zero in the proof similarly to the previous section. Therefore in the following D A = D and X(D) will be equal to the H 1 (D) .

Hypothesis 3.
A is C 1 in a neighborhood V of ∂D and if either of both conditions apply: is positive definite and there exists α ≥ 0, c > 0 and 0 < η ≤ 1 such that Theorem 3.6. If D n = ∅, D = D A and hypothesis 3 holds true, there exists µ > 0 such that the operator T defined in (5) verifies for all ψ ∈ R(H).
Proof. We introduce A 0 such that A 0 = A inside V a neighborhood of ∂D and A 0 verifies hypothesis 3 in all D. Since we suppose that A is C 1 inside V we can choose A 0 to be C 1 inside all D. We also introduce Ω = supp(A − A 0 ), by construction Ω D.
The solution w satisfying (1) with v = ψ weakly converges in H 1 (D) to w ∈ H 1 (R d ) satisfying (1) with v = ψ. Lemma 3.4 implies that w ∞ → 0 in L 2 (S d−1 ) and therefore w ∞ = 0. The Rellich theorem and unique continuation theorem imply that w = 0 outside D. Thus we have that u = ψ + w and v = ψ solve the interior transmission eigenvalue problem (7). Hypothesis 1 implies that that ψ = w = 0. Let us introduce the intermediate (scattered) field u s 0, that solves: We denote by u 0, = u s 0, + ψ the total field. We also introduce u s that solves: We have Since u 0, ∈ H 1 (D) satisfies div(A 0 ∇u 0, ) + k 2 u 0, = 0 in D, we infer by interior elliptic regularity that u 0, ∈ H 2 (Ω) (from [12] and the fact that A 0 is C 1 ). Due to compact embeddings from H 2 to H 1 , we deduce that u 0, strongly converges to zero in H 1 (Ω). For the same reasons we deduce that ψ strongly converges to zero in H 1 (Ω).
By continuity of the forward scattering problem verified by u s and the strong convergence of u 0, in H 1 (Ω), we deduce that u s strongly converges to zero in H 1 loc (R d ). We therefore deduce that for large enough (14) becomes: To treat |(T 0 ψ , ψ )| = D (A 0 − Id)∇u 0, ∇ψ dx we need to consider two cases depending on the compatibility of the sign of A 0 − Id and Id (as in [7]). First we consider the case when there exist α ≥ 0 and c > 0 such that (A 0 −Id)−α (A 0 ) ≥ c > 0. Since u s 0, solves (12) we deduce that: The weak convergence of u s 0, in H 1 loc (R d ) and u 0, in H 1 (D) imply the strong convergence in L 2 loc (R d ) and L 2 (D) respectively. Therefore the last three terms in the equality above go to zero. Moreover (15) implies that |(T 0 ψ , ψ )| go to zero. Therefore the first term in (16) goes also to zero and the hypothesis on A 0 implies that Therefore u 0, 2 H 1 (D) goes to zero as well as u s . This implies that ψ 2 H 1 (D) → 0 which is a contradiction. Then we consider the case when (A 0 ) is positive definite. We cannot use (16) since the terms involving u 0, and u s 0, do not have the same sign. From the definition of T 0 we have: Using equation (1) verified by u s 0, we have: The last two terms go to zero because of regularity and compact embedding. The real part of the remaining term is both will go to zero. Those two terms can be combined through a positive parameter λ in order to form the following quantity: If we denote the quantity under the integral over D in this identity by M (∇ψ , ∇u s 0, ). We observe that : Our assumption on A 0 implies that for we have This implies that ψ 2 goes to zero which is a contradiction.
Remark 2. One can weaken the regularity assumption on A in V (e.g. example piecewise C 1 ) as long as one obtain an interior regularity property (e.g. u 0, ∈ H s (Ω) where s > 1) which implies strong convergence through compact embeddings [13].

