Lamé Parameter Estimation from Static Displacement Field Measurements in the Framework of Nonlinear Inverse Problems

Lamé Parameter Estimation from Static Displacement Field Measurements in the Framework of Nonlinear Inverse Problems

Simon Hubmer111Johannes Kepler University Linz, Doctoral Program Computational Mathematics, Altenbergerstraße 69, A-4040 Linz, Austria (simon.hubmer@dk-compmath.jku.at), corresponding author., Ekaterina Sherina222Technical University of Denmark, Department of Applied Mathematics and Computer Science, Asmussens Allé, 2800 Kongens Lyngby, Denmark (sershe@dtu.dk), Andreas Neubauer333Johannes Kepler University Linz, Industrial Mathematics Institute, Altenbergerstraße 69, A-4040 Linz, Austria (neubauer@indmath.uni-linz.ac.at), Otmar Scherzer444University of Vienna, Computational Science Center, Oskar Morgenstern-Platz 1, 1090 Vienna, Austria (otmar.scherzer@univie.ac.at) 555Johann Radon Institute Linz, Altenbergerstraße 69, A-4040 Linz, Austria (otmar.scherzer@univie.ac.at)
Abstract

The problem of estimating Lamé parameters from full internal static displacement field measurements is formulated as a nonlinear operator equation. The Fréchet derivative and the adjoint of the nonlinear operator are derived. The main theoretical result is the verification of a nonlinearity condition guaranteeing convergence of iterative regularization methods, which is proven in an infinite dimensional context. Furthermore, numerical examples for recovery of the Lamé parameters from simulated displacement data are presented, simulating a static elastography experiment.

Keywords: Elastography, Inverse Problems, Nonlinearity Condition, Linearized Elasticity, Lamé Parameters, Parameter Identification, Landweber Iteration

AMS: 65J22, 65J15, 74G75

1 Introduction

The inverse problem of quantitative elastography consists in estimating material parameters from measurements of displacement data.

In this paper we assume that the model of linearized elasticity, describing the relation between forces and displacements, is valid. Then, quantitative elastography consists in estimating the spatially varying Lamé parameters from displacement field measurements induced by external forces.

There exist a vast amount of literature on identifiability of the Lamé parameters, stability, and different reconstruction methods. See for example [6, 8, 9, 10, 11, 14, 15, 18, 20, 22, 25, 26, 32, 37, 38, 43, 31, 30] and the references therein. Many of the above works deal with the time-dependent equations of linearized elasticity, since the resulting inverse problem is arguably more stable and better to solve. However, in many application including the ones we have in mind, no dynamic, i.e., time-dependent displacement field data is available and hence, one has to work with the static elasticity equations.

In this paper we consider the inverse problem of identifying the Lamé parameters from static displacement field measurements. We reformulate this problem as a nonlinear operator equation

 F(λ,μ)=u, (1.1)

and provide the Fréchet derivative and its adjoint of . For dynamic measurement data of the displacement field , similar investigation have been performed in [31, 30].

The main result of this paper is the verification of the (strong) nonlinearity condition [21] in an infinite dimensional setting, which is the basic assumption guaranteeing convergence of iterative regularization methods. Finally, we present some sample reconstructions with iterative regularization methods from numerically simulated displacement field data.

2 Mathematical Model of Linearized Elasticity

In this section we introduce the basic notation and recall the basic equation of linearized elasticity:

Notation.

denotes a non-empty bounded, open and connected set in , , with a Lipschitz continuous boundary , which has two subsets and , satisfying , and .

Definition 2.1.

Given body forces , displacement , surface traction and Lamé parameters and , the forward problem of linearized elasticity with displacement-traction boundary conditions consists in finding satisfying

 −div(σ(~u))=f,inΩ,~u|ΓD=gD,σ(~u)→n|ΓT=gT, (2.1)

where is an outward unit normal vector of and the stress tensor defining the stress-strain relation in is defined by

 σ(u):=λdiv(u)I+2μE(u),E(u):=12(∇u+∇uT), (2.2)

where is the identity matrix and is called the strain tensor.

It is convenient to homogenize problem (2.1) in the following way: Taking a such that , one then seeks such that

 −div(σ(u))=f+div(σ(Φ)),inΩ,u|ΓD=0,σ(u)→n|ΓT=gT−σ(Φ)→n|ΓT. (2.3)

Throughout this paper, we make the following

Assumption 2.1.

Let , , and . Furthermore, let be such that .

