Faddeev Eigenfunctions for Multipoint Potentials

Grinevich, P. G.; Novikov, Roman

doi:10.32523/2306-6172-2013-1-2-76-91

Cited by 10 publications

(13 citation statements)

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“…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.…”

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confidence: 84%

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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confidence: 84%

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

et al. 2016

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“…We remark that the functions ψ of (4) essentially differ from the Faddeev eigenfunctions found in [4,5] for the Schrödinger operators with multi-point delta-type potentials. The reason is that in [4,5] the operator with such a potential is replaced by its regularization going back to [6], whereas in the present note we work formally with the original potentials considering the regularization, of the equation Hθ = 0, given by the Moutard system (3).…”

mentioning

confidence: 77%

“…This equation has the Manakov form H t = HA + BH where A and B are differential operators. If U satisfies (5) and ω meets (1) and the equation…”

mentioning

confidence: 99%

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The Moutard transformation and two-dimensional multipoint delta-type potentials

Novikov¹,

Тайманов²

2013

Russ. Math. Surv.

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“…We consider the model of point scatterers in three dimensions, which goes back to the classical works [4], [6], [9], [3] and presented in detail in the book [1]. For more recent results on such models, see [5], [2], [7] and references therein. More precisely, we consider the stationary Schrödiger equation…”

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confidence: 99%