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2016 year, number 3

1.
Using a randomized method of a maximum cross-section for simulating random structure systems with distributed transitions

Tat'yana Aleksandrovna Averina1,2
1Institute of Computational Mathematics and Mathematical Geophysics SB RAS, pr. Acad. Lavrentieva 6, Novosibirsk, 630090, Russia
2Novosibirsk State University, Pirogova st., 2, Novosibirsk, 630090, Russia
Keywords: статистическое моделирование, системы со случайной структурой, стохастические дифференциальные уравнения, пуассоновский поток, метод «максимального сечения», statistical simulation, systems with a random structure, stochastic differential equations, Poisson flow, numerical methods, maximum cross-section method

Abstract >>
In this paper we consider the random structure systems with distributed transitions. A theorem about the form of conditional structure distributions has been proved. To simulatethese distributions a statistical algorithm using a randomized method of maximum cross-section is constructed. Also, a modified version of this algorithm using the simulation of one random number has been constructed. The algorithms developed were used for the simulation of the numerical solution of random structure systems with distributed transitions. The theorem about a weak convergence of the numerical solution, obtained by the algorithms developed has been proved.



2.
The search for admissible solutions by the interior point algorithms

Valerii Ivanovich Zorkal'tsev
L. A. Melentiev Energy Systems Institute, Siberian Branch of the Russian Academy of Sciences, Lermontova st., 130, Irkutsk, 664033, Russia
Keywords: метод внутренних точек, линейное программирование, ввод в область допустимых решений, interior point algorithm, linear programming, techniques of arriving at the feasible solutions region

Abstract >>
A family of interior point algorithms for the linear programming problems is considered. In these algorithms, the entering into the domain of admissible solution of the original problem is represented as optimization process of the extended problem. This extension is realized by adding just one new variable. The main objective of the paper is to give a theoretical justification of the proposed procedure of entering into the feasible domain of the original problem, under the assumption of non-degeneracy of the extended problem. Particularly, we prove that given the constraints of the original problem being consistent, the procedure leads to a relative interior point of the feasible solutions domain.



3.
Preconditioning of GMRES by the skew-Hermitian iterations

Lev A. Krukier, Tatiana S. Martynova
SFU, Institute of mathematics, mechanics and computing science, Stachki Ave., Bld. 2, Rostov-on-Don, 344090, Russia
Keywords: эрмитово и косоэрмитово расщепление матрицы, итерационные методы, предобусловливание, методы подпространств Крылова, система уравнений с седловой матрицей, Hermitian and skew-Hermitian splitting, iterative methods, preconditioning, Krylov subspace method, saddle point linear system

Abstract >>
A class of preconditioners for solving non-Hermitian positive definite systems of linear algebraic equations is proposed and investigated. It is based on the Hermitian and skew-Hermitian splitting of the initial matrix. The generalization for saddle point systems which have semidefinite or singular (1,1) blocks is given. Our approach is based on an augmented Lagrangian formulation. It is shown that such preconditioners are effective for the iterative solution of systems of linear algebraic equations by the GMRES.



4.
Mathematical study of two-variable systems with adaptive numerical methods

Kolade M. Owolabi1,2
1University of the Western Cape Private, Bag X17, Bellville 7535, South Africa
2Federal University of Technology, Akure PMB 704, Akure, Ondo State, Nigeria
Keywords: модель хищник-жертва, ЭВР-методы, нелинейный, образование структур, реакция-диффузия, устойчивость, зависящие от времени дифференциальные уравнения в частных производных (ДУЧП), неустойчивость по Тьюрингу, predator-prey model, ETD methods, nonlinear, pattern formation, reaction-diffusion, stability, time-dependent PDE, Turing instability

Abstract >>
In this paper, we consider reaction-diffusion systems arising from two-component predator-prey models with Smith growth functional response. The mathematical approach used here is twofold, since the time-dependent partial differential equations consist of both linear and nonlinear terms. We discretize the stiff or moderately stiff term with a fourth-order difference operator, advance the resulting nonlinear system of ordinary differential equations with a family of two competing exponential time differencing (ETD) schemes, and analyze them for stability. A numerical comparison of these two methods for solving various predator-prey population models with functional responses is also presented. Numerical results show that the techniques require less computational work. Also in the numerical results, some emerging spatial patterns are unveiled.



