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Open accessJournal ArticleDOI: 10.1080/17415977.2020.1795151

Lipschitz stability estimate and reconstruction of Lamé parameters in linear elasticity

04 Mar 2021-Inverse Problems in Science and Engineering (Taylor & Francis)-Vol. 29, Iss: 3, pp 396-417
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 ...

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Topics: Lipschitz continuity (63%), Linear elasticity (58%), Lamé parameters (52%) ... read more
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10 results found


Open accessJournal ArticleDOI: 10.1007/S00211-020-01162-8
Bastian Harrach1Institutions (1)
Abstract: We derive a simple criterion that ensures uniqueness, Lipschitz stability and global convergence of Newton’s method for the finite dimensional zero-finding problem of a continuously differentiable, pointwise convex and monotonic function. Our criterion merely requires to evaluate the directional derivative of the forward function at finitely many evaluation points and for finitely many directions. We then demonstrate that this result can be used to prove uniqueness, stability and global convergence for an inverse coefficient problem with finitely many measurements. We consider the problem of determining an unknown inverse Robin transmission coefficient in an elliptic PDE. Using a relation to monotonicity and localized potentials techniques, we show 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. We derive a constructive method to identify these boundary measurements, calculate the stability constant and give a numerical example.

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Topics: Uniqueness (57%), Lipschitz continuity (55%), Newton's method (55%) ... read more

14 Citations


Open accessJournal ArticleDOI: 10.1088/1361-6420/ABC8A9
Sarah Eberle1, Bastian Harrach1Institutions (1)
Abstract: In this paper, we deal with the inverse problem of the shape reconstruction of inclusions in elastic bodies. The main idea of this reconstruction is based on the monotonicity property of the Neumann-to-Dirichlet operator presented in a former article of the authors. Thus, we introduce the so-called standard as well as linearized monotonicity tests in order to detect and reconstruct inclusions. In addition, we compare these methods with each other and present several numerical test examples.

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Topics: Inverse problem (51%)

6 Citations


Open accessJournal ArticleDOI: 10.1088/1361-6420/ABD29A
01 Apr 2021-Inverse Problems
Abstract: We treat an inverse electrical conductivity problem which deals with the reconstruction of nonlinear electrical conductivity starting from boundary measurements in steady currents operations. In this framework, a key role is played by the Monotonicity Principle, which establishes a monotonic relation connecting the unknown material property to the (measured) Dirichlet-to-Neumann operator (DtN). Monotonicity Principles are the foundation for a class of non-iterative and real-time imaging methods and algorithms. In this article, we prove that the Monotonicity Principle for the Dirichlet Energy in nonlinear problems holds under mild assumptions. Then, we show that apart from linear and p-Laplacian cases, it is impossible to transfer this Monotonicity result from the Dirichlet Energy to the DtN operator. To overcome this issue, we introduce a new boundary operator, identified as an Average DtN operator.

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Topics: Operator (computer programming) (53%), Dirichlet's energy (53%), Nonlinear system (52%) ... read more

5 Citations


Open accessJournal ArticleDOI: 10.1088/1361-6420/ABC8A9
01 Apr 2021-Inverse Problems
Abstract: In this paper, we deal with the inverse problem of the shape reconstruction of inclusions in elastic bodies. The main idea of this reconstruction is based on the monotonicity property of the Neumann-to-Dirichlet operator presented in a former article of the authors. Thus, we introduce the so-called standard as well as linearized monotonicity tests in order to detect and reconstruct inclusions. In addition, we compare these methods with each other and present several numerical test examples.

... read more

Topics: Inverse problem (51%)

4 Citations


Open accessPosted Content
Abstract: In this paper, we consider the inverse problem of recovering a diffusion and absorption coefficients in steady-state optical tomography problem from the Neumann-to-Dirichlet map. We first prove a Global uniqueness and Lipschitz stability estimate for the absorption parameter provided that the diffusion is known. Then, we prove a Lipschitz stability result for simultaneous recovery of diffusion and absorption. In both cases the parameters belong to a known finite subspace with a priori known bounds. The proofs relies on a monotonicity result combined with the techniques of localized potentials. To numerically solve the inverse problem, we propose a Kohn-Vogeliustype cost functional over a class of admissible parameters subject to two boundary value problems. The reformulation of the minimization problem via the Neumann-toDirichlet 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 illustrate some numerical results.

