Monotonicity-Based Regularization for Shape Reconstruction in Linear Elasticity
arXiv:2105.01941 · doi:10.1007/s00466-021-02121-2
Abstract
We deal with the shape reconstruction of inclusions in elastic bodies. For solving this inverse problem in practice, data fitting functionals are used. Those work better than the rigorous monotonicity methods from [5], but have no rigorously proven convergence theory. Therefore we show how the monotonicity methods can be converted into a regularization method for a data-fitting functional without losing the convergence properties of the monotonicity methods. This is a great advantage and a significant improvement over standard regularization techniques. In more detail, we introduce constraints on the minimization problem of the residual based on the monotonicity methods and prove the existence and uniqueness of a minimizer as well as the convergence of the method for noisy data. In addition, we compare numerical reconstructions of inclusions based on the monotonicity-based regularization with a standard approach (one-step linearization with Tikhonov-like regularization), which also shows the robustness of our method regarding noise in practice.
arXiv comment26 pages, 15 figures
References in corpus (8)
- Monotonicity based shape reconstruction in electrical impedance tomography
- Enhancing residual-based techniques with shape reconstruction features in Electrical Impedance Tomography
- Lamé Parameter Estimation from Static Displacement Field Measurements in the Framework of Nonlinear Inverse Problems
- Shape Reconstruction in Linear Elasticity: Standard and Linearized Monotonicity Method
- Lipschitz stability estimate and reconstruction of Lamé parameters in linear elasticity
- Lipschitz continuous dependence of piecewise constant Lamé coefficients from boundary data: the case of non flat interfaces
- An introduction to finite element methods for inverse coefficient problems in elliptic PDEs
- Boundary determination of the Lamé moduli for the isotropic elasticity system