Since we want to consider weak solutions of (2.3), we make the following

Definition 2.2.

Let Assumption 2.1 hold. We define the space

 V:=H10,ΓD(Ω)N,whereH10,ΓD(Ω):={u∈H1(Ω)|u|ΓD=0},

the linear form

 l(v):=⟨f,v⟩H−1(Ω),H1(Ω)+⟨gT,v⟩H−12(ΓT),H12(ΓT), (2.4)

and the bilinear form

 aλ,μ(u,v):=∫Ω(λdiv(u)div(v)+2μE(u):E(v))dx, (2.5)

where the expression denotes the Frobenius product of the matrices and , which also induces the Frobenius norm .

Note that both and are also well defined for .

Definition 2.3.

A function satisfying the variational problem

 aλ,μ(u,v)=l(v)−aλ,μ(Φ,v),∀v∈V, (2.6)

is called a weak solution of the linearized elasticity problem (2.3).

From now on, we only consider weak solutions of (2.3) in the sense of Definition 2.3.

Definition 2.4.

The set of admissible Lamé parameters is defined by

 M(μ––):={(λ,μ)∈L∞(Ω)2|∃0<ε≤μ––c2KN+2c2K:λ≥−ε,μ≥μ––−ε>0}.

Concerning existence and uniqueness of weak solutions, we get the following

Theorem 2.1.

Let the Assumption 2.1 hold and assume that the Lamé parameters for some . Then there exists a unique weak solution of (2.3). Moreover, there exists a constant such that

 ∥u∥H1(Ω)≤cLM(∥f∥H−1(Ω)+cT∥gT∥H−12(ΓT)+(N∥λ∥L∞(Ω)+2∥μ∥L∞(Ω))∥Φ∥H1(Ω)).
Proof.

Using the Cauchy-Schwarz inequality yields

 (2.7)

for all . From this and the trace inequality (5.1), it follows that satisfies the estimate:

 |L(v)|≤|l(v)|+∣∣aλ,μ(Φ,v)∣∣≤(∥f∥H−1(Ω)+cT∥gT∥H−12(ΓT)+(N∥λ∥L∞(Ω)+2∥μ∥L∞(Ω))∥Φ∥H1(Ω))∥v∥H1(Ω).

Since , there exists an such that and . Together with Korn’s inequality (5.4) and (5.3), for all we have

 aλ,μ(v,v)=∫Ωλ(div(v))2+2μ∥E(v)∥2Fdx≥(−εN+2(μ––−ε))∫Ω∥E(v)∥2Fdx≥μ––c2K∥∇v∥2L2(Ω)≥μ––c2K1+c2F∥v∥2H1(Ω),

which shows the coercivity of . Hence, the assertion follows from the Lax-Milgram Lemma applied to and with . ∎

3 The Inverse Problem

After considering the forward problem of linearized elasticity, we now turn to the inverse problem, which is to estimate the Lamé parameters by measurements of the displacement field . More precisely, we are facing the following

Problem.

Let Assumption 2.1 hold and let be a measurement of the true displacement field satisfying

 ∥∥u−uδ∥∥L2(Ω)≤δ, (3.1)

where is the noise level. Given the model of linearized elasticity (2.1) in the weak form (2.6), the problem is to find the Lamé parameters .

The problem of linearized elastography can be formulated as the solution of the operator equation (1.1) with the operator

 F:D(F):={(λ,μ)∈L∞(Ω)2|λ≥0,μ≥μ––>0}→L2(Ω)N,(λ,μ)↦u(λ,μ), (3.2)

where is the solution of (2.6) and hence, we can apply all results from classical inverse problems theory [16], given that the necessary requirements on hold. For showing them, it is necessary to write in a different way: We define the space

 V∗:=(H10,ΓD(Ω)N)∗, (3.3)

which is the dual space of . Next, we introduce the operator connected to the bilinear form , defined by

 ~Aλ,μ:H1(Ω)N→V∗,~v↦(v↦aλ,μ(~v,v)), (3.4)

and its restriction to , i.e., , namely

 Aλ,μ:V→V∗,v↦(¯v↦aλ,μ(v,¯v)). (3.5)

Furthermore, for and , we define the canonical dual

Next, we collect some important properties of and . For ease of notation,

 (3.6)
Proposition 3.1.

