A reliable estimation method of a dipole for three-dimensional Poisson equation

Yamatani, Katsu; Ohnaka, Kohzaburo

doi:10.1016/s0377-0427(98)00082-x

Cited by 6 publications

(6 citation statements)

References 9 publications

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“…Uniqueness [5] and stability [6,7] results are available. For some numerical results, see [8][9][10][11] in two dimensions and [12][13][14][15][16] in three dimensions. We will show that some of these problems for balls can be solved exactly by analytical methods.…”

Section: Introductionmentioning

confidence: 99%

Finding a source inside a sphere

Tsitsas¹,

Martin²

2011

Inverse Problems

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A sphere excited by an interior point source or a point dipole gives a simplified yet realistic model for studying a variety of applications in medical imaging. We suppose that there is an exterior field (transmission problem) and that the total field on the sphere is known. We give analytical inversion algorithms for determining the interior physical characteristics of the sphere as well as the location, strength and orientation of the source/dipole. We start with static problems (Laplace's equation) and then proceed to acoustic problems (Helmholtz equation).

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Section: Introductionmentioning

confidence: 99%

Finding a source inside a sphere

Tsitsas¹,

Martin²

2011

Inverse Problems

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“…There are many algorithm to solve (3), for example [26,28,8]. In geophysical community an almost universally adopted strategy is to reduce problem (3) to the Fredholm integral equation of the first kind [20,25,14,3,1,23,12, among many others].…”

Section: Introductionmentioning

confidence: 99%

Source identification in the self-potential method and its connection to Stokes type systems

Malovichko¹,

Yavich²

2019

Preprint

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This paper develops a novel approach to the problem of source current identification for the diffusion equation in connection with geophysical self-potential measurements. The problem is split into two subproblems: (a) the scalar source identification, and (b) solution of the divergence equation. For subproblem (a), we design an algorithm for reconstructing the scalar source function, which does not require solving the Fredholm integral equation of the first kind. Instead, the problem is reformulated as a linear operator equation, which is solved by a projection method. The dimension of the subspace, in which the source function is sought, is independent of the dimension of the forward problem, leading to reduction of the size of the inverse problem. Numerical experiments with exact and noisy data are presented. For subproblem (b), the divergence equation was posed as a minimization problem, which, by means of Lagrangian formalism, was reduced to a system of partial differential equation of Stokes type with a unique solution. To demonstrate how this framework can be used in a practical application, we implemented the algorithm in the two-dimensional physical space, using a finite-different discretization on staggered grids.

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“…Other possible applications of our inverse source problem can be bioluminescence tomography [20,33] or the physical problem of locating dense masses modeled by a finite linear combination of monopole sources F = m k=1 λ k δ C k , where C k are points in and λ k scalar quantities [32,34]. In the bioluminescence case, equation (1.1) is replaced by the following one:…”

Section: Introductionmentioning

confidence: 99%

“…Ohe and Ohnaka [32] derived such relations using the Fourier expansion of the boundary potential in twodimensional (2D) space. Yamatani and Ohnaka [34] derived such relations using the Green's theorem and harmonic function in 3D, and estimated source positions with known moments. He and Romanov [21] also used the Green's theorem when the source is a single dipole.…”

Section: Introductionmentioning

confidence: 99%

A stable recovering of dipole sources from partial boundary measurements

El-Badia¹,

Farah²

2010

Inverse Problems

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In this paper we consider an inverse dipole source problem for an elliptic equation from boundary measurements and an application to the inverse electroencephalography problem. An identification method based on the so-called Kohn and Vogelius cost function is presented, for which a robustness result is established. Some numerical experiments and a comparison with the leastsquares method are shown in order to illustrate the efficiency of this method. They were performed using both a spherical and a realistic geometry of the head.

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A reliable estimation method of a dipole for three-dimensional Poisson equation

Cited by 6 publications

References 9 publications

Finding a source inside a sphere

Finding a source inside a sphere

Source identification in the self-potential method and its connection to Stokes type systems

A stable recovering of dipole sources from partial boundary measurements

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