On the randomised stability constant for inverse problems

Alberti, Giovanni S.; Capdeboscq, Yves; Privat, Yannick

doi:10.3934/mine.2020013

Cited by 4 publications

(4 citation statements)

References 45 publications

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“…The first result states that the presence of an interior critical point for a nonconstant solution u forces oscillations in its boundary data. The result is known for H 1/2 boundary data, see [AC18,proposition 6.7], but we give an extension to the case where the boundary data can be slightly worse that H 1/2 (e.g. piecewise smooth).…”

Section: Resultsmentioning

confidence: 83%

“…Proof. We follow the argument in [AC18,proposition 6.7]. By [Sch90, theorem 2.3.3] the interior regularity of u is C 1,α loc (Ω).…”

Section: Resultsmentioning

confidence: 99%

“…For instance, [BMN01] require that g = (g 1 , g 2 ) is a C 1 diffeomorphism and maps ∂Ω onto the boundary of a convex domain for the condition (1) to hold. A discussion of results of this type is given in [AC18] (see also [Alb22] for recent work on random boundary data).…”

Section: Introductionmentioning

confidence: 99%

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Jacobian of solutions to the conductivity equation in limited view

Salo¹,

Schlüter²

2022

Inverse Problems

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The aim of hybrid inverse problems such as Acousto-Electric Tomography or Current Density Imaging is the reconstruction of the electrical conductivity in a domain that can only be accessed from its exterior. In the inversion procedure, the solutions to the conductivity equation play a central role. In particular, it is important that the Jacobian of the solutions is non-vanishing. In the present paper we address a two-dimensional limited view setting, where only a part of the boundary of the domain can be controlled by a non-zero Dirichlet condition, while on the remaining boundary there is a zero Dirichlet condition. For this setting, we propose sufficient conditions on the boundary functions so that the Jacobian of the corresponding solutions is non-vanishing. In that regard we allow for discontinuous boundary functions, which requires the use of solutions in weighted Sobolev spaces. We implement the procedure of reconstructing a conductivity from power density data numerically and investigate how this limited view setting affects the Jacobian and the quality of the reconstructions.

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Section: Resultsmentioning

confidence: 83%

“…Proof. We follow the argument in [AC18,proposition 6.7]. By [Sch90, theorem 2.3.3] the interior regularity of u is C 1,α loc (Ω).…”

Section: Resultsmentioning

confidence: 99%

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Jacobian of solutions to the conductivity equation in limited view

Salo¹,

Schlüter²

2022

Inverse Problems

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“…We note, however, that no mathematical results for imaging with data of the form (2c) and (2d) seem to be available in the literature. See also [25,26] for more on the mathematics of hybrid data inverse problems.…”

Section: Introductionmentioning

confidence: 99%

Numerical conductivity reconstruction from partial interior current density information in three dimensions

Yazdanian¹,

Knudsen²

2021

Inverse Problems

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This paper focuses on conductivity imaging of a three dimensional object using interior current density information (ICDI). Applications include the emerging hybrid tomographic methods known as magnetic resonance electrical impedance tomography and current density impedance imaging, which potentially have both high contrast and high resolution. For all possible forms of ICDI data, the Fréchet derivative of the map between conductivity and ICDI is derived. Then, an iterative reconstruction method is formulated based on the Newton scheme. The method is implemented numerically and its properties are investigated on simulated data obtained from two different phantoms. The method is also benchmarked against the J-substitution method. We systematically study the possibilities, challenges, shortcomings, and artifacts due the different forms of full and partial ICDI data and one or several boundary conditions. The results establish that at least two components of two non-parallel interior current densities are required to obtain good reconstructions; this is an important outcome for conductivity imaging methods which use only one component of the magnetic field. The results hold promise for the near real-time and high resolution conductivity reconstruction in practical applications.

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