Resolution guarantees for the reconstruction of inclusions in linear elasticity based on monotonicity methods

Eberle, Sarah; Harrach, Bastian

doi:10.1088/1361-6420/accb07

Cited by 3 publications

(3 citation statements)

References 38 publications

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“…Under suitable assumptions, such testing can be mathematically formulated as a quest for information on the Lamé parameters inside the investigated object in the framework of linear elasticity. We refer to [8,10,11,13,23,31,32,33,34,41,42,43,44] for theoretical results and to [7,15,20,18,19,22,25,29,35,37,39,40,47,51,52,53] for reconstruction methods related to the inverse problem of linear elasticity. In this work, we acknowledge that any practical measurement setting related to nondestructive testing in the framework of linear elasticity allows only a finite number of boundary pressure activations.…”

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confidence: 99%

Monotonicity-based regularization for shape reconstruction in linear elasticity

Eberle

Harrach

2022

Comput Mech

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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 Eberle and Harrach (Inverse Probl 37(4):045006, 2021), 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.

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confidence: 99%

Monotonicity-based regularization for shape reconstruction in linear elasticity

Eberle

Harrach

2022

Comput Mech

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“…Thus, this will build the basis for further examinations and the development of the 'linearised monotonicity methods' (see [EH21]) for the stationary case) which will allow us a faster implementation and as such, a testing with more test inclusions. A further extension could be the consideration of a monotonicity-based regularisation as well as the examination of the resolution guarantee (similar as considered in for the stationary elastic inverse problem in [EH22,EBH23], respectively).…”

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confidence: 99%

The monotonicity method for inclusion detection and the time harmonic elastic wave equation

Eberle-Blick,

Pohjola

2024

Inverse Problems

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“…The p-transferred boundary pressure g p defined by (23) is not strictly speaking compatible with the construction in Section 2.2 since the unit normal in (23) is not translated by −p. However, it is easy to check that the analysis of Section 2.2 remains valid for such g p if the p-derivative of the pressure field g ′ p is (re)defined by only differentiating and translating the scalar multiplier of the unit normal in (22) and leaving the unit normal itself untouched. What remains to be chosen is the parametrization for η, i.e., for the perturbations in λ and µ around their background values.…”

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confidence: 99%

Bayesian experimental design for linear elasticity

Eberle-Blick,

Hyvönen

2024

IPI

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This work considers Bayesian experimental design for the inverse boundary value problem of linear elasticity in a two-dimensional setting. The aim is to optimize the positions of compactly supported pressure activations on the boundary of the examined body in order to maximize the value of the resulting boundary deformations as data for the inverse problem of reconstructing the Lamé parameters inside the object. We resort to a linearized measurement model and adopt the framework of Bayesian experimental design, under the assumption that the prior and measurement noise distributions are mutually independent Gaussians. This enables the use of the standard Bayesian A-optimality criterion for deducing optimal positions for the pressure activations. The (second) derivatives of the boundary measurements with respect to the Lamé parameters and the positions of the boundary pressure activations are deduced to allow minimizing the corresponding objective function, i.e., the trace of the covariance matrix of the posterior distribution, by gradient-based optimization algorithms. Two-dimensional numerical experiments are performed to test the functionality of our approach: all introduced algorithms are able to improve experimental designs, but only exhaustive search reliably finds a global minimizer.

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Resolution guarantees for the reconstruction of inclusions in linear elasticity based on monotonicity methods

Cited by 3 publications

References 38 publications

Monotonicity-based regularization for shape reconstruction in linear elasticity

Monotonicity-based regularization for shape reconstruction in linear elasticity

The monotonicity method for inclusion detection and the time harmonic elastic wave equation

Bayesian experimental design for linear elasticity

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