An inverse problem in elastography involving Lamé systems

J. Inverse Ill-Posed Probl. 2018; 26(5): 589–605 Research Article Enrique Fernández-Cara and Faustino Maestre* An inverse problem in elastography involving Lamé systems https://doi.org/10.1515/jiip-2017-0065 Received June 21, 2017; revised December 9, 2017; accepted January 25, 2018 Abstract: This paper deals with some inverse problems for the linear elasticity system with origin in elastography: we try to identify the material coefficients from some extra information on (a part of) the boundary. In our main result, we assume that the total variation of the coefficient matrix is a priori bounded. We reformulate the problem as the minimization of a function in an appropriate constraint set. We prove that this extremal problem possesses at least one solution with the help of some regularity results. Two crucial ingredients are a Meyers-like theorem that holds in the context of linear elasticity and a nonlinear interpolation result by Luc Tartar. We also perform some numerical experiments that provide satisfactory results. To this end, we apply the Augmented Lagrangian algorithm, completed with a limited-memory BFGS subalgorithm. Finally, on the basis of these experiments, we illustrate the influence of the starting guess and the errors in the data on the behavior of the iterates. Keywords: Inverse problems, linear elasticity, Lamé systems, bounded variation coefficients, elastography MSC 2010: 35L55, 35R30, 49N45, 74B05 1 Introduction This paper is concerned with some inverse problems for linear elastic materials. These problems are found when we try to apply elastography techniques for instance for breast tumor detection. Elastography is a technique that intends to detect elastic properties of tissue. The basic motivation is that tumor tissue is considerably stiffer than normal soft tissue and, consequently, its resulting deformation after a mechanical action is much smaller. Elastography can be described by the action of three elements: ∙an acoustic wave generator, conceived on the basis of low frequency mechanical excitations, ∙acaptor that detects these waves, ∙a mathematical solver, able to identify tissue stiffness from related measurements. For a more detailed description, see for instance [26, 29, 33, 36]. From the mathematical viewpoint, our main task is to consider and try to solve an inverse problem. The situation is as follows. Let Ω⊂ℝNbe an open, bounded and regular domain which represents an elastic body (N=2or N=3) and let a positive time Tbe given. Let us set Q:=Ω×(0,T)and Σ:=∂Ω×(0,T). Assume that the external sources fm, the initial displacement-velocity couples (u0 m,u1 m)and the boundary data Bm are given for m=1,...,M, where the number Mrepresents the number of measurements. Let us consider Enrique Fernández-Cara, Department of Differential Equations and Numerical Analysis, University of Sevilla, Aptdo. 1160, 41080 Sevilla, Spain, e-mail: c[email protected]. http://orcid.org/0000-0001-6293-9249 *Corresponding author: Faustino Maestre, Department of Differential Equations and Numerical Analysis, University of Sevilla, Aptdo. 