On uniqueness and non-uniqueness for current reconstruction from magnetic fields

Hauer, Karl-Heinz; Kuhn, Lars T.; Potthast, Roland

doi:10.1088/0266-5611/21/3/010

Cited by 40 publications

(42 citation statements)

References 7 publications

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“…The non-uniqueness issue has been examined by several researchers [19,16,13,22,25,30]. Recently, in the vector static case (zero frequency), Hauer, Kühn and Potthast [22] studied the limitations to reconstruct the current density from its magnetic field by characterizing the nullspace of the Biot-Savart operator for anisotropic conductivity. In [2], the non-radiating sources are considered when the background medium is non-constant for the full Maxwell system.…”

Section: Introductionmentioning

confidence: 99%

A multi-frequency inverse source problem

Bao

Lin

Triki

2010

Journal of Differential Equations

178

172

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International audienceThis paper is concerned with an inverse source problem that determines the source from measurements of the radiated fields away at multiple frequencies. Rigorous stability estimates are established when the background medium is homogeneous. It is shown that the ill-posedness of the inverse problem decreases as the frequency increases. Under some regularity assumptions on the source function, it is further proven that by increasing the frequency, the logarithmic stability converts to a linear one for the inverse source problem

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Section: Introductionmentioning

confidence: 99%

A multi-frequency inverse source problem

Bao

Lin

Triki

2010

Journal of Differential Equations

178

172

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“…Re (x, y, ω k ) G [12] Re (x, y, ω k ) G [21] Re (x, y, ω k ) G [22] Re (x, y, ω k ) , where G [11] Re (x, y, ω…”

Section: Integral Equationsmentioning

confidence: 99%

“…Mathematically, the inverse source scattering problems have been widely examined for acoustic and electromagnetic waves by many researchers [1-3, 8, 9, 15, 17, 29, 32]. For instance, it is known that the inverse source problem does not have a unique solution at a fixed frequency due to the existence of non-radiating sources [16,21]; it is ill-posed as small variations in the measured data can lead to huge errors in the reconstructions [7].…”

mentioning

confidence: 99%

Inverse Random Source Scattering for Elastic Waves

Bao¹,

Chen²,

Li³

2017

SIAM J. Numer. Anal.

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Abstract. This paper is concerned with the direct and inverse random source scattering problems for elastic waves where the source is assumed to be driven by an additive white noise. Given the source, the direct problem is to determine the displacement of the random wave field. The inverse problem is to reconstruct the mean and variance of the random source from the boundary measurement of the wave field at multiple frequencies. The direct problem is shown to have a unique mild solution by using a constructive proof. Based on the explicit mild solution, Fredholm integral equations of the first kind are deduced for the inverse problem. The regularized Kaczmarz method is presented to solve the ill-posed integral equations. Numerical experiments are included to demonstrate the effectiveness of the proposed method.

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“…[12], [8]. There is a large variety of current distributions j in some domain Ω which generate the same magnetic field H in the open exterior Ω e = R 3 \ Ω of Ω.…”

Section: Roland Potthast and Martin Wannertmentioning

confidence: 99%

Uniqueness of Current Reconstructions for Magnetic Tomography in Multilayer Devices

Potthast¹,

Wannert²

2009

SIAM J. Appl. Math.

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Magnetic tomography is an important emerging technique for the nondestructive investigation and monitoring of electrical devices. Measurements of the magnetic field of the currents in a device are used to reconstruct the current distribution. Here we investigate the uniqueness problem for current reconstructions from multilayer devices. The general magnetic tomography problem is well known to be highly nonunique and unstable. Here, as a new result for the single-layer and multilayer device case we will study the splitting procedure and equivalence of the full nullspace to the nullspaces of operators supported on lower-dimensional subsets of the device. We will base our results on the uniqueness of wave source splitting combined with tools from potential theory and explicit estimates for particular surface integrals involving the Biot-Savart integral operator.

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On uniqueness and non-uniqueness for current reconstruction from magnetic fields

Cited by 40 publications

References 7 publications

A multi-frequency inverse source problem

A multi-frequency inverse source problem

Inverse Random Source Scattering for Elastic Waves

Uniqueness of Current Reconstructions for Magnetic Tomography in Multilayer Devices

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