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4A 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
vector case; three and six for 2-tensor case. The recovery algorithms are based on the approximate inverse
method [2, 3].
This research was partially supported by RFBR and DFG according to the research project 19-51-12008.
References
1. Sharafutdinov V. A. Slice-by-slice reconstruction algorithm for vector tomography with incomplete data. Inverse
Problems. 2007. V. 23, No 6. P. 2603�2627.
2. Louis A. K., Maass P. A mollifier method for linear operator equations of the first kind. Inverse Problems. 1990.
V. 6, No 3. P. 427�440.
3. Derevtsov E. Yu., Louis A. K., Maltseva S. V., Polyakova A. P., Svetov I. E. Numerical solvers based on the method of
approximate inverse for 2D vector and 2-tensor tomography problems. 2017. Inverse Problems. V 33, No 12, 124001.
An inverse problem for a system of nonlinear parabolic equations
E. V. Tabarintseva
South Ural State University, Chelyabinsk
Email: eltab@rambler.ru
DOI 10.24412/cl-35065-2021-1-02-16
An inverse problem for a system of nonlinear parabolic equations is considered in the present paper.
Namely, it is required to restore the initial condition by a given time-average value of the solution to the
system of the nonlinear parabolic equations. An exact in the order error estimate of the optimal method for
solving the inverse problem through the error estimate for the corresponding linear problem is obtained. A
stable approximate solution to the unstable nonlinear problem under study is constructed by means of the
projection regularization method which consists of using the representation of the approximate solution as a
partial sum of the Fourier series and the auxiliary boundary conditions method. An exact in the order estimate
for the error of the projection regularization method is obtained on one of the standard correctness classes. As
a consequence, it is proved the optimality of the projection regularization method. As an example of a
nonlinear system of parabolic equations, which has important practical applications, a spatially distributed
model of blood coagulation is considered. Numerical examples are given to confirm the theoretical results.
This work was supported by Act 211 Government of the Russian Federation, contract � 02.A03.21.0011.
References
1. Ivanov V. K., Vasin V.V., Tanana V.P. The Theory of Linear Ill-Posed Problems and Its Applications. VCP, 2002.
2. Denisov A. M. Inverse problem for a quasilinear system of partial differential equations with a nonlocal boundary
condition // Comput. Math. Math. Phys. 2014. V. 54 (10), P. 1513-1521.
3. Lobanov A. I., Starogilova T.K., Guria G.T. Numerical investigation of pattern formation in blood coagulation //
Matematicheskoe modelirovanie. 1997. V. 9 (3), P. 83-95.
Determination of the variable density of the rod from the natural frequencies of longitudinal vibrations
I. M. Utyashev
Mavlyutov Institute of Mechanics UFRC RAS, Ufa
Email: utyashevim@mail.ru
DOI 10.24412/cl-35065-2021-1-02-17
Rods of various configurations are elements of many structures and machines. Therefore, the acoustic and
vibration diagnostics of such parts has been widely developed [1-3]. The paper considers the problem of de-
termining the variable density of the rod from the natural frequencies of longitudinal vibrations. It is assumed