1160, 41080 Sevilla, Spain, e-mail: [email protected]. http://orcid.org/0000-0001-6644-0277 Brought to you by | Biblioteca de la Universidad de Sevilla Authenticated Download Date | 11/13/18 8:42 AM 590 |E. Fernández-Cara and F. Maestre, An inverse problem in elastography involving Lamé systems the following systems, which model the behavior of the body under small displacements: { { { { { { { um,tt −∇⋅(Ae(um))=fm,(x,t)∈Q, um=Bm,(x,t)∈Σ, um(x,0)=um,0(x),um,t(x,0)=um,1(x),x∈Ω. (1.1) Here, e(um):=1 2(∇um+∇uT m)and A={Aijkl}1≤i,j,k,l≤nis a fourth-order tensor-valued function on Ω. Then the inverse problem is to find Asuch that usolves (1.1) together with the additional boundary condition Ae(um)⋅ν=Υmon S×(0,T),(1.2) where ν=ν(x)is the outwards directed unit normal vector at points x∈∂Ω,S⊂∂Ωand the Υmare prescribed. In particular, if we assume that Aξ =2μ ξ +λtr(ξ)Id for some scalar functions μand λand all ξ∈ℝN×N, then (1.1) reduces to the usual Lamé system { { { { { { { um,tt −∇⋅(μ(x)(∇um+∇uT m)+λ(x)(∇⋅um)Id)=fm,(x,t)∈Q, u=Bm,(x,t)∈Σ, u(x,0)=um,0(x),ut(x,0)=um,1(x),x∈Ω, (1.3) and (1.2) reads (μ(x)(∇um+∇uT m)+λ(x)(∇⋅um)Id)⋅ν=Υmon S×(0,T)(1.4) and the associated inverse problem is to find μand λsuch that the solution to (1.3) satisfies (1.4). The literature concerning inverse problems of these kinds is large, in particular in the case of (1.3)–(1.4); see for instance [2, 3, 18, 19, 21, 36]. There are more sophisticated models that take into account visco-elastic effects, porosity, etc. and have been proposed in other papers; see for instance [22, 34, 37]. The main questions concerning these inverse problems are uniqueness, stability and reconstruction. In this paper, we will focus mainly on the third one. In general terms, the uniqueness problem is as follows: Problem. Assume that Υmand Υ󸀠 mare given and let (μ,λ)and (μ󸀠,λ󸀠)solve the corresponding associated inverse problems. Do we necessarily have (μ,λ)=(μ󸀠,λ󸀠)? Here, the number Mof measurements or experiments plays a fundamental role. Thus, in [15], the authors established the uniqueness of determining one single coefficient using three measurements (M=3). Later, in [20], with the same amount of measurements, the uniqueness of all the coefficients of the system was established. On the other hand, the stability of (1.3)–(1.4) has been analyzed recently in several papers. In a typical stability result, it is proved that the “distance” between solutions (μ,λ)and (μ󸀠,λ󸀠)can be bounded by a function of the “distance” between data {Υm: 1 ≤m≤M}and {Υ󸀠 m: 1 ≤m≤M}in a neighborhood of a fixed {Υ0 m: 1 ≤m≤M}. Since stability implies uniqueness, this is an important question in applications to geophysics, material sciences or medicine. In [16], conditional stability and uniqueness for all the coefficients of the system with two measurements (M=2) is demonstrated. In [17], the authors proved conditional stability results with one single interior measurement, provided the initial data satisfy some nondegeneracy condition. In [4], the authors proved a logarithmic stability estimate for the Lamé coefficients again with M=1assuming that the data are known in a