Well-posedness of the conductivity reconstruction from an interior current density in terms of Schauder theory

Kim, Yong-Jung; Lee, Min Gi

doi:10.1090/qam/1368

Cited by 6 publications

(7 citation statements)

References 20 publications

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“…When the entire current density field is known, a stability result has been obtained in [16]. For the anisotropic conductivities we mention the work in [7] which a proved stability in two dimensions from full knowledge of four current density fields.…”

Section: Introductionmentioning

confidence: 98%

Stability in conductivity imaging from partial measurements of one interior current

Montalto¹,

Tamasan²

2017

IPI

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We prove a stability result in the hybrid inverse problem of recovering the electrical conductivity from partial knowledge of one current density field generated inside a body by an imposed boundary voltage. The region where interior data stably reconstructs the conductivity is well defined by a combination of the exact and perturbed data.2000 Mathematics Subject Classification. 35R30, 35J65, 65N21.

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Section: Introductionmentioning

confidence: 98%

Stability in conductivity imaging from partial measurements of one interior current

Montalto¹,

Tamasan²

2017

IPI

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“…We use the characteristic method (see, for instance, [22,29,30]) for solving (4.20). For any x 0 ∈ , consider the Cauchy problem:…”

Section: Orthogonal Field Methodsmentioning

confidence: 99%

A mathematical and numerical framework for magnetoacoustic tomography with magnetic induction

Ammari

Boulmier

Millien

2015

Journal of Differential Equations

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We provide a mathematical analysis and a numerical framework for magnetoacoustic tomography with magnetic induction. The imaging problem is to reconstruct the conductivity distribution of biological tissue from measurements of the Lorentz force induced tissue vibration. We begin with reconstructing from the acoustic measurements the divergence of the Lorentz force, which is acting as the source term in the acoustic wave equation. Then we recover the electric current density from the divergence of the Lorentz force. To solve the nonlinear inverse conductivity problem, we introduce an optimal control method for reconstructing the conductivity from the electric current density. We prove its convergence and stability. We also present a point fixed approach and prove its convergence to the true solution. A new direct reconstruction scheme involving a partial differential equation is then proposed based on viscosity-type regularization to a transport equation satisfied by the electric current density field. We prove that solving such an equation yields the true conductivity distribution as the regularization parameter approaches zero. Finally, we test the three schemes numerically in the presence of measurement noise, quantify their stability and resolution, and compare their performance.

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“…Theorem 1. Let σ satisfy (12), b 0 , c 0 , u 0 , and a 0 be as above and G(•; a 0 ) be as in (17) with ǫ = 0.Then…”

Section: Existence and Uniqueness Of A Minimizermentioning

confidence: 99%

“…Continuous dependence on σ on a (for a given unperturbed Dirichlet data) can be found in [18], and, for partial data in [19]. For further references on determining the isotropic conductivity based on measurements of current densities see [35,11,13,14,10,15,12], and for reconstructions on anisotropic conductivities from multiple measurements see [9,6,1,2].…”

Section: Introductionmentioning

confidence: 99%

A regularized weighted least gradient problem for conductivity imaging

Tamasan¹,

Timonov²

2019

Inverse Problems

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We propose and study a regularization method for recovering an approximate electrical conductivity solely from the magnitude of one interior current density field. Without some minimal knowledge of the boundary voltage potential, the problem has been recently shown to have nonunique solutions, thus recovering the exact conductivity is impossible. The method is based on solving a weighted least gradient problem in the subspace of functions of bounded variations with square integrable traces. The computational effectiveness of this method is demonstrated in numerical experiments.

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Well-posedness of the conductivity reconstruction from an interior current density in terms of Schauder theory

Cited by 6 publications

References 20 publications

Stability in conductivity imaging from partial measurements of one interior current

Stability in conductivity imaging from partial measurements of one interior current

A mathematical and numerical framework for magnetoacoustic tomography with magnetic induction

A regularized weighted least gradient problem for conductivity imaging

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