3.2.3.
A final coercivity result. We introduce D = i D i n ∪ i D i A where the D i are simply connected disjoint components. We assume that A − Id is not zero in the neighborhood of the boundary D i A and A − Id equals zero in the neighborhood of the boundary D i n . With those notation and the result of Theorems 3.6 and 3.5 we can give the final result under Hypothesis 1 in the case of many disjoint scatters.
Theorem 3.7. Assume A has C 1 regularity in D i A ∩ V and that either conditions apply : • there exist c > 0 and α > 0 such that either We have that T defined by (5) verifies: where ψ ∈ R(H).
Proof. We set D 1 = i D i A and D 2 = i D i n . In this case we have that By the linearity of the forward scattering problem, if we introduce the two total fields associated to the two incidents waves ψ 1 = ψ| D1 in D 1 and 0 in D 2 and ψ 2 = ψ| D2 in D 2 and 0 in D 1 , denoted u 1 = u s 1 + ψ 1 and u 2 = u s 2 + ψ 2 . Then we have: where T 1 and T 2 are the operators corresponding to D 1 and D 2 respectively. We clearly see that the last two terms go to zero (using a compactness argument). Therefore if T 1 and T 2 have the same sign, we obtain that T is coercive. The sign of T 1 and T 2 are given in the proofs of Theorems 3.6 and 3.5 respectively, which allows us to conclude.
Remark 3. We concentrate on sign changing contrast but we believe that both the results and the methods of the proofs could be straightforwardly extended to inclusion of any kind (sound-soft, sound-hard, Robin condition,...) strictly included inside the penetrable obstacle.
Remark 4. We concentrate on the wave equation but the FM and GLSM might be applied to Electrical Impedance Tomography (EIT) [11]. Despite the similarity between sampling methods for EIT and inverse scattering our proofs do not extend straightforwardly to the EIT equation. As far as we know the case where the contrast changes sign strictly inside is still an open question for EIT.
4. Application to the GLSM and factorization methods.

4.1.
Application to the GLSM. We recall that the farfield pattern of the green function Φ z is given by, and that that lemma 3.1 give a range characterization of D. In order to use this range characterization the GLSM framework introduce the cost functional J α , defined for g ∈ L 2 (S d−1 ) by From the results of [2], [1] and [3] (partly based on lemmas 3.3, 3.2 and 3.1), we obtain the following characterization of D.
Moreover, there exists δ 0 (α) such that for all δ(α) ≤ δ 0 (α), Hg z,α,δ(α) converges strongly to the solution v of (7) with (f, g) = (Φ z , ∂Φz ∂ν ) as α goes to zero. 4.2. Application to the factorization method. From [16] we have the following theorem for the factorization method: The first three assumptions are direct consequences of lemmas 3.3, 3.2 and 3.1. For the last assumption the application of the Factorization method is more restrictive than the GLSM as it relies on the fact that the real part of T have to be of the form "coercive + compact". In section 3 we have proven that T is actually of the form T 0 + K, where K is compact and T 0 extend assumptions on the contrast in a neighborhood of the boundary ∂D to all D. Therefore for the factorization method to work we need to find a set of assumptions on the contrasts inside all D that ensure that | (T 0 )| is of the form "coercive+compact". Such hypothesis for D A can be found in [7] (Theorem 4.8) and in [16] for D n . Those results allow us to state the following theorem, Numerical experiments. We restrict ourselves to the two dimensional isotropic case (A is the identity matrix) and will introduce the algorithms for the discrete version of the GLSM and FM. We indentify S 1 with the interval [0, 2π[. In order to collect the data of the inverse problem we solve numerically (1) for N incident fields u i ( 2πj N , ·), j ∈ 0, ..., N − 1 using a finite element solver. The discrete version of F is the matrix F := (u ∞ ( 2πj N , 2πk N ) 0≤j,k≤N . We add some noise to the data to build a noisy farfield matrix F δ where (F δ ) j,k = F j,k (1 + σN j,k ) for σ > 0 and N j,k an uniform complex random variable in [−1, 1) 2 . Similarly we consider the discrete version of the green function Φ z (j) = φ z ( 2πj N ) for j ∈ 0, ..., N . We apply both the factorization method and the GLSM to kite shape obstacle where with n = 0.2 except within a disk stricly inside the kite wehe n = 2. We choose N = 100 and a wavelength λ = 2π k = 0.5. We fix the regularization parameter α as explained in [2] for the GLSM and using the Morozov discrepancy principle for the factorization method. Figure 1 shows that there is no significant change in the ability of the methods to reconstruct the inclusion when the contrast changes sign or not. The axes of the figure are measured in λ.