neighborhood of the boundary of the spatial domain. See [1, 3, 5, 12, 13, 18, 23, 24, 27] for other results. We will work with systems of the form (1.1), with M=2,fm,fm,t∈L2(Q)N,um,0=0,um,1∈H1 0(Ω)N,B=0,Υ∈L2(Σ)N(1.5) and A∈𝕄(α,β;Ω)∩𝔹𝕍(Ω),(1.6) for some 0<α<β. Here, we have denoted by 𝕄(α,β;Ω)the family of all measurable tensors Asatisfying ∙A(x)=A(x)T,A(x)Λ⋅Λ≥α|Λ|2, ∙|A(x)Λ|≤β|Λ|a.e. in Ωfor all symmetric Λ∈ℝN×N; see the beginning of Section 2 for the notation. Brought to you by | Biblioteca de la Universidad de Sevilla Authenticated Download Date | 11/13/18 8:42 AM E. Fernández-Cara and F. Maestre, An inverse problem in elastography involving Lamé systems |591 It is known that, under assumptions (1.5)–(1.6), there exists at most one solution uto the inverse problem (1.1)–(1.2); see [4, 16]. On the other hand, the direct problem associated to (1.1) is well posed. More precisely, under the same assumptions, for every mthere exists a unique solution um, with um,um,t∈C0([0,T];H1 0(Ω)N),um,tt,∇⋅(Ae(um))∈C0([0,T];L2(Ω)N).(1.7) This is a consequence of the standard semigroup theory. Indeed, it suffices to introduce the Hilbert space Y:=H1 0(Ω)N×L2(Ω)N, the unbounded operator A:D(A)⊂Y󳨃→Ywith {D(A):={(v0,v1)∈Y:∇⋅(Ae(v0))∈L2(Ω)N,v1∈H1 0(Ω)N}, A(v0,v1):=(v1,∇⋅(Ae(v0))), the initial data y0:=(0,um,1)(which belongs to D(A)) and the right-hand side F:=(0,fm)and consider the Cauchy problem in Y {lyt=Ay+F(t),t∈(0,T), y(0)=y0. Moreover, we have (Ae(um))⋅ν∈L2(0,T;H−1/2(∂Ω)N),m=1,2; we will justify this below, in Section 3. Therefore, in order to solve the inverse problem associated to (1.1) and (1.2), we can introduce the cost function I(A):=1 2 T ∫ 0 2 ∑ m=1‖Ae(um)⋅ν|S−Υm‖2dt (1.8) (where Asatisfies (1.6), umsolve the associated system (1.1) and ‖⋅‖is the norm in H−1/2(S)N) and formulate the following (direct) extremal problem: { { { Minimize I(A) subject to (1.6) and (1.1).(1.9) The following assertions are obvious: ∙If Ais a solution to the inverse problem associated to (1.1)–(1.2), then Aalso solves (1.9). ∙Conversely, if Asolves (1.9) and I(A)=0(which can be expected for realistic data Υm), then Ais also the unique solution to the original inverse problem. Therefore, under the uniqueness assumptions (1.5)–(1.6) for the inverse problem, it is completely meaningful to try to solve (1.9). This will be the adopted viewpoint and the goal of this paper. The rest of this paper is organized as follows. In Section 2, we introduce the appropriate notation, we recall some preliminary results and we state our main result; this is an existence theorem for a modified version of (1.9). Section 3 is devoted to the proof of this result. The main ingredients are a regularity result for the linear elasticity system (1.1) of the Meyers kind and an (abstract) nonlinear interpolation result by Luc Tartar. In Section 4, we present some numerical results. Finally, Section 5 deals with some additional comments and questions. 