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9control. The solvability of a multiplicative control problem under consideration was proved in [2]. For concrete
reaction and mass transfer coefficients we obtain optimality systems and based on their analysis we establish
local stability estimates for optimal solutions.
This work was carried out in the framework of the State Order for the Institute of Applied Mathematics of the Far
East Branch of the Russian Academy of Science, topic no. 075-01095-20-00.
References
1. Brizitskii R. V., Saritskaya Zh. Yu. Optimization analysis of the inverse coefficient problem for the nonlinear
convection-diffusion-reaction equation // J. Inv. Ill-Posed Probl. 2018. V. 26, N 6. P. 821-833.
2. Brizitskii R. V., Bystrova V. S., Saritskaia Zh. Yu. Analysis of boundary value and extremum problems for a nonlinear
reaction-diffusion-convection equation // Differential Equation. 2021. V. 57, N 5. P. 615-629.
On the inverse boundary value problem solution for parabolic equation in a cylindrical coordinate system
A. I. Sidikova
South Ural State University, Chelyabinsk
Email: sidikovaai@susu.ru
DOI 10.24412/cl-35065-2021-1-02-12
This work solves the problem the inner wall temperature determination of a hollow cylinder consisting of
composite materials. The problem is reduced to an ordinary differential equation using the Fourier transform
in time and the Fourier transform is found for the exact solution of the desired inverse boundary value
problem.
The projection regularization method [1] is used to solve it. This method makes it possible to obtain a sta-
ble solution to the problem and to estimate the error of the approximate solution. Working with Bessel func-
tions [2] complicates the technique of obtaining the error estimate. This problem is of known interest in con-
nection with the theory of thermocouples and devices for measuring current.
The work was supported by the Ministry of Science and Higher Education of the Russian Federation (government or-
der FENU-2020-0022).
References
1. V.K. Ivanov, V. V. Vasin and V.P. Tanana Theory of Linear Ill-Posed problems and its Applications. The Netherlands,
VSP, 2002.
2. Gray A., Mathews G.B. A Treatise on Bessel Functions and their Applications to Physics. M.: Nauka 1953.
Direct method for solving the problem of identifying the coefficients of an elliptic equation
S. B. Sorokin
Institute of Computational Mathematics and Mathematical Geophysics SB RAS
Novosibirsk State University
Email: sorokin@sscc.ru
DOI 10.24412/cl-35065-2021-1-02-13
A direct method for solving the inverse coefficient problem for an elliptic equation with piecewise con-
stant coefficients is presented. The method allows much faster, in comparison with the implementation of it-
erative procedures for this problem, to restore the unknown coefficients of thermal conductivity. The values
(measurements) of the solution at the break points of the coefficients are used as additional information.
A numerical study of the algorithm has shown its efficiency. For unperturbed additional information,
piecewise constant coefficients are reconstructed exactly regardless of the number of parts that make up the
object under study. An analysis of the sensitivity of the algorithm to the disturbance of additional information
showed that for bodies consisting of a small number of parts, with a disturbance of up to 5 %, the result of re-
storing the coefficients can be considered satisfactory. With an increase in the number of breakpoints of the
coefficients, the sensitivity to data perturbation increases.
This work was (partially) supported by state assignment of the Institute of Computational Mathematics and Mathe-
matical Geophysics of the SBRAS No. 0315-2019-0001.
Numerical solution of the inverse problems of magnetic cloaking shell design
J. E. Spivak
Institute of Applied Mathematics FEB RAS
Far Eastern Federal University, Vladivostok
Email: u3l3i3y3a3@mail.ru
DOI 10.24412/cl-35065-2021-1-02-14
In recent years, the inverse problems of designing devices serving to cloak material bodies from static
magnetic fields have been actively studied. The first solutions obtained in this field have a number of disad-
vantages. The main one is the difficulty of technical implementation. To simplify this difficulty, firstly, the work
assumes that the designed device consists of a finite number of concentric layers, each of which is filled with a
homogeneous isotropic medium. Second, the inverse problems under consideration are reduced by the opti-
mization method [1] to finite-dimensional extremum problems in which the role of controls is played by mag-
netic permeabilities of each layer. To find the required controls, a numerical algorithm developed on the parti-
cle swarm optimization method is used [2]. The obtained optimal solutions correspond to highly efficient and
technically easily realizable devices in the considered class of inverse problems [3].
This work was supported by the Russian Foundation for Basic Research, project no. 19-31-90039.
References
1. Tikhonov A.N., Arsenin V.Ya. Methods for solving ill-posed problems. M.: Nauka, 1974.
2. Alekseev G.V., Tereshko D.A. Particle swarm optimization-based algorithms for solving inverse problems of
designing thermal cloaking and shielding devices // Int. J. Heat Mass Transf. 2019. V. 135. P. 1269-1277.
3. Alekseev G.V., Spivak Y.E. Numerical analysis of two-dimensional magnetic cloaking problems based on an
optimization method // Diff. Equation. 2020. V. 56 (9). P. 1219-1229.
On solution of the slice-by-slice three-dimensional vector and 2-tensor tomography problems
by the approximate inverse method
I. E. Svetov1, S. V. Maltseva1, A. P. Polyakova1, A. K. Louis2
1Sobolev Institute of Mathematics SB RAS
2Saarland University, Saarbrucken, Germany
Email: svetovie@math.nsc.ru
DOI 10.24412/cl-35065-2021-1-02-15
A numerical solution of the problems of recovering the solenoidal part of a three-dimensional vector and
symmetric 2-tensor fields using the incomplete tomographic data are proposed. The initial data of the prob-
lems are values of the ray transforms for all straight lines, which are parallel to at least one of the planes from
a finite set of planes [1]. We consider two sets of planes, the number of planes in which are: two and three for
