Simultaneous numerical determination of a corroded boundary and its admittance

Karageorghis, Andréas; Bin‐Mohsin, Bandar; Lesnic, D.; Marin, Liviu

doi:10.1080/17415977.2014.969728

Cited by 4 publications

(5 citation statements)

References 30 publications

(26 reference statements)

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“…where ν is the outward unit normal to ∂D and 0 < λ ∈ C(∂D) or L ∞ (∂D) is a Robin coupling real function usually called the impedance function, [15], or admittance, [8]. When λ → 0 or λ → ∞ we obtain the particular cases of a sound-hard or sound-soft scatterer, respectively.…”

Section: Mathematical Formulationmentioning

confidence: 99%

The method of fundamental solutions for the identification of a scatterer with impedance boundary condition in interior inverse acoustic scattering

Karageorghis

Lesnic

Marin

2018

Engineering Analysis with Boundary Elements

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Article:Karageorghis, A, Lesnic, D orcid.org/0000-0003- 3025-2770 and Marin, L (2018) The method of fundamental solutions for the identification of a scatterer with impedance boundary condition in interior inverse acoustic scattering. Engineering Analysis with Boundary Elements, 92. pp. 218-224.Abstract. We employ the method of fundamental solutions (MFS) for detecting a scatterer surrounding a host acoustic homogeneous medium D due to a given point source inside it. On the boundary of the unknown scatterer (assumed to be star-shaped), allowing for the normal velocity to be proportional to the excess pressure, a Robin impedance boundary condition is considered. The coupling Robin function λ may or may not be known. The additional information which is supplied in order to compensate for the lack of knowledge of the boundary ∂D of the interior scatterer D and/or the function λ is given by the measurement of the scattered field (generated by the interior point source) on a curve inside D. These measurements may be contaminated with noise so their inversion requires regularization. This is enforced by minimizing a penalised least-squares functional containing various regularization parameters to be prescribed. In the MFS, the unknown scattered field u s is approximated with a linear combination of fundamental solutions of the Helmholtz operator with their singularities excluded from the solution domain D and this yields the discrete version of the objective functional. Physical constraints are added and the resulting constrained minimization problem is solved using the MATLAB c ⃝ toolbox routine lsqnonlin. Numerical results are presented and discussed.

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Section: Mathematical Formulationmentioning

confidence: 99%

The method of fundamental solutions for the identification of a scatterer with impedance boundary condition in interior inverse acoustic scattering

Karageorghis

Lesnic

Marin

2018

Engineering Analysis with Boundary Elements

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“…There is a considerable amount of literature available, dealing with both theoretical and computational aspects of both the individual problems, i.e. estimation of the Robin coefficient or estimation of the domain shape, see e.g [1][2][3][5][6][7][8][9][10][11][12][13][14][15], as well as of the problem of joint estimation [16][17][18][19][20][21][22][23][24][25][26]. We also mention the works [27,28], in which a similar problem is considered using a generalized impedance boundary condition.…”

Section: Introductionmentioning

confidence: 99%

Joint estimation of Robin coefficient and domain boundary for the Poisson problem

Nicholson¹,

Niskanen²

2021

Inverse Problems

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We consider the problem of simultaneously inferring the heterogeneous coeﬃcient ﬁeld for a Robin boundary condition on an inaccessible part of the boundary along with the shape of the boundary for the Poisson problem. Such a problem arises in, for example, corrosion detection, and thermal parameter estimation. We carry out both linearised uncertainty quantiﬁcation, based on a local Gaussian approximation, and full exploration of the joint posterior using Markov chain Monte Carlo (MCMC) sampling. By exploiting a known invariance property of the Poisson problem, we are able to circumvent the need to re-mesh as the shape of the boundary changes. The linearised uncertainty analysis presented here relies on a local linearisation of the parameter-to-observable map, with respect to both the Robin coeﬃcient and the boundary shape, evaluated at the maximum a posteriori (MAP) estimates. Computation of the MAP estimate is carried out using the Gauss-Newton method. On the other hand, to explore the full joint posterior we use the Metropolis-adjusted Langevin algorithm (MALA), which requires the gradient of the log-posterior. We thus derive both the Fréchet derivative of the solution to the Poisson problem with respect to the Robin coeﬃcient and the boundary shape, and the gradient of the log-posterior, which is eﬃciently computed using the so-called adjoint approach. The performance of the approach is demonstrated via several numerical experiments with simulated data.

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“…When D is a rigid inclusion ( ) λ = ∞ or a cavity ( ) λ = 0 ; see, e.g. [27] for the physical explanation, we can formulate the initialboundary value problem and define the Neumann-to-Dirichlet map analogously. The inverse problem for active thermography is to reconstruct D from the measured data Λ D .…”

Section: Introductionmentioning

confidence: 99%

“…In [19,20], the inverse problem is form ulated as a shape optimization problem and is solved by optimization methods. A meshless method of fundamental solutions is proposed in [12], where | = u t 0 is nonzero and D can depend on t. This method is also applied to simultaneously reconstruct D and λ for the Laplace equation case; see [27]. As for non-iterative reconstruction methods, we refer to [10,11,23,25,26,29,39] and the references therein, where the dynamical probe method and the enclosure method are developed.…”

Section: Introductionmentioning

confidence: 99%

Numerical reconstruction of unknown Robin inclusions inside a heat conductor by a non-iterative method

Nakamura

Wang

2017

Inverse Problems

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Consider the problem of reconstructing unknown Robin inclusions inside a heat conductor from boundary measurements. This problem arises from active thermography and is formulated as an inverse boundary value problem for the heat equation. In our previous works, we proposed a sampling-type method for reconstructing the boundary of the Robin inclusion and gave its rigorous mathematical justification. This method is non-iterative and based on the characterization of the solution to the so-called Neumannto-Dirichlet map gap equation. In this paper, we give a further investigation of the reconstruction method from both the theoretical and numerical points of view. First, we clarify the solvability of the Neumann-to-Dirichlet map gap equation and establish a relation of its solution to the Green function associated with an initial-boundary value problem for the heat equation inside the Robin inclusion. This naturally provides a way of computing this Green function from the Neumann-to-Dirichlet map and explains what is the input for the linear sampling method. Assuming that the Neumann-to-Dirichlet map gap equation has a unique solution, we also show the convergence of our method for noisy measurements. Second, we give the numerical implementation of the reconstruction method for two-dimensional spatial domains. The measurements for our inverse problem are simulated by solving the forward problem via the boundary integral equation method. Numerical results are presented to illustrate the efficiency and stability of the proposed method. By using a finite sequence of transient input over a time interval, we propose a new sampling method over the time interval by single measurement which is most likely to be practical.

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Simultaneous numerical determination of a corroded boundary and its admittance

Cited by 4 publications

References 30 publications

The method of fundamental solutions for the identification of a scatterer with impedance boundary condition in interior inverse acoustic scattering

The method of fundamental solutions for the identification of a scatterer with impedance boundary condition in interior inverse acoustic scattering

Joint estimation of Robin coefficient and domain boundary for the Poisson problem

Numerical reconstruction of unknown Robin inclusions inside a heat conductor by a non-iterative method

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