2 Preliminaries, notation and main result Let us first recall some definitions and properties needed to analyze the PDEs of linear elasticity. In the sequel, it will be assumed that Ω⊂ℝNpossesses a regular boundary, at least of class W2,∞. Suppose that A=(Aijkl)1≤ijkl≤Nis a fourth-order tensor, Λ, Ξ ∈ℝN×Nand u=(u1,...,uN)is a vector-valued function. The following notation will be used: ∙AΛstands for the second-order tensor whose (i,j)component is (AΛ)ij :=∑1≤k,ℓ≤NAijkℓΛkℓ. ∙Ξ⋅Λ:=∑1≤i,j≤NΞijΛij. Brought to you by | Biblioteca de la Universidad de Sevilla Authenticated Download Date | 11/13/18 8:42 AM 592 |E. Fernández-Cara and F. Maestre, An inverse problem in elastography involving Lamé systems ∙We set (∇u)ij :=∂ui ∂xj ,(∇⋅Ξ)i:=∑ 1≤j≤N ∂Ξij ∂xj ,(DA)ijkℓm:=∂Aijkℓ ∂xm . ∙For 1≤p<+∞, we consider the usual Banach spaces Lp(Ω),Lp(Ω)Nand Lp(Ω)N×N, endowed with the norms ‖φ‖Lp(Ω):=(∫ Ω|φ|pdt)1 p , ‖u‖Lp(Ω)N:=(∑ 1≤i≤N∫ Ω|ui|pdt)1 p , ‖Ξ‖Lp(Ω)N×N:=(∑ 1≤i,j≤N∫ Ω|Ξij|pdt)1 p . We denote by W1,p(Ω),W1,p(Ω)Nand W1,p(Ω)N×Nthe similar Sobolev spaces. ∙By definition, 𝕃p(Ω),𝕎1,p(Ω)and 𝔹𝕍(Ω)are the spaces of x-dependent fourth-order tensors with components in Lp(Ω),W1,p(Ω)and BV(Ω), respectively. We will use the index Sto denote symmetry; thus, for example, 𝕃2 S(Ω)denotes the space of symmetric matrix-valued functions whose components belong to L2(Ω). For any A∈𝔹𝕍(Ω), we denote by TV(A)the corresponding total variation of A. ∙Finally, for any Banach space X, any T>0and any p∈[1,+∞),Lp(0,T;X)stands for the usual space of measurable and p-integrable (classes of) functions u:[0,T]󳨃→X. In this paper, our main goal is to analyze a problem of the kind (1.9), where Iis given by (1.8). Our main result is the following: Theorem 2.1. Let us set 𝕂(R)={A∈𝕄(α,β;Ω)∩𝔹𝕍(Ω):TV(A)≤R}(2.1) and let us assume that the data of the problem satisfy (1.5). Then, for any R>0, the extremal problem { { { Minimize I(A) subject to A∈𝕂(R)and (1.1) (2.2) possesses at least one solution A∗ R. Remark 2.2. Obviously, the crucial assumption in Theorem 2.1 is the uniform bound of TV(A). This is just what we need to get compactness in the appropriate space (see below). We can consider other similar extremal problems for which the existence of a solution can be established; see Section 5 for more details. Remark 2.3. In Theorem 2.1, we assert that at least one A∗ Rexists, but no uniqueness is ensured. However, if I(A∗ R)=0, in view of the uniqueness of the inverse problem, A∗ Ris also the unique minimizer of Iin 𝕂(R). Remark 2.4. Let us consider again the particular case of an isotropic media and the associated inverse problem (1.3)–(1.4). It makes sense to introduce L(α,β;Ω):={a∈L∞(Ω):α≤a(x)≤βa.e.}, 𝕂0(R):={(μ,λ)∈L(α,β;Ω)×L(α,β;Ω):TV(μ),TV(λ)≤R} and the corresponding (simplified) extremal problem { { { Minimize I0(μ,λ) subject to (μ,λ)∈𝕂0(R)and (1.3). (2.3) The arguments in the proof of Theorem 2.1 can also be used to establish the existence of a minimizer (μ∗ R,λ∗ R). Again, if I0(μ∗ R,λ∗ R)=0, then (μ∗ R,λ∗ R)is the unique solution to the inverse problem (1.3)–(1.4). Brought to you by | Biblioteca de la Universidad de Sevilla Authenticated Download Date | 11/13/18 8:42 AM E. Fernández-Cara and F. Maestre, An inverse problem in elastography involving Lamé systems |593 3 Proof of Theorem 2.1 Let us recall that, for any solution to the state equation (1.1), Ae(um)⋅νis a well-defined object in H−1/2(∂Ω)N for each t∈[0,T]. In fact, Ae(um)⋅νis well defined by duality in H−1/2(∂Ω)Nthrough the identities ⟨Ae(um)⋅ν,z⟩H−1/2,H1/2=⟨∇⋅(Ae(um)),z⟩+∫ Ω Ae(um)⋅∇zfor all z∈H1(Ω)N, which are completely meaningful, in view of the regularity of the solutions to (1.1) and, more precisely, (1.7). Let {An}be a minimizing sequence for Iin 𝕂(R). Then, at least for a subsequence, we have the following for some A∗∈𝕂(R): { { { An→A∗weakly-∗in 𝔹𝕍(Ω), An→A∗strongly in 𝕃p(Ω)for all p∈[1,+∞)and a.e. (3.1) Let us denote by un mthe states associated to An. Then, from the standard energy estimates, it can also be assumed that {un m→u∗ mweakly-∗in L∞(0,T;H1 0(Ω)N), un m,t→u∗ m,tweakly-∗in L∞(0,T;L2(Ω)N).(3.2) Let us prove that the u∗ mare the states associated to A∗, i.e. { { { { { { { u∗ m,tt −∇⋅(A∗e(u∗ m))=fm(x,t),(x,t)∈Q, u∗ m=0,(x,t)∈Σ, u∗ m(x,0)=0,u∗ m,t(x,0)=um,1(x),x∈Ω. (3.3) From the second assertion in (3.1) and the first one in (3.2), we see that Ane(un m)→A∗e(u∗ m)weakly in Lp1(0,T;Lp2(Ω)N)for all p1∈[1,+∞)and all p2∈[1,2). Consequently, we can pass to the limit in the equations and boundary and initial conditions satisfied by un m and deduce that u∗ mis the unique solution to (3.3). In order to achieve the proof of Theorem 2.1, we have to show that lim inf n→+∞I(An)≥I(A∗). Obviously, this will be the case if we are able to prove that Ane(un m)⋅ν→A∗e(u∗ m)⋅νweakly in L2(0,T;H−1/2(∂Ω)N). Taking into account the definition of Ane(un m)⋅ν, we have ⟨Ane(un m)⋅ν,z⟩=⟨∇⋅(Ane(un m)),z⟩+∬ Q Ane(un m)⋅∇zfor all z∈L1(0,T;H1(Ω)N). Therefore, since ∇⋅(Ane(un m))=un m,tt −fmconverges weakly-∗in L∞(0,T;L2(Ω)N), it will suffice to show that the ∇un m(and therefore the e(un m)) belong to a compact set in L2(Q)N×N. In fact, we are now going to prove a slightly stronger property: that un mbelongs to a compact set in C0([0,T];X)for a Hilbert space X that is compactly embedded in H1 0(Ω)N. To this end, let us recall some (classical) notation: for any couple of Banach spaces E0and E1, any θ∈(0,1)and any p∈[1,+∞],[E0,E1]θ,pwill stand for the usual associated interpolation space of Petree, whenever this makes sense. We will need the following result: Lemma 3.1. Assume that A∈𝕂(R). There exists δ∈(0,1), only depending on α,βand R, such that, for any h∈L2(Ω)N, the elliptic system {−∇⋅(Ae(w))=h,x∈Ω, w=0,x∈∂Ω, (3.4) Brought to you by | Biblioteca de la Universidad de Sevilla Authenticated Download Date | 11/13/18 8:42 AM 594 |E. Fernández-Cara and F. Maestre, An inverse problem in elastography involving Lamé systems possesses exactly one solution wh∈Xδ:=[D(∆),H1 0(Ω)N]δ,∞, where D(∆)=H1 0(Ω)N∩H2(Ω)Nis the domain of the Dirichlet Laplacian. Furthermore, the mapping h󳨃→whis linear and continuous, i.e. ‖wh‖Xδ≤C(N, Ω, α,β,R)‖h‖L2for all h∈L2(Ω)N.