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.
Funding statement: Partially supported by grants MTM2016-76990-P (DGI-MICINN, Spain) and MTM2014-53309-P (DGI-MINECO, Spain).
References
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Articles in the same Issue
- Frontmatter
- Identification of mathematical model of bacteria population under the antibiotic influence
- On the determination of differential pencils with nonlocal conditions
- An inverse problem in elastography involving Lamé systems
- Solution of the inverse seismic problem in a layered elastic medium by means of the τ-p Radon transform
- An adaptive multigrid conjugate gradient method for the inversion of a nonlinear convection-diffusion equation
- Ambarzumyan-type theorems on a time scale
- A converse result for Banach space convergence rates in Tikhonov-type convex regularization of ill-posed linear equations
- Lipschitz stability estimates in inverse source problems for a fractional diffusion equation of half order in time by Carleman estimates
- Inverse dynamic and spectral problems for the one-dimensional Dirac system on a finite tree
- Phaseless inverse problems with interference waves
- Quasi-solution of linear inverse problems in non-reflexive Banach spaces
Articles in the same Issue
- Frontmatter
- Identification of mathematical model of bacteria population under the antibiotic influence
- On the determination of differential pencils with nonlocal conditions
- An inverse problem in elastography involving Lamé systems
- Solution of the inverse seismic problem in a layered elastic medium by means of the τ-p Radon transform
- An adaptive multigrid conjugate gradient method for the inversion of a nonlinear convection-diffusion equation
- Ambarzumyan-type theorems on a time scale
- A converse result for Banach space convergence rates in Tikhonov-type convex regularization of ill-posed linear equations
- Lipschitz stability estimates in inverse source problems for a fractional diffusion equation of half order in time by Carleman estimates
- Inverse dynamic and spectral problems for the one-dimensional Dirac system on a finite tree
- Phaseless inverse problems with interference waves
- Quasi-solution of linear inverse problems in non-reflexive Banach spaces