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

Abstract

We consider a problem of quantitative static elastography, the estimation of the Lamé parameters from internal displacement field data. This problem is formulated as a nonlinear operator equation. To solve this equation, we investigate the Landweber iteration both analytically and numerically. The main result of this paper is the verification of a nonlinearity condition in an infinite dimensional Hilbert space context. This condition guarantees convergence of iterative regularization methods. Furthermore, numerical examples for recovery of the Lamé parameters from displacement data simulating a static elastography experiment are presented.

Characterization of partial derivatives with respect to material parameters in a fluid–solid interaction problem

“…The interaction between the fluid and the solid is described by the kinematic interface condition (8), and the dynamic one (9), both defined along the interface Γ. To guarantee uniqueness of solutions, the outgoing Sommerfeld radiation condition (10) is imposed at infinity.…”

“…8 The sign convention for T in [34] is different from [12,31]. In either convention, the properties of T are shown by using the multipole expansions of solutions to the Helmholtz equation.…”

“…Figure 1(a). For numerical evaluation using volumed-based discretization, the infinite domain is truncated using absorbing boundary condition (ABC), and the problem we focus on, denoted by OP, is defined on a finite convex domain Ω finite , scattering with penetrable obstacles 1 , to be distinguished with elastic problems on bounded domains such as [8] for elastostatics, [9] in time-harmonic elastodynamics, or [10,11] in elastodynamics.…”

“…Note that most references assume at least C 2 regularity, cf. [6,2], with the exception of [5] for acoustic scattering with Lipschitz obstacles, and [14,15,16] for acoustic-elastic transmission (in fact OP ∞ ) with polygonal-shaped obstacles, and the aforementioned [11,8] for elasticity on bounded Lipschitz domain. Under Lipschitz assumption, the auxiliary problems can contain singular boundary or interface terms that do not fit in the canonical boundary Sobolev spaces, and to give sense to the problem is not trivial (cf.…”

“…These justifications seem to be overlooked in literature, when one works directly with the variational forms, e.g. the aforementioned references [10,11,8], cf. Remark 5.…”