Uniqueness of an inverse problem with single measurement data generated by a plane wave in partial finite differences

Klibanov, Michael V.

doi:10.1088/0266-5611/27/11/115005

Cited by 6 publications

(5 citation statements)

References 21 publications

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“…Finally, note that multi-channel inverse problems and their applications to inverse problems in greater dimensions were initially considered for the one-dimensional multi-channel case, see [1], [26]. As one of the most recent result in this direction see [14].…”

Section: Introductionmentioning

confidence: 99%

Monochromatic Reconstruction Algorithms for Two-dimensional Multi-channel Inverse Problems

Novikov

Santacesaria

2012

International Mathematics Research Notices

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Abstract. We consider two inverse problems for the multi-channel two-dimensional Schrödinger equation at fixed positive energy, i.e. the equation −∆ψ + V (x)ψ = Eψ at fixed positive E, where V is a matrixvalued potential. The first is the Gel'fand inverse problem on a bounded domain D at fixed energy and the second is the inverse fixed-energy scattering problem on the whole plane R 2 . We present in this paper two algorithms which give efficient approximate solutions to these problems: in particular, in both cases we show that the potential V is reconstructed with Lipschitz stability by these algorithms up to O(E −(m−2)/2 ) in the uniform norm as E → +∞, under the assumptions that V is m-times differentiable in L 1 , for m ≥ 3, and has sufficient boundary decay.

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Section: Introductionmentioning

confidence: 99%

Monochromatic Reconstruction Algorithms for Two-dimensional Multi-channel Inverse Problems

Novikov

Santacesaria

2012

International Mathematics Research Notices

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“…All these theorems were proven by the method, which was originated in 1981 in three papers [19,20,28]; also see, e.g. [21,29,30,31,34,35] as well as sections 1.10, 1.11 of the book [11] and references cited there for some follow up publications of those authors about this method. This method is based on Carleman estimates.…”

Section: The Local Strong Convexity Of the Tikhonov Functional (216)mentioning

confidence: 99%

“…We refer to surveys [35,48] for more references. Lifting the above assumption is a long standing and well known open question, see [34] for a recent partial answer to this question. Nevertheless, because of applications, it makes sense to develop numerical methods for the above CIP, regardless on the absence of proper uniqueness theorems.…”

Section: The Local Strong Convexity Of the Tikhonov Functional (216)mentioning

confidence: 99%

The adaptivity refines approximate solutions of ill-posed problems due to the relaxation property

Beilina,

Klibanov

2012

Preprint

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Adaptive Finite Element Method (adaptivity) is known to be an effective numerical tool for some ill-posed problems. The key advantage of the adaptivity is the image improvement with local mesh refinements. A rigorous proof of this property is the central part of this paper. In terms of Coefficient Inverse Problems with single measurement data, the authors consider the adaptivity as the second stage of a two-stage numerical procedure. The first stage delivers a good approximation of the exact coefficient without an advanced knowledge of a small neighborhood of that coefficient. This is a necessary element for the adaptivity to start iterations from. Numerical results for the two-stage procedure are presented for both computationally simulated and experimental data.

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“…The Carleman estimate is an inequality for a solution to a PDE with the weighted L 2 -norm and is effectively applied for proving the unique continuation for the solution of PDE. In [6], Bukhgeim and Klibanov generalize this elegant technique to the proof of uniqueness for inverse coefficient problems for hyperbolic, parabolic and elliptic equations with the lateral data; see also [7][8][9]32] and references therein for related publications. Since then it was widely used by many researchers for proving stability and uniqueness results for inverse problems of recovering coefficients of PDEs; for example, see [10][11][12][13][14][15][16][17][18][19][20].…”

Section: Introductionmentioning

confidence: 99%

Determination of an unknown source for a thermoelastic system with a memory effect

Liu

2012

Inverse Problems

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We study an inverse problem of determining a spatially varying source term in a thermoelastic medium with a memory effect. The coupling phenomena between elasticity and heat as well as the memory effect make such an inverse problem very complicated. We firstly prove a pointwise Carleman estimate for a general strongly coupled hyperbolic system, and then obtain a Carleman estimate for the hyperbolic thermoelastic system. Based on this estimate, we finally establish a Hölder stability for the inverse source problem only by making a displacement measurement on a given subdomain for sufficiently large times, provided the source be known near the boundary. The uniqueness for such an inverse problem is yielded as a direct result.

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Uniqueness of an inverse problem with single measurement data generated by a plane wave in partial finite differences

Cited by 6 publications

References 21 publications

Monochromatic Reconstruction Algorithms for Two-dimensional Multi-channel Inverse Problems

Monochromatic Reconstruction Algorithms for Two-dimensional Multi-channel Inverse Problems

The adaptivity refines approximate solutions of ill-posed problems due to the relaxation property

Determination of an unknown source for a thermoelastic system with a memory effect

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