Lipschitz stability estimate and reconstruction of Lam\'e parameters in linear elasticity
Reads0
Chats0
TLDR
In this article, the authors consider the inverse problem of recovering an isotropic elastic tensor from the Neumann-to-Dirichlet map and prove a Lipschitz stability estimate for Lame parameters with certain regularity assumptions.Abstract:
In this paper, we consider the inverse problem of recovering an isotropic elastic tensor from the Neumann-to-Dirichlet map. To this end, we prove a Lipschitz stability estimate for Lame parameters with certain regularity assumptions. In addition, we assume that the Lame parameters belong to a known finite subspace with a priori known bounds and that they fulfill a monotonicity property. The proof relies on a monotonicity result combined with the techniques of localized potentials. To numerically solve the inverse problem, we propose a Kohn-Vogelius-type cost functional over a class of admissible parameters subject to two boundary value problems. The reformulation of the minimization problem via the Neumann-to-Dirichlet operator allows us to obtain the optimality conditions by using the Frechet differentiability of this operator and its inverse. The reconstruction is then performed by means of an iterative algorithm based on a quasi-Newton method. Finally, we give and discuss several numerical examples.read more
Citations
More filters
Posted Content
Monotonicity-based inversion of the fractional Schr\"odinger equation
Bastian Harrach,Yi-Hsuan Lin +1 more
TL;DR: In this article, if-and-only-if monotonicity relations between positive bounded potentials and their associated nonlocal Dirichlet-to-Neumann maps are provided.
Journal ArticleDOI
Monotonicity-based inversion of the fractional schodinger equation ii. general potentials and stability
Bastian Harrach,Yi-Hsuan Lin +1 more
TL;DR: This work uses monotonicity-based methods for the fractional Schrodinger equation with general potentials q in L^\infty(Omega) in a Lipschitz bounded open set Omega \subset R^n in any dimensi...
Journal ArticleDOI
Monotonicity-based Inversion of the Fractional Schrödinger Equation I. Positive Potentials
Bastian Harrach,Yi-Hsuan Lin +1 more
TL;DR: This work provides if-and-only-if monotonicity relations between positive bounded potentials and their associated nonlocal Dirichlet-to-Neumann maps and can prove uniqueness for the nonlocal Calderon problem in a constructive manner.
Journal ArticleDOI
Uniqueness, stability and global convergence for a discrete inverse elliptic Robin transmission problem
TL;DR: It is shown that a piecewise-constant coefficient on an a-priori known partition with a-Priori known bounds is uniquely determined by finitely many boundary measurements and that it can be uniquely and stably reconstructed by a globally convergent Newton iteration.
Journal ArticleDOI
Monotonicity-based inversion of the fractional Schr\"odinger equation II. General potentials and stability
Bastian Harrach,Yi-Hsuan Lin +1 more
TL;DR: In this article, if-and-only-if monotonicity relations between potentials and the Dirichlet-to-Neumann map hold up to a finite dimensional subspace.
References
More filters
Book
The Finite Element Method for Elliptic Problems
Philippe G. Ciarlet,J. T. Oden +1 more
TL;DR: The finite element method has been applied to a variety of nonlinear problems, e.g., Elliptic boundary value problems as discussed by the authors, plate problems, and second-order problems.
Book
Finite Element Method for Elliptic Problems
TL;DR: In this article, Ciarlet presents a self-contained book on finite element methods for analysis and functional analysis, particularly Hilbert spaces, Sobolev spaces, and differential calculus in normed vector spaces.
Journal ArticleDOI
Mathematical elasticity: volume i: three-dimensional elasticity
Book
Three-dimensional elasticity
TL;DR: In this article, the equations of equilibrium and the principle of virtual work for three-dimensional elasticity have been discussed and the boundary value problems of 3D elasticity has been studied.
Related Papers (5)
Monotonicity-based shape reconstruction in electrical impedance tomography ∗
Bastian Harrach,Marcel Ullrich +1 more
Enhancing residual-based techniques with shape reconstruction features in electrical impedance tomography
Bastian Harrach,Mach Nguyet Minh +1 more