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9considered for non-scalar fields. This report contains a generalizations of this method for the case of vector
and 2-tensor fields. The results of numerical simulations are presented.
This work was supported by the Russian Foundation for Basic Research (RFBR) and the German Science Foundation
(DFG) according to the joint German-Russian research project 19-51-12008.
References
1. Natterer F 1986. The Mathematics of Computerized Tomography (New York: John Wiley & Sons).
2. Defrise M and De Mol C 1983 A regularized iterative algorithm for limited-angle inverse Radon transform // Opt.
Acta: Int. J. Opt 30. 403-408.
Comparative analysis of the Gerchberg � Papoulis method in the problems of scalar and vector tomography
S. V. Maltseva 1, V. V. Pickalov 2
1Sobolev Institute of Mathematics SB RAS
2Khristianovich Institute of Theoretical and Applied Mechanics SB RAS
Email: maltsevasv@math.nsc.ru; pickalov@itam.nsc.ru
DOI 10.24412/cl-35065-2021-1-02-03
The paper discusses the methods of solving the inverse problems of a few-views and limited angle data of
two-dimensional tomography using iterative procedures in Fourier space. Partial absence of angular data can
be approximately compensated by extrapolation in the Gerchberg � Papoulis iterative method [1]. In particu-
lar, these algorithms should be complemented by a priori information and regularization procedures [2�3].
The computational experiment compare reconstruction errors of mathematical models of scalar and vector
problems.
This research was partially supported by the Russian Foundation for Basic Research, project RFBR-DFG No. 19-51-
12008.
References
1. Defrise M., De Mol C. A regularized iterative algorithm for limited-angle inverse Radon transform // Optica Acta.
1983. V. 30, N. 4. P. 403-408.
2. Maltseva S. V., Svetov I. E., and Louis A. K. An iterative algorithm for reconstructing a 2D vector field by its limited-
angle ray transform // J. of Physics: Conference Series. 2021. V. 1715, art. 12037.
3. Derevtsov E.Yu., Pickalov V.V. Reconstruction of vector fields and their singularities from ray transforms //
Numerical Analysis and Applications. 2011. V. 4, N. 1. P. 21-35.
The problem of identification an unknown substance by the radiographic method
V. G. Nazarov
Institute of Applied Mathematics FEB RAS
Email: naz@iam.dvo.ru
DOI 10.24412/cl-35065-2021-1-02-04
The paper considers the problem of partial identification of the chemical composition of unknown medi-
um by the method of multiple transillumination of this medium with X-ray radiation. Sample of unknown sub-
stance is assumed to be homogeneous in chemical composition, and the photon flux, collimated both in direc-
tion and in energy. A mathematical model is formulated for the identification problem. The proposed ap-
proach to solving the problem is based on the method of singular value decomposition of a matrix [1, 2]. At
the first stage of solving the problem is reduced to finding singular numbers and singular vectors for the series
systems of algebraic equations linear with respect to products of unknown quantities. Then, based on the re-
ceived data, a special function is built, called an indicator of the distinguishability of substances, which enables
the sufficient conditions for the distinguishability of various substances [3]. Based on tabular data [4], calcula-
tions were made for a number of specific groups of chemical elements.
This work was supported by the Russian Foundation for Basic Research (project no. 20-01-00173).
References
1. Godunov S.K., Antonov A.G., Kiriluyk O.P., Kostin V.I. Guaranteed accuracy of solving systems of linear equations in
Euclidean spaces, Novosibirsk, Nauka, 1988, (in Russian).
2. Forsythe G.E., Malcolm M.A., Moler C.B. Computer methods for mathematical computations, Prentice-Hall, Inc.
Englewood Cliffs, N.J. 07632, 1977.
3. Nazarov V.G. Estimation of the Calculation Accuracy in the Problem of Partial Identification of a Substance // J. of
Applied and Industrial Mathematics, 2020. 14(3), pp. 555-565.
4. Berger M.J., Hubbell J.H., Seltzer S.M., Chang J., Coursey J.S., Sukumar R., Zucker D.S. XCOM: Photon Cross Section
Database, National Institute of Standards and Technology, Gaithersburg, 2005, URL: http://www.physics.nist.gov/xcom.
Optimization of solution of inverse problem for stochastic differential equation
A. V. Neverov1, O. I. Krivorotko1,2
1Novosibirsk State University
2Institute of Computational Mathematics and Mathematical Geophysics SB RAS
Email: a.neverov@g.nsu.ru
DOI 10.24412/cl-35065-2021-1-03-04
The problem of drift and volatility parameters identification in stochastic differential equations (SDEs) us-
ing additional measurement of single trajectory of stochastic process is investigated. The classical way for solv-
ing such a problem is to reduce it to a Fokker-Planck equation [1] and minimize a data fidelity functional, that
is unstable. For higher-dimensional systems of SDEs, the numerical solution of the Fokker-Planck equations
becomes infeasible.
We propose regularized Landweber iteration algorithm [2] with fidelity functional based on mathematical
expectations, for easier parallelization of solution. Parameters are implemented in solution-dependent form,
for implicit time-dependency, with Fourier series for reducing number of variables. In addition to that adjoint
problem is deterministic. We conduct this process on synthetic data for validation of an algorithm and regular-
ization for variety of input data with Tikhonov regularization.
This work was supported by the Russian Science Foundation (project no. 18-71-10044).
References
1. Risken H.: The Fokker-Planck equation. Methods of solution and applications. Springer Series in Synergetics,
Springer-Verlag, 1989.
2. Kaltenbacher B., Schopfer F., and Schuster T.: Iterative methods for non-linear ill-posed problems in Banach spac-
es: convergence and applications to parameter identification problems. Inverse Problems, 2009.
Numerical solution of inverse problem for acoustic equation, based on a modified version
of GLK-approach
N. S. Novikov
Institute of Computational Mathematics and Mathematical Geophysics SB RAS
Novosibirsk State University
Email: novikov-1989@yandex.ru
DOI 10.24412/cl-35065-2021-1-02-05
In this talk we consider the coefficient inverse problems for the acoustic equation in 1D and 2D cases
[1�3]. We propose the new version of the Gelfand � Levitan � Krein approach to reduce the non-linear inverse
