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5We present the visualization of halos and anti-halos generated by the cirrus clouds for different shapes and
orientation of ice crystals.
The study was carried out under the CSC (China Scholarship Council) and the State Contract with ICMMG SB RAS
(0251-2021-0002).
References
1. Marchuk G. A., Mikhailov G. I., Nazaraliev M. A., Darbinian R. A., Kargin B. A., Elepov B. S. Monte Carlo methods in
atmospheric optics. Springer-Verlag, Berlin, Heidelberg, New York, 1980.
2. Tape W. Atmospheric Halos. M.: American Geophysical Union, 1994.
Study of asymptotics of particle transfer process with multiplication in a random medium
S. A. Rozhenko
The Institute of Computational Mathematics and Mathematical Geophysics SB RAS
Email: sergroj@mail.ru
DOI 10.24412/cl-35065-2021-1-00-87
Simulation is carried out using weight modeling and double randomization in order to estimate the aver-
age particle flow from a random medium in which particle multiplication occurs.
The main goal of this work is to study the possibility of superexponential asymptotics being realized for a
standard model of an isotropic random field of density of the medium.
At the same time, for small correlation radii, a radical reduction in the complexity of calculations could be
achieved by replacing double randomization with randomized modeling of trajectories taking into account the
value of the correlation length.
This work was carried out under state contract with ICMMG SB RAS number 0251-2021-0002.
A vector Monte Carlo algorithm for large systems of linear equations
K. K. Sabelfeld1,2, A. E. Kireeva1
1Institute of Computational Mathematics and Mathematical Geophysics SB RAS
2Novosibirsk State University
Email: karl@osmf.sscc.ru, kireeva@ssd.sscc.ru
DOI 10.24412/cl-35065-2021-1-00-89
A Monte Carlo randomization algorithm for solving large systems of linear algebraic equations is present-
ed. This algorithm combines the randomized stochastic matrix based algorithms proposed in [1], and an itera-
tive method of solving integral equations suggested in [2] which has no spectral restrictions on its convergens
in contrast to the conventional Neumann series based method. We develop a special transform of the original
non-negative matrix to a column stochastic matrix which is then conveniently used for calculation of matrix
iterations. The algorithm of randomized calculation of matrix iterations proposed in [1] operates by sampling
random rows and columns instead of matrix-matrix and matrix-vector multiplications. To solve a system of
linear algebraic equations with a matrix whose eigenvalues are greater than 1, we apply the transformation
and the relevant iterative procedure given in [2]. We analyze the correctness, laboriousness and efficiency of
the method for various matrix sizes. As a byproduct, a vector random walk on grids and a modified random
walk on boundary algorithms for three-dimensional potential problems are constructed.
This work was supported by the Russian Science Foundation under grant � 19-11-00019, and the Russian Fund of
Fundamental Studies under Grant 20-51-18009 in the part of random walk process implementations.
References
1. Sabelfeld Karl K. Vector Monte Carlo stochastic matrix-based algorithms for large linear systems // Monte Carlo
Methods and Applications, De Gruyter, 2016, v.22 (3), 259-264.
2. Polozhy G. N. On a method for solving integral equations // Izv. Academy of Sciences of the USSR. Ser. Mat. 1959,
v.23, Is. 2, 295-312 [In Russian], (Polozhij G. N. Ob odnom metode resheniya integral'nyh uravnenij // Izv. AN SSSR. Ser.
matem., 1959, Tom. 23, vypusk 2, 295-312).
A randomized iterative method for solving integral equations of the second kind
K. K. Sabelfeld1,2, I. A. Shalimova1,2
1Institute of Computational Mathematics and Mathematical Geophysics SB RAS
2Novosibirsk State University
Email: karl@osmf.sscc.ru, ias@osmf.sscc.ru
DOI 10.24412/cl-35065-2021-1-00-90
In this presentation we deal with an extension of the conventional Neumann series based Monte Carlo
method for solving integral equations of the second kind. This approach is based on a randomized evaluation
of the iterative procedure proposed by Polozhy in [1]. In contrast to the simple iteration method, this iterative
procedure converges without any spectral restriction on the integral operator. The major challenge encoun-
tered in the present study is the analysis of the variance behavior as a function of the number of iterations.
Preliminary simulations carried out for boundary integral equations of the potential theory indicates that this
behavior is most likely not linear. A discrete version of this stochastic algorithm implementation of the Polozhy
iterative method has been developed and presented in [3] where a vector randomization of the matrix itera-
tions suggested in [2] has been applied.
This work is supported by the Russian Science Foundation under grant � 19-11-00019, and the Russian Fund of Fun-
damental Studies under Grant 20-51-18009 in the part of random walk process implementations.
References
1. Polozhy G. N. On a method for solving integral equations // Izv. Academy of Sciences of the USSR. Ser. Mat. 1959,
v.23, Is. 2, 295-312 [In Russian], (Polozhij G. N. Ob odnom metode resheniya integral'nyh uravnenij // Izv. AN SSSR. Ser.
matem., 1959, Tom. 23, vypusk 2, 295-312).
2. Sabelfeld Karl K. Vector Monte Carlo stochastic matrix-based algorithms for large linear systems // Monte Carlo
Methods and Applications, De Gruyter, 2016, v.22 (3), 259-264.
Using DSMC calculations to estimate heterogeneous reaction constants based on experimental data
E. V. Shkarupa1, M.Yu. Plotnikov2
1Institute of Computational Mathematics and Mathematical Geophysics SB RAS
2Kutateladze Institute of Thermophysics of SB RAS
Email: sev@osmf.sscc.ru
DOI 10.24412/cl-35065-2021-1-00-91
The direct simulation Monte Carlo method (DSMC) is widely used in solving problems of the rarefied gas
dynamics. At the present stage, one of the promising areas of its use is the study of the interaction of gas with
surfaces as applied to the problem of gas "activation" on catalytic surfaces. Various approaches to modeling
heterogeneous reactions by the DSMC method are being developed. By virtue of the structure of the DSMC
method, it is fundamental for all the approaches to use microscopic reaction probability when a particle col-
