Lamé Parameter Estimation from Static Displacement Field Measurements in the Framework of Nonlinear Inverse Problems
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
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
The main result of this paper is the verification of the (strong) nonlinearity condition  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:
denotes a non-empty bounded, open and connected set in , , with a Lipschitz continuous boundary , which has two subsets and , satisfying , and .
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
where is an outward unit normal vector of and the stress tensor defining the stress-strain relation in is defined by
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
Throughout this paper, we make the following
Let , , and . Furthermore, let be such that .
Since we want to consider weak solutions of (2.3), we make the following
Let Assumption 2.1 hold. We define the space
the linear form
and the bilinear form
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 .
A function satisfying the variational problem
is called a weak solution of the linearized elasticity problem (2.3).
The set of admissible Lamé parameters is defined by
Concerning existence and uniqueness of weak solutions, we get the following
Using the Cauchy-Schwarz inequality yields
for all . From this and the trace inequality (5.1), it follows that satisfies the estimate:
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
The problem of linearized elastography can be formulated as the solution of the operator equation (1.1) with the operator
where is the solution of (2.6) and hence, we can apply all results from classical inverse problems theory , given that the necessary requirements on hold. For showing them, it is necessary to write in a different way: We define the space
which is the dual space of . Next, we introduce the operator connected to the bilinear form , defined by
and its restriction to , i.e., , namely
Furthermore, for and , we define the canonical dual
Next, we collect some important properties of and . For ease of notation,
The boundedness and linearity of and for all are immediate consequences of the boundedness and bilinearity of and we have
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
which, by the definition of and , can be written as
and is equivalent to the variational problem
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
inequality (3.8) implies
showing that is a continuous operator.
3.1 Calculation of the Fréchet Derivative
In this section, we compute the Fréchet derivative of using the representation (3.10).
The operator defined by (3.10) and considered as an operator from for some is Fréchet differentiable for all with
We start by defining
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
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
which, together with
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:
Calculate as the solution of the variational problem (2.6).
Calculate as the solution of the variational problem
Note that for classical results on iterative regularization methods (see ) 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 , the Sobolev space embeds compactly into for , i.e., there exists a constant such that
This suggests to consider as an operator from
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)
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
In order to give an explicit form of we need the following
The linear operators , defined by
respectively, are well-defined and bounded for all .
Using the Cauchy-Schwarz inequality it is easy to see that is bounded with . Furthermore, due to (3.15),
and the Lax-Milgram Lemma also is bounded for . ∎
Using this, we can now proof the main result of this section.
Concerning the calculation of , note that it can again be carried out in independent steps, namely:
Calculate as the solution of the variational problem (2.6).
Compute , i.e., find the solution of the variational problem
Compute the functions given by
Calculate the functions and as the solutions of the variational problems
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
we define the operator
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,
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 . In the theorem below we show a version of this nonlinearity condition sufficient for proving convergence of iterative methods for the operator .
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