The operators and defined by (3.4) and (3.5), respectively, are bounded and linear for all . In particular, for all

 ∥∥A¯λ,¯μ−Aλ,μ∥∥V,V∗≤∥∥~A¯λ,¯μ−~Aλ,μ∥∥H1(Ω),V∗≤∥∥(¯λ,¯μ)−(λ,μ)∥∥∞. (3.7)

Furthermore, for all with , the operator is bijective and has a continuous inverse satisfying . In particular, for all and

 ∥∥A−1¯λ,¯μ¯v∗−A−1λ,μv∗∥∥V≤cLM(∥∥(¯λ,¯μ)−(λ,μ)∥∥∞∥∥A−1λ,μv∗∥∥V+∥¯v∗−v∗∥V∗). (3.8)
Proof.

The boundedness and linearity of and for all are immediate consequences of the boundedness and bilinearity of and we have

 ∥∥~Aλ,μ−~A¯λ,¯μ∥∥H1(Ω),V∗=∥∥~A¯λ−λ,¯μ−μ∥∥H1(Ω),V∗=supu∈H1(Ω),u≠0∥∥~A¯λ−λ,¯μ−μu∥∥V∗∥u∥H1(Ω)=supu∈H1(Ω),u≠0supv∈V,v≠0∣∣a¯λ−λ,¯μ−μ(u,v)∣∣∥u∥H1(Ω)∥v∥V≤∥∥(¯λ,¯μ)−(λ,μ)∥∥∞,

which also translates to , since . Moreover, due to the Lax-Milgram Lemma and Theorem 2.1, is bijective for with and therefore, by the Open Mapping Theorem, exists and is linear and continuous. Again by the Lax-Milgram Lemma, there follows .

Let and with be arbitrary but fixed and consider and . Subtracting those two equations, we get

 Aλ,μu−A¯λ,¯μ¯u=v∗−¯v∗,

which, by the definition of and , can be written as

 A¯λ,¯μ(u−¯u)=A¯λ−λ,¯μ−μu+v∗−¯v∗.

and is equivalent to the variational problem

 a¯λ,¯μ((u−¯u),v)=a¯λ−λ,¯μ−μ(u,v)+⟨v∗−¯v∗,v⟩V∗,V,∀v∈V. (3.9)

Now since is bounded, the right hand side of (3.9) is bounded by

Hence, due to the Lax-Milgram Lemma the solution of (3.9) is unique and depends continuously on the right hand side, which immediately yields the assertion. ∎

Using and , the operator can be written in the alternative form

 F(λ,μ)=A−1λ,μ(l−~Aλ,μΦ), (3.10)

with defined by (2.4). Now since, due to (3.7),

inequality (3.8) implies

 ∥∥F(¯λ,¯μ)−F(λ,μ)∥∥V≤cLM∥∥(¯λ,¯μ)−(λ,μ)∥∥∞(∥F(λ,μ)∥H1(Ω)+∥Φ∥H1(Ω)), (3.11)

showing that is a continuous operator.

Remark.

Note that can also be considered as an operator from to , in which case Theorem 2.1 and Proposition 3.1 guarantee that it remains well-defined and continuous, which we use later on.

3.1 Calculation of the Fréchet Derivative

In this section, we compute the Fréchet derivative of using the representation (3.10).

Theorem 3.2.

The operator defined by (3.10) and considered as an operator from for some is Fréchet differentiable for all with

 F′(λ,μ)(hλ,hμ)=−A−1λ,μ(Ahλ,hμu(λ,μ)+~Ahλ,hμΦ). (3.12)
Proof.

We start by defining

 Gλ,μ(hλ,hμ):=−A−1λ,μ(Ahλ,hμu(λ,μ)+~Ahλ,hμΦ).

Due to Proposition 3.1, is a well-defined, bounded linear operator which depends continuously on with respect to the operator-norm. Hence, if we can prove that is the Gateâux derivative of it is also the Fréchet derivative of . For this, we look at

 F(λ+thλ,μ+thμ)−F(λ,μ)t−Gλ,μ(hλ,hμ)=1t(A−1λ+thλ,μ+thμ(l−~Aλ+thλ,μ+thμΦ)−A−1λ,μ(l−~Aλ,μΦ))+A−1λ,μ(Ahλ,hμu(λ,μ)+~Ahλ,hμΦ). (3.13)

Note that it can happen that . However, choosing small enough, one can always guarantee that , in which case remains well-defined as noted above. Applying to (3.13) we get