(3.5) In view of this lemma, un mis uniformly bounded in L∞(0,T;Xδ). For any ρ∈(δ,1), one has Xδ󳨅→Xρ:=[D(∆),H1 0(Ω)N]ρ,∞󳨅→H1 0(Ω)N, where the embeddings are compact. On the other hand, un m,tis uniformly bounded in L∞(0,T;L2(Ω)N). Therefore, from well-known compactness results in spaces of the form C0([0,T];B)(see for instance [35]), we deduce that the un mbelong to a compact set of C0([0,T];Xρ)for any ρ∈(δ,1). This proves our assertion and ends the proof of Theorem 2.1. Proof of Lemma 3.1. We will use the following facts: Proposition. When A∈𝕄(α,β;Ω), then (3.4) is uniquely solvable in H1 0(Ω)Nfor each h∈L2(Ω)N, with estimates involving only Ω,αand ‖h‖L2(Ω): ‖wh‖H1 0(Ω)N≤C(Ω, α)‖h‖L2(Ω)Nfor all h∈L2(Ω)N. In fact, from Meyers-like estimates for elasticity systems, we can even get something better; see (3.7). Let us explain this. First, let us recall these estimates: Theorem 3.2. Assume that A∈𝕄(α,β;Ω)and g∈L2(Ω)N×Nand let ube the solution to {−∇⋅(Ae(u))=∇⋅g,x∈Ω, u=0,x∈∂Ω. (3.6) There exists pM>2, depending only on N,Ω,αand β, such that, for all p∈[2,pM)and any g∈Lp S(Ω)N×N, the corresponding solution to (3.6) belongs to W1,p 0(Ω)Nand satisfies ‖u‖W1,p(Ω)N≤C(p,N, Ω, α,β)‖g‖Lp(Ω)N×N. Recall that the original Meyers’ Theorem deals with scalar elliptic problems, see [28]. It has been established in the context of linear elasticity in [6] and [10]. The scalar version was used in [14] to prove a result similar to Theorem 2.1, concerning an inverse problem for the wave equation. In order to use this result in the context of (3.4), it will be enough to check that, for each h∈L2(Ω)N, there exist tensor-valued functions g∈Lp S(Ω)N×Nwith p>2such that ∇⋅g=h,‖g‖Lp(Ω)N×N≤C(p,N, Ω)‖h‖L2(Ω)N. But this is true. Indeed, it suffices to take g=e(ϕ), where ϕis, for instance, the unique solution to {−∇⋅(e(ϕ))=h,x∈Ω, ϕ=0,x∈∂Ω. Thus, it is clear that there exists p>2such that, for all h∈L2(Ω)N, the associated solution to (3.4) satisfies wh∈W1,p 0(Ω)Nand ‖wh‖W1,p(Ω)N≤C(p,N, Ω, α,β)‖h‖L2(Ω)N.(3.7) Now, let us introduce r=2p p−2(3.8) and let us fix h∈L2(Ω)N. Assume that Aand A󸀠are given in 𝕄(α,β;Ω)and let us denote by wh(respectively, w󸀠 h) the solution to the elliptic problem (3.4) corresponding to A(respectively, A󸀠). Then the following holds: α‖e(wh−w󸀠 h)‖2 L2(Ω)N≤∫ Ω A󸀠e(wh−w󸀠 h)⋅e(wh−w󸀠 h) =∫ Ω(A󸀠−A)e(wh)⋅e(wh−w󸀠 h) ≤‖A󸀠−A‖𝕃r(Ω)‖e(wh)‖Lp(Ω)N‖e(wh−w󸀠 h)‖L2(Ω)N. Brought to you by | Biblioteca de la Universidad de Sevilla Authenticated Download Date | 11/13/18 8:42 AM E. Fernández-Cara and F. Maestre, An inverse problem in elastography involving Lamé systems |595 From Korn’s inequality, we also get ‖w󸀠 h−wh‖H1 0(Ω)N≤C(p,N, Ω, α,β)‖A󸀠−A‖𝕃r(Ω)‖h‖L2(Ω)N.(3.9) This estimate will be used below. Proposition. When ris given by (3.8) and A∈𝕄(α,β;Ω)∩𝕎1,r(Ω), then system (3.4) is uniquely solvable in D(∆)=H2(Ω)N∩H1 0(Ω)Nfor all h∈L2(Ω), with estimates involving only p,N,Ω,α,β,‖DA‖𝕃r(Ω)Nand ‖h‖L2(Ω)N: ‖wh‖H2(Ω)N≤C(p,N, Ω, α,β)(1+‖DA‖𝕃r(Ω)N)‖h‖L2(Ω)N.(3.10) This is a consequence of (3.7) and the standard linear elasticity regularity theory, see for instance [11, 32]. The basic ideas of the argument are the following. After introducing a partition of unit, the first task is to prove that any difference quotient τi,ε(ϕwh)with (τi,εwh)(x):=1 ε(wh(x+εei)−wh(x)) and ϕ∈D(Ω)satisfies an estimate of the form ‖∇τi,ε(ϕwh)‖L2(Ω)N×N≤C(p,N, Ω, α,β)(1+‖DA‖𝕃r(Ω)N)‖h‖L2(Ω)N for any sufficiently small ε>0. Notice that −∇⋅(A(x)e(τi,ε(ϕwh)))=τi,ε(ϕh)+∇⋅(τi,εA e(τi,ε(ϕwh))+⋅⋅⋅, where the dots contain