Solution of inverse problems in elasticity imaging using the adjoint method

Abstract

We consider the problem of determining the shear modulus of a linearelastic, incompressible medium given boundary data and one component of the displacement field in the entire domain. The problem is derived from applications in quantitative elasticity imaging. We pose the problem as one of minimizing a functional and consider the use of gradient-based algorithms to solve it. In order to calculate the gradient efficiently we develop an algorithm based on the adjoint elasticity operator. The main cost associated with this algorithm is equivalent to solving two forward problems, independent of the number of optimization variables. We present numerical examples that demonstrate the effectiveness of the proposed approach.

Elastic modulus imaging: some exact solutions of the compressible elastography inverse problem

“…In that case, only two components of displacement are needed in the plane; see section 6 below. Finally, the exact solution may be useful in benchmarking other computational approaches based, for example, on iterative optimization [3,4,5,6,7].…”

Combined distributed parameters and source estimation in tokamak plasma heat transport

“…This led to a purely nonlinear parameter estimation problem. Note also that, in previous works [10], [11] where the problem was tackled in the infinite dimensional framework, the source term was considered as a known variable and the estimation was computed using nonlinear optimization algorithms performing well for small scaled systems. The main contribution of our formulation is first to carry out the projection only in space, which allows to finally deal with a linear state-space system with time-varying coefficients.…”

Non-identifiability of the Rayleigh damping material model in Magnetic Resonance Elastography

“…In MRE, the number of measurements corresponds to the number of measured displacements. The minimisation is performed by updating θ using the conjugate-gradient (CG) method [40]. Calculation of u c i (θ) is referred as the forward computational problem, and the process of iterative estimation of the material property parameters θ that minimize Eq.…”