5.
The influence of the shelf zone relief and the coastline geometry on coastal trapped waves

Gennadii Alekseevich Platov
Institute of Computational Mathematics and Mathematical Geophysics SB RAS, pr. Lavrentieva, 6, Novosibirsk, 630090, Russia
Keywords: береговые захваченные волны, шельфовая зона, окраинные моря, coastal trapped waves, shelf zone, marginal seas

Abstract >>
This paper presents the results of numerical experiments with a model of the coastal trapped waves, which made it possible to identify two features that are important in terms of the regional modeling of the shelf zone interaction with the open ocean. The first feature is the fact that the wave train of this type may be formed as a result of the wind action at a considerable distance from the place where their impact may occur. The propagation of waves along the coastline takes place without significant loss of wave energy, provided that the coastline and topography of the shelf zone contain no features comparable to the Rossby radius. However, the wave loses its energy while passing capes, submarine canyons and in the case when the width of a shelf decreases. For the regional modeling, the possibility of remote wave generation should be well understood and taken into account. The second feature is that a propagating wave is able to spend part of its energy on the formation of density anomalies on a shelf by raising the intermediate waters of the adjoining offshore areas of the open ocean. Thus, the coastal trapped waves carry the wind energy from the areas of the wind impact to other coastal areas, where it can bring about the formation of density anomalies and other types of motion.



6.
Evaluation of statistical error when calculating velocity and temperature components by the direct simulation Monte Carlo method

Mikhail Yu. Plotnikov1, Elena Valer'evna Shkarupa2
1S.S. Kutateladze Institute of Thermophysics, Siberian Division of the Russian Academy of Sciences, pr. Lavrent'eva, 1, Novosibirsk, 630090, Russia
2Institute of Computational Mathematics and Mathematical Geophysics SB RAS, pr. Lavrentieva, 6, Novosibirsk, 630090, Russia
Keywords: прямое статистическое моделирование, статистическая погрешность, равновесная статистическая физика, direct simulation Monte Carlo method, statistical error, equilibrium statistical physics

Abstract >>
The direct simulation Monte Carlo method is now widely used to solve the problems of rarefied gas dynamics. While solving stationary problems a special feature of the method is using dependent sample values of random variables to calculate macroparameters of a gas flow. In this paper, the possibility of using the results of statistical physics to estimate the statistical error of the DSMC method is theoretically analyzed. A simple approach to approximate evaluating the statistical error while calculating components of the velocity and temperature is proposed. The approach is tested on a number of problems.



7.
On an algorithm of bilateral restrictions smoothing with spline

Aleksandr Iosifovich Rozhenko1, Egor A. Fedorov2
1Institute of Computational Mathematics and Mathematical Geophysics SB RAS, pr. Acad. Lavrentieva, 6, Novosibirsk, 630090, Russia
2ООО «Data Ist», pr. Acad. Lavrentieva, 22, Novosibirsk, 630090, Russia
Keywords: сглаживание, сплайн, гильбертово пространство, выпуклое программирование, воспроизводящее отображение, радиальная базисная функция, smoothing, spline, Hilbert space, convex programming, reproducing mapping, radial basis function

Abstract >>
In this paper, the problem of constructing a spline σ in the Hilbert space satisfying bilateral restrictions z- ≤ A σ ≤ z+ with a linear operator A and minimizing a squared Hilbert seminorm is studied. A solution to this problem could be obtained with the convex programming iterative methods, in particular, with the gradient projection method. A modification of the gradient projection method allowing one to reveal a set of active restrictions in a smaller number of iterations is offered. The efficiency of the modification proposed is shown on the problem of approximation with a pseudo-linear bivariate spline.