 Aλ,μ(F(λ+thλ,μ+thμ)−F(λ,μ)t−Gλ,μ(thλ,thμ))=1t(Aλ,μA−1λ+thλ,μ+thμ(l−~Aλ+thλ,μ+thμΦ)−(l−~Aλ,μΦ))+(Ahλ,hμu(λ,μ)+~Ahλ,hμΦ),

which, together with

 Aλ,μA−1λ+thλ,μ+thμ(l−~Aλ+thλ,μ+thμΦ)=(l−~Aλ+thλ,μ+thμΦ)−tAhλ,hμA−1λ+thλ,μ+thμ(l−~Aλ+thλ,μ+thμΦ),

yields

 Aλ,μ(F(λ+thλ,μ+thμ)−F(λ,μ)t−Gλ,μ(thλ,thμ))=−Ahλ,hμA−1λ+thλ,μ+thμ(l−~Aλ+thλ,μ+thμΦ)+Ahλ,hμu(λ,μ)=−Ahλ,hμ(u(λ+thλ,μ+thμ)−u(λ,μ)). (3.14)

By the continuity of and and due to (3.11) we can deduce that is indeed the Gateâux derivative and, due to the continuous dependence on , also the Fréchet derivative of , which concludes the proof. ∎

Concerning the calculation of , note that it can be carried out in two distinct steps, requiring the solution of two variational problems involving the same bilinear form (which can be used for efficient implementation) as follows:

1. Calculate as the solution of the variational problem (2.6).

2. Calculate as the solution of the variational problem

 aλ,μ(^u,v)=−ahλ,hμ(u,v)−ahλ,hμ(Φ,v),∀v∈V.
Remark.

Note that for classical results on iterative regularization methods (see [28]) to be applicable, one needs that both the definition space and the image space are Hilbert spaces. However, the operator given by (3.2) is defined on . Therefore, one could think of applying Banach space regularization theory to the problem (see for example [40, 29, 41]). Unfortunately, a commonly used assumption is that the involved Banach spaces are reflexive, which excludes . Hence, a commonly used approach is to consider a space which embeds compactly into , for example the Banach space or the Hilbert space with and large enough, respectively. Although it is preferable to assume as little smoothness as possible for the Lamé parameters, we focus on the setting in this paper, since the resulting inverse problem is already difficult enough to treat analytically.

Due to Sobolev’s embedding theorem [1], the Sobolev space embeds compactly into for , i.e., there exists a constant such that

 ∥v∥L∞(Ω)≤csE∥v∥Hs(Ω),∀v∈Hs(Ω). (3.15)

This suggests to consider as an operator from

 Ds(F):={(λ,μ)∈Hs(Ω)2|λ≥0,μ≥μ––>0}→L2(Ω)N, (3.16)

for some . Since due to (3.15) there holds , our previous results on continuity and Fréchet differentiability still hold in this case. Furthermore, it is now possible to consider the resulting inverse problem in the classical Hilbert space framework. Hence, in what follows, we always consider as an operator from for some .

3.2 Calculation of the Adjoint of the Fréchet Derivative

We now turn to the calculation of , the adjoint of the Fréchet derivative of , which is required below. For doing so, note first that for defined by (3.5)

 (3.17)

This follows immediately from the definition of and the symmetry of the bilinear form . Moreover, as an immediate consequence of (3.17), and continuity of it follows

 ⟨v∗,A−1λ,μ¯v∗⟩V∗,V=⟨¯v∗,A−1λ,μv∗⟩V∗,V,∀v∗,¯v∗∈V∗. (3.18)

In order to give an explicit form of we need the following

Lemma 3.3.

The linear operators , defined by

 Tw:=⎛⎜⎝v↦∫Ωw⋅vdx⎞⎟⎠, (3.19)

and ,

 ⟨Esu,v⟩Hs(Ω)=∫Ωuvdx,∀v∈Hs(Ω), (3.20)

respectively, are well-defined and bounded for all .

Proof.

Using the Cauchy-Schwarz inequality it is easy to see that is bounded with . Furthermore, due to (3.15),

 ∫Ωuvdx≤∥u∥L1(Ω)∥v∥L∞(Ω)≤csE∥u∥L1(Ω)∥v∥Hs(Ω),∀v∈Hs(Ω),

and the Lax-Milgram Lemma also is bounded for . ∎

Using this, we can now proof the main result of this section.

Theorem 3.4.

Let with given as in (3.16) for some . Then the adjoint of the Fréchet derivative of is given by

 (3.21)

where and are defined by (3.19) and (3.20), respectively.

