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5singular value decomposition method. The singular value decompositions of the ray transforms operators act-
ing on vector and 2-tensor fields were constructed earlier [1, 2].
This research was partially supported by RFBR and DFG according to the research project 19-51-12008.
References
1. Derevtsov E. Yu., Efimov A. V., Louis A. K. and Schuster T. Singular value decomposition and its application to
numerical inversion for ray transforms in 2D vector tomography. J. of Inverse and Ill-posed Problems. 2011. V. 19, No 4�5.
P. 689�715.
2. Derevtsov E. Yu., Polyakova A. P. Solution of the Integral Geometry Problem for 2-Tensor Fields by the Singular
Value Decomposition Method. J. of Mathematical Sciences. 2014. V. 202, No 1. P. 50�71.
On the singular value decomposition of the dynamic ray transforms operators acting on 2D tensor fields
A. P. Polyakova1, I. E. Svetov1, B. Hahn2
1Sobolev Institute of Mathematics SB RAS
2University of Wurtzburg, Germany
Email: apolyakova@math.nsc.ru
DOI 10.24412/cl-35065-2021-1-02-07
We consider the problems of the dynamic 2D vector and 2-tensor tomography. The initial data are values
of the longitudinal and/or transverse and/or mixed dynamic ray transforms. An object motion is known and
consist of rotation and shifting [1]. Properties of the dynamic ray transforms operators are investigated. The
singular value decompositions of the operators with usage of the classic orthogonal polynomials are con-
structed [2, 3].
This research was partially supported by RFBR and DFG according to the research project 19-51-12008.
References
1. Hahn B. Null space and resolution in dynamic computerized tomography. Inverse Problems. 2016. V. 32, No 2,
025006.
2. Polyakova A. P., Svetov I. E., Hahn B. N. The Singular Value Decomposition of the Operators of the Dynamic Ray
Transforms Acting on 2D Vector Fields. Lect. Notes in Computer Science. 2020. V. 11974. P. 446�453.
3. Polyakova A. P., Svetov I.E. The singular value decomposition of the dynamic ray transforms operators acting on
2-tensor fields in R2. J. of Physics: Conference Series. 2021. V. 1715. 012040.
Regularization methods for solving the continuation problem
A. Prikhodko1,2, M. Shishlenin1,2
1Institute of Computational Mathematics and Mathematical Geophysics SB RAS
2Novosibirsk State University
Email: a.prikhodko@g.nsu.ru
DOI 10.24412/cl-35065-2021-1-03-08
When conducting experiments for heat and mass transfer, the main studied values are the heat flux densi-
ty and temperature. The most commonly used field and local methods that have high accuracy, in particular
infrared thermography.
There are no direct methods for measuring the heat flux density at a distance. There is a need to estimate
the flow density from temperature measurements.
A method for inverting a finite-difference scheme is developed and implemented, a gradient method for
solving the inverse problem is implemented, and a gradient formula is obtained. A comparative analysis of the
developed methods is carried out. The methods are tested on experimental data.
The work was carried out with the financial support of the Russian Foundation for Basic Research (project code
20-51-54004).
References
1. Belonosov A., Shishlenin M., Klyuchinskiy D. A comparative analysis of numerical methods of solving the continua-
tion problem for 1D parabolic equation with the data given on the part of the boundary // Advances in Computational
Mathematics. 2019. V. 45, No 2. P. 735�755.
Regularization of inclusions of differential equations solutions based on the kinematics of a vector field
in problems of stability of a set of trajectories
A. N. Rogalev
Institute of Computational Modelling SB RAS, Krasnoyarsk
Email: rogalyov@icm.krasn.ru
DOI 10.24412/cl-35065-2021-1-02-08
In the papers of specialists on stability problems, it is proposed to investigate stability problems for a set
of trajectories of nonlinear dynamics under the influence of arbitrary perturbing influences [1]. In this report,
to estimate the stability of the sets of solutions, the boundaries of the sets of solutions will be efficiently and
accurately computed applying symbolic formulas for solutions of ordinary differential equations [2] and evalu-
ating the parameters of the kinematics of the vector field. The parameters of the vector field kinematics per-
form regularization, which eliminates the excessive growth of the boundaries of the inclusions. The examples
show the advantages of this approach to solving stability problems for a set of trajectories.
This work was carried out within the framework of the state assignment of the Federal Research Center of the KSC
SB RAS, project No. 0287-2021-0002.
References
1. Martynyuk A. Stability of a Set of Trajectories of Nonlinear Dynamics // Doklady Akademii Nauk. 2007. V. 414,
No. 3. P. 299-303.
2. Rogalev A.N., Rogalev A.A., Feodorova N.A. Numerical Computations of the Safe Boundaries of Complex Technical
Systems and Practical Stability// J. of Physics: Conference Series 1399 (2019) 033112. doi:10.1088/1742-6596/1399/3/
033112.
Regularization of the global error estimation for numerical solutions of differential equations based
on tracking approximate trajectories
A. N. Rogalev
Institute of Computational Modelling SB RAS, Krasnoyarsk
Email: rogalyov@icm.krasn.ru
DOI 10.24412/cl-35065-2021-1-02-09
The main idea of the regularization of the global error estimate is that new variables or new relations are
introduced that make it possible to reduce (or even eliminate) the influence of the Lyapunov instability on the
bounds of the numerical solutions and to refine these estimates. Such a regularization was performed in sev-
eral works of the author, for example, in [1]. In this report, it is proposed to consider the perturbed system as
a modified model based on pseudo-trajectories (approximate) trajectories, the effect of numerical errors on
