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Roman Novikov

Publications and source records attributed to Roman Novikov.

44 records · Page 3Linked to original sources
arXiv2010

On the reconstruction of conductivity of bordered two-dimensional surface in R^3 from electrical currents measurements on its boundary

An electrical potential U on bordered surface X (in Euclidien three-dimensional space) with isotropic conductivity function sigma>0 satisfies equation d(sigma d^cU)=0, where d^c is real operator associated with complex (conforme) structure on X induced by Euclidien metric of three-dimensional space. This paper gives exact reconstruction of conductivity function sigma on X from Dirichlet-to-Neumann mapping (for aforementioned conductivity equation) on the boundary of X. This paper extends to the case of the Riemann surfaces the reconstruction schemes of R.Novikov (1988) and of A.Bukhgeim (2008) given for the case of domains in two-dimensional Euclidien space. The paper extends and corrects the statements of Henkin-Michel (2008), where the inverse boundary value problem on the Riemann surfaces was firstly considered.

math.AP↗
arXiv2009

On iterative reconstruction in the nonlinearized polarization tomography

We give uniqueness theorem and reconstruction algorithm for the nonlinearized problem of finding the dielectric anisotropy f of the medium from non-overdetermined polarization tomography data. We assume that the medium has uniform background parameters and that the anisotropic (dielectric permeability) perturbation is described by symmetric and sufficiently small matrix-function f . On a pure mathematical level this article contributes to the theory of non-abelian Radon transforms and to iterative methods of inverse scattering.

math-ph↗
arXiv2008

An effectivization of the global reconstruction in the Gel'fand-Calderon inverse problem in three dimensions

By developing the d-bar approach to global "inverse scattering" at zero energy we give a principal effectivization of the global reconstruction method for the Gel'fand-Calderon inverse boundary value problem in three dimensions. This work goes back to results published by the author in 1987, 1988 and proceeds from recent progress in the d-bar approach to inverse scattering in 3D published by the author in 2005, 2006.

math.AP↗
arXiv2007

On problem of polarization tomography, I

The polarization tomography problem consists of recovering a matrix function f from the fundamental matrix of the equation $Dη/dt=π_{\dotγ}fη$ known for every geodesic $γ$ of a given Riemannian metric. Here $π_{\dotγ}$ is the orthogonal projection onto the hyperplan $\dotγ^{\perp}$. The problem arises in optical tomography of slightly anisotropic media. The local uniqueness theorem is proved: a $C^1$- small function f can be recovered from the data uniquely up to a natural obstruction. A partial global result is obtained in the case of the Euclidean metric on $R^3$.

math-ph↗
arXiv2006

On non-overdetermined inverse scattering at zero energy in three dimensions

We develop the d-bar -approach to inverse scattering at zero energy in dimensions d>=3 of [Beals, Coifman 1985], [Henkin, Novikov 1987] and [Novikov 2002]. As a result we give, in particular, uniqueness theorem, precise reconstruction procedure, stability estimate and approximate reconstruction for the problem of finding a sufficiently small potential v in the Schrodinger equation from a fixed non-overdetermined ("backscattering type") restriction h on $Γ$ of the Faddeev generalized scattering amplitude h in the complex domain at zero energy in dimension d=3. For sufficiently small potentials v we formulate also a characterization theorem for the aforementioned restriction h on $Γ$ and a new characterization theorem for the full Faddeev function h in the complex domain at zero energy in dimension d=3. We show that the results of the present work have direct applications to the electrical impedance tomography via a reduction given first in [Novikov, 1987, 1988].

math.AP↗
arXiv2005

The d-bar approach to approximate inverse scattering at fixed energy in three dimensions

We develop the d-bar approach to inverse scattering at fixed energy in dimensions $d\ge 3$ of [Beals, Coifman 1985] and [Henkin, Novikov 1987]. As a result we propose a stable method for nonlinear approximate finding a potential $v$ from its scattering amplitude $f$ at fixed energy $E>0$ in dimension $d=3$. In particular, in three dimensions we stably reconstruct n-times smooth potential $v$ with sufficient decay at infinity, $n>3$, from its scattering amplitude $f$ at fixed energy $E$ up to $O(E^{-(n-3-ε)/2})$ in the uniform norm as $E\to +\infty$ for any fixed arbitrary small $ε>0$ (that is with almost the same decay rate of the error for $E\to +\infty$ as in the linearized case near zero potential).

math-ph↗
arXiv2004

Formulas and equations for finding scattering data from the Dirichlet-to-Neumann map with nonzero background potential

For the Schrodinger equation at fixed energy with a potential supported in a bounded domain we give formulas and equations for finding scattering data from the Dirichlet-to-Neumann map with nonzero background potential. For the case of zero background potential these results were obtained in [R.G.Novikov, Multidimensional inverse spectral problem for the equation -Δψ+(v(x)-Eu(x))ψ=0, Funkt. Anal. i Ego Prilozhen 22(4), pp.11-22, (1988)].

cond-mat.other↗