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10References
1. Petrashen G. I. Solutions of vector limit problems of mathematical physics in the case of a sphere // Reports of the
academy of sciences. 1945. V. 46. N. 7 P. 291-294.
Preconditioning methods based on spanning tree algorithms and Schur complement techniques
G. A. Omarova1,2, D. V. Perevozkin1
1Institute of Computational Mathematics and Mathematical Geophysics SB RAS
2Novosibirsk State University
Email: foxillys@gmail.com
DOI 10.24412/cl-35065-2021-1-00-42
This work continues the evaluation of tree-based preconditioners started in [1, 2]. The new approach sug-
gests using Schur complement [3] to invert the preconditioning matrix instead of using direct solvers for build-
ing the preconditioner. This allows to reduce both memory footprint and computation time. Also, this ap-
proach is a good candidate for parallelization. We compare its convergence rate and performance with some
well-known preconditioners and solvers. The comparison is performed using SLAEs from the University of Flor-
ida�s sparse matrix collection [4].
References
1. Pravin M. Vaidya. Solving linear equations with symmetric diagonally dominant matrices by constructing good
preconditioners. Unpublished manuscript. A talk based on the manuscript was presented at the IMA Workshop on Graph
Theory and Sparse Matrix Computation, October 1991, Minneapolis.
2. Chen D, Toledo S. Implementation and evaluation of Vaidya�s preconditioners. Preconditioning 2001. 2001.
3. Saad, Y. Iterative methods for sparse linear systems. Society for Industrial and Applied Mathematics, 2003.
4. SuiteSparse Matrix Collection Formerly the University of Florida Sparse Matrix Collection. [Electron. resource].
URL: https://sparse.tamu.edu/ (date of access: 27.05.2021).
Comparative analysis of methods for solving SLAE in three-dimensional initial-boundary value problem
M. S. Pekhterev, V. P. Ilin, V. S. Gladkikh
Institute of Computational Mathematics and Mathematical Geophysics SB RAS
Email: maxim-pekhterev@mail.ru
DOI 10.24412/cl-35065-2021-1-00-43
An experimental study of the performance of rendering methods for systems of linear algebraic equations
(SLAE) in implicit approximation schemes for three-dimensional boundary value problems in modeling non-
stationary heat conduction processes with phase transitions is carried out [1, 4]. Algorithms of finite volumes
on an unstructured grid [2] and various approaches to temporal approximation [5] are considered. To improve
the performance of the preconditioned iterative processes used in the Krylov subspaces [3], the technique of
choosing the optimal initial residuals at each time step is used. The effectiveness of the proposed approaches
is demonstrated in a representative series of methodological experiments.
This work was supported by the RSF-19-11-00048.
References
1. Vabishchevich P. N., Samarsky A.A. Computational heat transfer // M. Editorial URSS, 2003.
2. Petukhov A.V. Solution of the three-dimensional complex Heimholtz equation by the method of barycentric finite
volumes // Avtometriya, publishing house SB RAS. 2007. T. 43. No. 2. 112�124.
3. Ilyin V.P. Methods of finite differences and finite volumes for elliptic equations // Novosibirsk, publishing house of
ICMiMG SB RAS. 2001.
4. Gladkikh V.S., Petukhov A.V. Numerical simulation non-stationary heat fields with taking into account phase
transition // Conference proceedings � XIX All-Russian Scientific Conference-School Modern Problems of Mathematical
Modeling".
5. Samarskiy A.A. Introduction to the theory of difference schemes // M.: Nauka, 1971.
Crystal structures and continued fractions
L. V. Pekhtereva, V. A. Seleznev
Novosibirsk State Technical University
Email: seleznev@corp.nstu.ru
DOI 10.24412/cl-35065-2021-1-00-44
We consider a model of the crystal structure based on the representation of finite continued fractions by
unimodular morphisms of a plane integer lattice. The specified representation and properties of these uni-
modular morphisms are obtained in [1]. The model constructed here allows us to explain the existing limita-
tions of the sets of Weiss parameters (the rational ratio of the lengths of the edges of the forming cell) of the
crystal lattice by the Gaussian distribution of natural numbers in the representation of continued fractions.
References
1. Seleznev V. A., Pekhtereva L. V. Matrix algorithm for calculating continued fractions // Marchuk Scientific
Readings � 2019. Advanced Mathematics, Computations and Applications: proceedings of the international conference,
Novosibirsk, July 1-5, 2019. Novosibirsk: PPC NSU, 2019. P. 444-447.
On the application of a modified genetic algorithm to optimize the parameters of methods for solving
systems of linear algebraic equations
A. A. Petrushov1,2, B. I. Krasnopolsky1
1Research Institute of Mechanics, Lomonosov Moscow State University
2Lomonosov Moscow State University
Email: petrushov.aa18@physics.msu.ru, krasnopolsky@imec.msu.ru
DOI 10.24412/cl-35065-2021-1-00-45
The solution of a system of linear algebraic equations (SLAE) is among the most frequently encountered
stages of mathematical physics problems. In a number of computational fluid dynamics (CFD) applications, this
stage takes more than 90% of the overall calculation time. Reducing the simulation time (and hence the time
for solving SLAEs) can significantly speed up the calculations.
This work presents a modified genetic algorithm for automated parameters selection for the SLAE solution
methods. The algorithm includes a neural network model that generalizes the SLAE solution statistics and lim-
its the search area for the optimal parameters set. The performance of the algorithm is shown on SLAEs for the
model Poisson equations and SLAEs from CFD calculations. An increase in search efficiency is shown when the
neural network model is included in the basic genetic algorithm. The quantitative results of solution accelerat-
ing for SLAEs of various sizes are presented (30 % acceleration in comparison with manual parameters selec-
tion).
This work was supported by the Russian Science Foundation, Grant No. 18-71-10075.