Proof.

Using Theorem 3.2 and (3.19) we get

 ⟨F′(λ,μ)(hλ,hμ),w⟩L2(Ω)=⟨−A−1λ,μ(Ahλ,hμu(λ,μ)+~Ahλ,hμΦ),w⟩L2(Ω)=⟨Tw,−A−1λ,μ(Ahλ,hμu(λ,μ)+~Ahλ,hμ)Φ)⟩V∗,V

Together with (3.18) and the definition of and we get

 ⟨Tw,−A−1λ,μ(Ahλ,hμu(λ,μ)+~Ahλ,hμ)Φ)⟩V∗,V=ahλ,hμ(u(λ,μ)+Φ,−A−1λ,μTw)=∫Ωhλdiv(u(λ,μ)+Φ)div(−A−1λ,μTw)dx+∫Ω2hμE(u(λ,μ)+Φ):E(−A−1λ,μTw)dx.

Together with the fact that the product of two functions is in , which applies to and , the statement of the theorem now immediately follows from the definition of (3.20). ∎

Concerning the calculation of , note that it can again be carried out in independent steps, namely:

1. Calculate as the solution of the variational problem (2.6).

2. Compute , i.e., find the solution of the variational problem

 aλ,μ(u(w),v)=∫Ωw⋅vdx,∀v∈V.
3. Compute the functions given by

 u1(w):=div(u+Φ)div(−u(w)),u2(w):=2E(u+Φ):E(−u(w)).
4. Calculate the functions and as the solutions of the variational problems

 ⟨^λ(w),v⟩Hs(Ω)=∫Ωu1(w)vdx,∀v∈Hs(Ω),⟨^μ(w),v⟩Hs(Ω)=∫Ωu2(w)vdx,∀v∈Hs(Ω).
5. Combine the results to obtain .

3.3 Reconstruction of compactly supported Lamé parameters

In many cases, the Lamé parameters are known in a small neighbourhood of the boundary and hence have to be reconstructed only on the remaining part. As a physical problem, we have in mind a test sample consisting of a known material with various inclusions of unknown location and Lamé parameters inside. The resulting inverse problem is better behaved than the original problem and we are even able to prove a nonlinearity condition guaranteeing convergence of iterative solution methods for nonlinear ill-posed problems in this case.

More precisely, assume that we are given a bounded, open, connected Lipschitz domain with and background functions and and assume that the searched for Lamé parameters can be written in the form , where both are compactly supported in . Hence, after introducing the set

 Ds(Fc):={(λ,μ)∈Hs(Ω)2|λ≥−λb,μ≥μ––−μb>0,supp((λ,μ))⊂Ω1},

we define the operator

 Fc:Ds(Fc)→L2(Ω)N,(λ,μ)↦Fc(λ,μ):=F(λb+λ,μb+μ), (3.22)

which is well-defined for . Hence, the sought for Lamé parameters can be reconstructed by solving the problem and taking .

Continuity and Fréchet differentiability of also transfer to . For example,

 F′c(λ,μ)(hλ,hμ)=−A−1(λb+λ,μb+μ)(Ahλ,hμu(λ,μ)+~Ahλ,hμΦ). (3.23)

Furthermore, a similar expression as for the adjoint of the Fréchet derivative of also holds for . Consequently, the computation and implementation of , its derivative and the adjoint can be carried out in the same way as for the operator and hence, the two require roughly the same amount of computational work. However, as we see in the next section, for the operator it is possible to prove a nonlinearity condition.

3.4 Strong Nonlinearity Condition

The so-called (strong) tangential cone condition or (strong) nonlinearity condition is the basis of the convergence analysis of iterative regularization methods for nonlinear ill-posed problems [28]. In the theorem below we show a version of this nonlinearity condition sufficient for proving convergence of iterative methods for the operator .

Theorem 3.5.

Let for some and let be a bounded, open, connected Lipschitz domain with . Then for each there exists a constant such that for all satisfying on and on there holds

 (3.24)
Proof.

Let with such that on and on . For the purpose of this proof, set and . By definition, we have

 ⟨F(λ,μ)−F(¯λ,¯μ)−F′(λ,μ)((λ,μ)−(¯λ,¯μ)),w⟩L2(Ω)=⟨(u−¯u)−A−1λ,μ(A¯λ−λ,¯μ−μu+~A¯λ−λ,¯μ−μΦ),w⟩L2(Ω).

Together with (3.19) and (3.18), we get