The nonlinear Fourier transform for two-dimensional subcritical potentials

Music, Michael

doi:10.3934/ipi.2014.8.1151

Cited by 14 publications

(35 citation statements)

References 17 publications

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“…Constant A in (15) can be chosen so large that all the exceptional points for potentials aQ, 0 ≤ a ≤ 1, are located in the disc |k| < A − 1. Then point k 0 (see (15), (20)) can be chosen independently of a and the matrix h a (ς, k), ς ∈ C, k ∈ C\D, is analytic in a ∈ (0, 1]. Moreover, the entries of the derivative ∂h o…”

Section: Proof Of Theorem 21mentioning

confidence: 99%

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On reconstruction of complex-valued once differentiable conductivities

Lakshtanov¹,

Vainberg²

2016

J. Spectr. Theory

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The classical ∂-method has been generalized recently [13], [14] to be used in the presence of exceptional points. We apply this generalization to solve Dirac inverse scattering problem with weak assumptions on smoothness of potentials. As a consequence, this provides an effective method of reconstruction of complex-valued one time differentiable conductivities in the inverse impedance tomography problem.

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Section: Proof Of Theorem 21mentioning

confidence: 99%

“…Proof. For each function g(·) in L 2 (R 2 ), the operator ∂ −1 k (g(k)·) is compact on L s (R 2 ) for all s > 2 (see, e.g., [15,Lemma 5.3]) and the following estimate holds (see the same lemma) ∂ −1…”

mentioning

confidence: 99%

On reconstruction of complex-valued once differentiable conductivities

Lakshtanov¹,

Vainberg²

2016

J. Spectr. Theory

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“…The following lemma concerns the exterior Faddeev problem (11). Let us introduce the following parameter ε = ε(k) :…”

Section: Reduction To Boundary Operatorsmentioning

confidence: 99%

“…The knowledge of exceptional points is particularly important when the Faddeev scattering problem is applied to solve the inverse problem of recovering the potential n from the Dirichlet-to-Neuman map F n at the boundary ∂O, which is defined by solutions of either equation (1) or (4) in O. For example, if the real potential is continued by zero in the exterior of O, then the following relation holds for the solution u of (1) under the condition of absence of exceptional points (e.g., see [12] and the discussion after formula (21) in [11])…”

Section: Introductionmentioning

confidence: 99%

“…In this case, equation (1) can be reduced to the equation ∇(q∇v) = 0 by the substitution u = √ qv. The case of so-called subcritical potentials was studied in [11]. Exceptional points for sign definite perturbations of conductive potentials were studied in [10], [18] under the condition that the potential and its perturbation are spherically symmetric.…”

Section: Introductionmentioning

confidence: 99%

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A test for the existence of exceptional points in the Faddeev scattering problem

Lakshtanov

Vainberg²

2017

Theor Math Phys

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Exceptional points are values of the spectral parameter for which the homogeneous Faddeev scattering problem has a non-trivial solution. We study the existence/absence of exceptional points for small perturbations of conductive potentials of arbitrary shape and show that problems with absorbing potentials do not have exceptional points in a neighborhood of the origin. A criterion for existence of exceptional points is given.

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Positive-energy D-bar method for acoustic tomography: a computational study

et al. 2016

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A new computational method for reconstructing a potential from the Dirichlet-to-Neumann map at positive energy is developed. The method is based on D-bar techniques and it works in absence of exceptional points -in particular, if the potential is small enough compared to the energy. Numerical tests reveal exceptional points for perturbed, radial potentials. Reconstructions for several potentials are computed using simulated Dirichlet-to-Neumann maps with and without added noise. The new reconstruction method is shown to work well for energy values between 10 −5 and 5, smaller values giving better results. Contentswhere u is the unique weak solution for the boundary value f , and v ∈ H 1 (Ω) with v| ∂Ω = g. As mentioned in [27] we then haveso our assumptions imply Λ q = h · Λ ω,κ,̺ . Assuming that h, k and ω are known, the inverse problem of AT then takes the following form: given Λ q and the energy E, reconstruct the potential q 0 . This is called the Gel'fand-Calderón problem posed by Gel'fand [8] and Calderón [6]. We can solve this problem using the D-bar method based of exponentially behaving Complex Geometric Optics (CGO) solutions first introduced by Faddeev [7] and later rediscovered in the context of inverse boundary-value problems by Sylvester and Uhlmann [38]. The D-bar method is based on the boundary integral equation proved by R. G. Novikov [31], the D-bar equation discovered by Beals and Coifman [1], and the relation of the CGO solution and the potential by R. G. Novikov [30]. See Nachman [28] for a discussion of the D-bar method applied to the AT problem.EIT and AT are related to the Gel'fand-Calderón problem by a transformation resulting to different energies: EIT is a zero-energy problem with E = 0 and AT is a positive energy problem with E > 0. In the zero-energy case in 2D, for conductivitytype potentials, Nachman [29] proved uniqueness and rigorously justified the D-bar reconstruction. The result was later generalized by Bukhgeim [3], who proved global uniqueness for general potentials at any fixed energy.The three novelties of this paper are:(1) We create a numerical algorithm for Faddeev Green's function for positive energy E > 0, a significant extension of the zero-energy case introduced in [36]. This is done in section 3 after which the function will be used throughout the numerical computations.(2) We investigate numerically the exceptional points which prevent the straightforward use of the D-bar method for reconstruction. This numerically complements the earlier works [25,26,16,17] focusing on the zero and non-zero energy cases. The results can be found in subsection 5.2. (3) We propose a new numerical algorithm for the D-bar method at positive energy and test it to reconstruct potentials using simulated DN-maps with and without added noise. In contrast to other methods, our algorithm is able to do reconstructions at low energies. See subsections 2.3 and 5.4 for comparisons with other reconstruction schemes ([34, 4, 3, 22]). The algorithm is detailed in section 4 and tested in subsection ...

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The nonlinear Fourier transform for two-dimensional subcritical potentials

Cited by 14 publications

References 17 publications

On reconstruction of complex-valued once differentiable conductivities

On reconstruction of complex-valued once differentiable conductivities

A test for the existence of exceptional points in the Faddeev scattering problem

Positive-energy D-bar method for acoustic tomography: a computational study

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