lower order terms. Consequently, α‖e(τi,ε(ϕwh))‖2 H1 0(Ω)N×N≤C‖ϕh‖L2(Ω)N‖e(τi,ε(ϕwh))‖H1 0(Ω)N×N +‖τi,εA‖𝕃r(Ω)‖e(τi,ε(ϕwh))‖W1,p(Ω)N×N‖e(τi,ε(ϕwh))‖H1 0(Ω)N×N+⋅⋅⋅ ≤C(p,N, Ω, α,β)(1+‖DA‖𝕃r(Ω))‖h‖L2(Ω)N‖e(τi,ε(ϕwh))‖H1 0(Ω)N×N+⋅⋅⋅. This, used together with the standard Korn’s inequality, furnishes a satisfactory estimate of ‖ϕwh‖H2(Ω)N. Similar estimates can be obtained near the boundary using that ∂Ω∈W2,∞. Collecting all together, we easily get (3.10). In the sequel, we will take ras in (3.8), where pis the exponent found above. Proposition. The assumptions in Lemma 3.1 imply in particular that Abelongs to a (Besov) space that can be obtained by applying the real interpolation method of Peetre to 𝕎1,r(Ω)and 𝕃r(Ω). More precisely, we have A∈𝔹𝕍(Ω)∩𝕃∞(Ω)⊂[𝕎1,p(Ω),𝕃p(Ω)]1/p󸀠,∞∩𝕃∞(Ω) for any p∈[1,+∞); see for instance [7, 9, 39]. In particular, A∈[𝕎1,r(Ω),𝕃r(Ω)]1/r󸀠,∞∩𝕃∞(Ω). Thus, it is reasonable to expect that the associated solutions belong to Xδ. In order to prove this, let us recall the following nonlinear interpolation result by Luc Tartar [38] (see also [39]): Theorem 3.3. Let Eiand Fibe Banach spaces, with E0⊂E1and F0⊂F1. Let U⊂E1be a nonempty open set and let S:U󳨃→F1be a mapping. We will assume that: (1) Smaps U∩E0into F0. (2) There exist λ∗∈(0,1]and μ∗>0such that, for any A∈U, we can find a neighborhood of VA⊂Uand a constant CAwith the following properties: (a) ‖S(A󸀠)−S(A)‖F1≤CA‖A󸀠−A‖λ∗ E1for all A󸀠∈VA. (b) ‖S(A󸀠)‖F0≤CA(1+‖A󸀠‖μ∗ E0)for all A󸀠∈VA∩E0. Then, for any θ∈(0,1), any p∈[1,+∞]and any A∈[E0,E1]θ,p∩U, one has that S(A)∈[F0,F1]η,q, where 1−η η=1−θ θ λ∗ μ∗,q=max (1,(1−θ μ∗+θ λ∗)p). Brought to you by | Biblioteca de la Universidad de Sevilla Authenticated Download Date | 11/13/18 8:42 AM 596 |E. Fernández-Cara and F. Maestre, An inverse problem in elastography involving Lamé systems Let us fix h∈L2(Ω)N, let us take in this result E0=𝕎1,r(Ω),E1=𝕃r(Ω),F0=D(∆),F1=H1 0(Ω)and U=𝕃r(Ω) and let us introduce the mapping S, with S(A)=w⇐⇒ {−∇⋅(Tα,β(A)e(w))=h,x∈Ω, w=0,x∈∂Ω. Here, Tα,β(A)stands for the following function: (Tα,β(A))ijkl ={ { { { { { { αif Aijkl <α, Aijkl if α≤Aijkl ≤β, βif Aijkl >β, 1≤i,j,k,l≤N. In view of (3.9) and (3.10), we see that all the hypotheses of Theorem 3.3 are satisfied with λ∗=μ∗=1, VA=U=𝕃r(Ω)and CA=C(Ω, N,α,β)‖h‖L2(Ω)Nfor any A∈𝕃r(Ω). Indeed, for any h∈L2(Ω)None has ‖S(A󸀠)−S(A)‖H1 0(Ω)N≤C(Ω, N,α,β)‖h‖L2(Ω)N‖Tα,β(A󸀠)−Tα,β(A)‖𝕃r(Ω) ≤C(Ω, N,α,β)‖h‖L2(Ω)N‖A󸀠−A‖𝕃r(Ω) and ‖S(A󸀠)‖H2(Ω)N≤C(Ω, N,α,β)‖h‖L2(Ω)N(1+‖DTα,β(A󸀠)‖𝕃r(Ω)) ≤C(Ω, N,α,β)‖h‖L2(Ω)N(1+‖DA󸀠‖𝕃r(Ω)) for all A,A󸀠∈𝕃r(Ω). Therefore, for any A∈𝔹𝕍(Ω)∩𝕄(α,β;Ω), the associated solution to (3.4) satisfies wh=S(A)∈[D(∆),H1 0(Ω)N]1/r󸀠,∞, i.e. wh∈Xδfor δ=1 r󸀠=p−2 2p. Recall that ‖wh‖Xδ=‖ς−δK(ς;wh)‖L∞(ℝ+), where for any ς≥0and w∈H1 0(Ω)None has K(ς;w):=inf{‖w0‖H2(Ω)N+ς‖w1‖H1 0(Ω)N:w=w0+w1,w0∈D(∆),w1∈H1 0(Ω)N}. On the other hand, for any A∈Eδ:=[𝕎1,r(Ω),𝕃r(Ω)]δ,∞, we have ‖A‖Eδ=‖ς−δK0(t;A)‖L∞(ℝ+) with K0(ς;A):=inf{‖A0‖𝕎1,r(Ω)+ς‖A1‖𝕃r(Ω):A=A0+A1,A0∈𝕎1,r(Ω),A1∈𝕃r(Ω)}. Taking into account that, for any A=A0+A1∈𝔹𝕍(Ω)∩𝕄(α,β;Ω), one has wh=S(A)=S(A0)+[S(A)−S(A0)] and, using again (3.10) and (3.9), we see that