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Topics: Lipschitz continuity (63%), Inverse problem (58%), Uniqueness (53%) ... read more

3 Citations


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52 results found


Open accessJournal ArticleDOI: 10.1088/0266-5611/21/2/R01
Marc Bonnet1, Andrei Constantinescu1Institutions (1)
23 Feb 2005-Inverse Problems
Abstract: This review is devoted to some inverse problems arising in the context of linear elasticity, namely the identification of distributions of elastic moduli, model parameters or buried objects such as cracks. These inverse problems are considered mainly for three-dimensional elastic media under equilibrium or dynamical conditions, and also for thin elastic plates. The main goal is to overview some recent results, in an effort to bridge the gap between studies of a mathematical nature and problems defined from engineering practice. Accordingly, emphasis is given to formulations and solution techniques which are well suited to general-purpose numerical methods for solving elasticity problems on complex configurations, in particular the finite element method and the boundary element method. An underlying thread of the discussion is the fact that useful tools for the formulation, analysis and solution of inverse problems arising in linear elasticity, namely the reciprocity gap and the error in constitutive equation, stem from variational and virtual work principles, i.e., fundamental principles governing the mechanics of deformable solid continua. In addition, the virtual work principle is shown to be instrumental for establishing computationally efficient formulae for parameter or geometrical sensitivity, based on the adjoint solution method. Sensitivity formulae are presented for various situations, especially in connection with contact mechanics, cavity and crack shape perturbations, thus enriching the already extensive known repertoire of such results. Finally, the concept of topological derivative and its implementation for the identification of cavities or inclusions are expounded.

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Topics: Linear elasticity (59%), Boundary element method (55%), Virtual work (55%) ... read more

381 Citations


Journal ArticleDOI: 10.1088/0266-5611/19/6/R01
01 Dec 2003-Inverse Problems
Abstract: This paper presents an overview and a detailed description of the key logic steps and mathematical-physics framework behind the development of practical algorithms for seismic exploration derived from the inverse scattering series. There are both significant symmetries and critical subtle differences between the forward scattering series construction and the inverse scattering series processing of seismic events. These similarities and differences help explain the efficiency and effectiveness of different inversion objectives. The inverse series performs all of the tasks associated with inversion using the entire wavefield recorded on the measurement surface as input. However, certain terms in the series act as though only one specific task, and no other task, existed. When isolated, these terms constitute a task-specific subseries. We present both the rationale for seeking and methods of identifying uncoupled task-specific subseries that accomplish: (1) free-surface multiple removal; (2) internal multiple attenuation; (3) imaging primaries at depth; and (4) inverting for earth material properties. A combination of forward series analogues and physical intuition is employed to locate those subseries. We show that the sum of the four task-specific subseries does not correspond to the original inverse series since terms with coupled tasks are never considered or computed. Isolated tasks are accomplished sequentially and, after each is achieved, the problem is restarted as though that isolated task had never existed. This strategy avoids choosing portions of the series, at any stage, that correspond to a combination of tasks, i.e., no terms corresponding to coupled tasks are ever computed. This inversion in stages provides a tremendous practical advantage. The achievement of a task is a form of useful information exploited in the redefined and restarted problem; and the latter represents a critically important step in the logic and overall strategy. The individual subseries are analysed and their strengths, limitations and prerequisites exemplified with analytic, numerical and field data examples.

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357 Citations


Journal ArticleDOI: 10.1137/S1052623401383455
Yu-Hong Dai1Institutions (1)
Abstract: The BFGS method is one of the most famous quasi-Newton algorithms for unconstrained optimization. In 1984, Powell presented an example of a function of two variables that shows that the Polak--Ribiere--Polyak (PRP) conjugate gradient method and the BFGS quasi-Newton method may cycle around eight nonstationary points if each line search picks a local minimum that provides a reduction in the objective function. In this paper, a new technique of choosing parameters is introduced, and an example with only six cyclic points is provided. It is also noted through the examples that the BFGS method with Wolfe line searches need not converge for nonconvex objective functions.

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198 Citations



Open accessJournal ArticleDOI: 10.1007/S00222-002-0276-1
Gen Nakamura1, Gunther Uhlmann2Institutions (2)
Abstract: We prove that we can determine the Lame parameters of an elastic, isotropic, inhomogeneous medium in dimensionsn≧3, by making measurements of the displacements and corresponding stresses at the boundary of the medium.

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Topics: Boundary value problem (59%), Uniqueness (53%), Lamé parameters (52%) ... read more

173 Citations


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No. of citations received by the Paper in previous years
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20216
20203
20191