On Inverse Problems for Characteristic Sources in Helmholtz Equations

Alves, Carlos J.S.; Mamud, Roberto; Martins, Nuno F. M.; Roberty, Nilson C.

doi:10.1155/2017/2472060

Cited by 9 publications

(3 citation statements)

References 25 publications

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“…In [15], recovering acoustic monopoles was investigated by means of point-wise acoustic-pressure measurements at a limited number of frequencies and formulation of associated sparse optimization problems for the Helmholtz equation. Besides, determining characteristic sources in the modified and classical Helmholtz equations based on external boundary measurements and a minimization scheme for an equivalent reciprocity functional was analyzed in [16]. In [17], three reconstruction algorithms were proposed for the Helmholtz equation, using near-field Cauchy data on the external boundary, to detect the number, location, size, and shape of hidden sources.…”

Section: Introductionmentioning

confidence: 99%

Detecting Line Sources inside Cylinders by Analytical Algorithms

Lazaridis

Tsitsas

2023

Mathematics

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Inverse problems for line sources radiating inside a homogeneous magneto-dielectric cylinder are investigated. The developed algorithms concern the determination of the location and the current of each source. These algorithms are mostly analytical and are based on proper exploitation of the moments obtained by integrating the product of the total field on the cylindrical boundary with complex exponential functions. The information on the unknown parameters of the problem is encoded in these moments, and hence all parameters can be recovered by means of relatively simple explicit expressions. The cases of one and two sources are considered and analyzed. Under certain conditions, the permittivity and permeability of the cylinder are also recovered. The results from two types of numerical experiments are presented: (i) for a single source, the effect of noise on the boundary data is studied, (ii) for two sources, the pertinent nonlinear system of equations is solved numerically and the accuracy of the derived solution is discussed.

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Section: Introductionmentioning

confidence: 99%

Detecting Line Sources inside Cylinders by Analytical Algorithms

Lazaridis

Tsitsas

2023

Mathematics

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“…This technique is concerned with the solution of a linear magnetic inverse problem of reconstructing the internal current profiles from the measurements of magnetic fields [3]. A large wealth of applications can be addressed by this technique and has provided many interesting results in different fields [4][5][6][7].…”

Section: Introductionmentioning

confidence: 99%

Inverse Problem Solution and Regularization Parameter Selection for Current Distribution Reconstruction in Switching Arcs by Inverting Magnetic Fields

Dong

Zhang

et al. 2018

Mathematical Problems in Engineering

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Current density distribution in electric arcs inside low voltage circuit breakers is a crucial parameter for us to understand the complex physical behavior during the arcing process. In this paper, we investigate the inverse problem of reconstructing the current density distribution in arcs by inverting the magnetic fields. A simplified 2D arc chamber is considered. The aim of this paper is the computational side of the regularization method, regularization parameter selection strategies, and the estimation of systematic error. To address the ill-posedness of the inverse problem, Tikhonov regularization is analyzed, with the regularization parameter chosen by Morozov's discrepancy principle, the L-curve, the generalized cross-validation, and the quasi-optimality criteria. The provided range of regularization parameter selection strategies is much wider than in the previous works. Effects of several features on the performance of these criteria have been investigated, including the signal-to-noise ratio, dimension of measurement space, and the measurement distance. The numerical simulations show that the generalized cross-validation and quasi-optimality criteria provide a more satisfactory performance on the robustness and accuracy. Moreover, an optimal measurement distance can be expected when using a planner sensor array to perform magnetic measurements.

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“…In [42] a method for the reconstruction of star-shaped characteristic sources was developped, by reducing the problem to an algebraic system of equations. The paper [6] traited the inverse characteristic source problem, in the case of Helmholtz equations, from the determination of the barycenter of the characteristic source and the recovery of its geometry from a class of star-shaped characteristic sources, using an algorithm based on an equivalent reciprocity functional formulation. [21] have also investigated the inverse source problem of the Helmholtz equation, where the source consists of multiple point sources, an algebraic algorithm was proposed to identify the number, locations and intensities of the point sources from boundary measurements.…”

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confidence: 99%

Shape optimization method for an inverse geometric source problem and stability at critical shape

Afraites¹,

Masnaoui²,

Nachaoui³

2022

DCDS-S

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This work deals with a geometric inverse source problem. It consists in recovering inclusion in a fixed domain based on boundary measurements. The inverse problem is solved via a shape optimization formulation. Two cost functions are investigated, namely, the least squares fitting, and the Kohn-Vogelius function. In this case, the existence of the shape derivative is given via the first order material derivative of the state problems. Furthermore, using the adjoint approach, the shape gradient of both cost functions is characterized. Then, the stability is investigated by proving the compactness of the Hessian at the critical shape for both considered cases. Finally, based on the gradient method, a steepest descent algorithm is developed, and some numerical experiments for non-parametric shapes are discussed.

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On Inverse Problems for Characteristic Sources in Helmholtz Equations

Cited by 9 publications

References 25 publications

Detecting Line Sources inside Cylinders by Analytical Algorithms

Detecting Line Sources inside Cylinders by Analytical Algorithms

Inverse Problem Solution and Regularization Parameter Selection for Current Distribution Reconstruction in Switching Arcs by Inverting Magnetic Fields

Shape optimization method for an inverse geometric source problem and stability at critical shape

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