K(ς;wh)=K(ς;S(A))≤C(Ω, N,α,β)‖h‖L2(Ω)N(1+K0(ς;A)). Therefore, ‖ς−δK(ς;wh)‖L∞(ℝ+)≤C(Ω, N,α,β)‖h‖L2(Ω)N(1+‖ς−δK0(ς;A)‖L∞(ℝ+)), which yields (3.5). This ends the proof of Lemma 3.1. Brought to you by | Biblioteca de la Universidad de Sevilla Authenticated Download Date | 11/13/18 8:42 AM E. Fernández-Cara and F. Maestre, An inverse problem in elastography involving Lamé systems |597 4 Some numerical results In this section, we present and apply a numerical method for the solution of (2.2). First, we will indicate the way we can compute the gradient of the cost function. Secondly, we will describe briefly an iterative algorithm that leads to good numerical results. This will be illustrated with some numerical experiments in the particular case of an elastic body governed by the Lamé system, i.e. for tensors Asatisfying Aξ =2μξ +λontr(ξ)Id for some real λand μand for all ξ∈ℝN×N. 4.1 The computation of the gradient Let R>0be given. Our aim is to solve numerically the extremal problem (2.2): { { { Minimize I(A) subject to A∈𝕂(R)and (1.1). Recall (1.8) and (2.1) for the definitions of I(A)and 𝕂(R), respectively. Let us assume that A∈𝕂(R)and A󸀠∈𝕃∞ S(Ω)∩𝔹𝕍(Ω)and let us compute formally the derivative of the cost function at Ain the direction A󸀠. More precisely, let us see that, under appropriate regularity hypotheses on Aand the associated states um, the derivative of Iwith respect to Ain the direction A󸀠exists and takes the form dI(A) dA ⋅A󸀠=2 ∑ m=1∫∫ Q A󸀠e(um)⋅e(pm),(4.1) where, for each m,pmis the unique solution to the adjoint system { { { { { { { pm.tt −∇⋅(Ae(pm))=0,(x,t)∈Q, pm=χ−1((Ae(um)⋅ν−Υm)1S),(x,t)∈Σ, pm(x,0)=0,pm,t(x,0)=0,x∈Ω. (4.2) Here, we have denoted by χthe canonical isomorphism from H1/2(∂Ω)Nonto H−1/2(∂Ω)N. Indeed, for any small s=0, let us set As:=A+sA󸀠, let us denote by us mthe solutions to the associated system (1.1) and let us put us m=um+szs m. Note that zs msatisfies { { { { { { { zs m,tt −∇⋅(Ase(zs m))=∇⋅(A󸀠e(um)),(x,t)∈Q, zs m=0,(x,t)∈Σ, zs m(x,0)=0,zs m,t(x,0)=0,x∈Ω. Consequently, zs mand zs m,tt are uniformly bounded in C0([0,T];H1 0(Ω)N)and C0([0,T];H−1(Ω)N), respectively. We will assume that ‖us m,tt −um,tt‖C0([0,T];L2(Ω)N)≤Cs for some constant C>0. Note that this is equivalent to supposing that zs m,tt is uniformly bounded in the space C0([0,T];L2(Ω)N). As a consequence, ∇⋅(A󸀠e(um)+Ase(zs m))is bounded in this space and the normal trace (A󸀠e(um)+Ase(zs m))⋅νis uniformly bounded in C0([0,T];H−1/2(∂Ω)N). Observe that 1 s(I(A+sA󸀠)−I(A))=1 2s 2 ∑ m=1 T ∫ 0[‖Ase(us m)⋅ν|S−Υm‖2−‖Ae(um)⋅ν|S−Υm‖2]dt =2 ∑ m=1 T ∫ 0((A󸀠e(um)+Ase(zs m))⋅ν|S,Ae(um)⋅ν|S−Υm)H−1/2dt +s 2 2 ∑ m=1 T ∫ 0‖(A󸀠e(um)+Ase(zs m))⋅ν|S‖2dt =B1(s)+B2(s).(4.3) Brought to you by | Biblioteca de la Universidad de Sevilla Authenticated Download Date | 11/13/18 8:42 AM 604 |E. Fernández-Cara and F. Maestre, An inverse problem in elastography involving Lamé systems [5] M. Bellassoued, D. Jellali and M. Yamamoto, Lipschitz stability in an inverse problem for a hyperbolic equation with a finite set of boundary data, Appl. Anal. 87 (2008), no. 10–11, 1105–1119. [6] A. Bensoussan, J.-L. Lions and G. 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