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Siberian Journal of Numerical Mathematics

2015 year, number 4

An algorithm of the simplex method using a dual basis

Gerard Idelfonovich Zabinyako
Institute of Computational Mathematics and Mathematical Geophysics SB RAS, pr. Lavrentieva, 6, Novosibirsk, 630090
Keywords: LU-разложения, обновление разложений, разреженные матрицы, симплекс-метод, линейное программирование, LU-decomposition, decomposition updating, sparse matrices, simplex method, linear programming

Abstract >>
An algorithm of the simplex method not requiring an explicit updating of the LU decomposition in iterations is considered. Solutions obtained with fixed LU factors are corrected using small auxiliary matrices. The results of numerical experiments are presented.

Oligopolistic interacting markets

Valery Ivanovich Zorkaltsev, Marina Alexandrovna Kiseleva
Melentiev Energy Systems Institute of Siberian Branch of the Russian Academy of Sciences, Lermontov St., 130, Irkutsk, 664033, Russia
Keywords: модель Курно, равновесие Нэша, потенциальная игра, Cournot model, Nash equilibria, potential game

Abstract >>
The model of several interacting Cournot markets is considered. The markets are named interacting because the same number of producers act on each of them. Every producer chooses his own supply volumes on every market using the price situations, his own costs and production and delivery limitations. It is proved that in the case of the linear demand functions the problem of finding the Nash equilibria in the interacting Cournot markets model represents a potential game, i. e. it is equivalent to a mathematical programming problem. Nonlinear demand functions linearization procedures and preferences of initial problem reduction to the potential game are discussed.

Normality conditions for semilinear matrix operators of the Stein type

Khakim Dododjanovich Ikramov
Lomonosov Moscow State University, GSP-1, Leninskie Gory, Moscow, 119991, Russian
Keywords: матричное уравнение Стейна, полулинейный оператор, сопряженный оператор, самосопряженность, нормальность, одновременное сингулярное разложение, сопряженно-нормальная матрица, Stein matrix equation, semilinear operator, adjoint operator, self-adjointness, normality, simultaneous singular value decomposition, conjugate-normal matrix

Abstract >>
Normality conditions are found for the operators associated with the semilinear analogs of the Stein matrix equation, namely, with the equations X A X B = C and  AX*B = C.

The assessment of tsunami heights above the bottom slope within the wave-ray approach

An.G. Marchuk
Institute of Computational Mathematics and Mathematical Geophysics SB RAS, pr. Lavrentieva, 6, Novosibirsk, 630090
Keywords: распространение цунами, уравнения мелкой воды, волновой луч, кинематика волнового фронта, tsunami propagation, shallow-water equations, wave ray, wave front kinematics

Abstract >>
In this paper, the kinematics of the tsunami wave ray and wave front above an uneven bottom is studied. The formula to determine the wave height along a ray tube is obtained. The exact analytical solution for the wave-ray trajectory above the bottom slope is derived. This solution gives the possibility to determine within the wave-ray approach the tsunami wave heights in an area with a sloping bottom relief. The distribution of the wave-height maxima in the area with the sloping bottom is compared to the one obtained by the numerical computation with a shallow-water model.

A new compact alternating group explicit iteration method for the solution of nonlinear time-dependent viscous Burgers' equation

Ranjan Kumar Mohanty1,2, Jyoti Talwar1
1South Asian University, Akbar Bhawan, Chanakyapuri, New Delhi, 110021, India
2University of Delhi, Delhi, 110007, India
Keywords: нелинейное параболическое уравнение, вязкий поток, компактный метод AGE, уравнение Бюргерса, число Рейнольдса, non-linear parabolic equation, viscous flow, Compact AGE Method, Burgers' equation, Reynolds number

Abstract >>
In this article, we discuss a new single sweep compact alternating group explicit method for the solution of time dependent viscous Burgers' equation both in Cartesian and polar coordinates. An error analysis for the new iterative method is discussed in detail. We have compared the results of the proposed iterative method with the results of a corresponding double sweep alternating group explicit (AGE) iterative method to demonstrate computationally the efficiency of the proposed method.

Solving the optimization problem of economic damage from environmental pollution by local sources

Ivan Sergeevich Novikov
Institute of Numerical Mathematics of the Russian Academy of Sciences, Gubkin str., 8, Moscow, 119333, Russia
Keywords: сопряженные уравнения, оптимальное управление, регуляризация Тихонова, экономический ущерб, численное моделирование загрязнений, adjoint equations, optimal control, Tikhonov regularization, economic damage, numerical modeling of pollution

Abstract >>
An economic damage optimization problem from local sources in a region has been formulated. An algorithm for solving the problem is proposed. Numerical experiments illustrating theoretical statements of the formulated problem and effectiveness of the algorithm proposed were carried out.

On applying Monte Carlo methods to analysis of nonlinear regression models

Georgy Igorevich Rudoy
Institution of Russian Academy of Sciences Dorodnicyn Computing Centre of RAS, Vavilov St. 40, 119333, Moscow, Russia
Keywords: символьная регрессия, нелинейные модели, устойчивость решений, дисперсия прозрачной среды, методы класса Монте-Карло, symbolic regression, nonlinear models, solution stability, transparent medium dispersion, Monte Carlo methods

Abstract >>
This paper presents a criterium, called the coefficients stability for inaccuracy in determining the coefficients of nonlinear regression models describing inexact data. A method for the coefficients stability estimation is also described. The proposed criterium is illustrated by a computational experiment with the data obtained by measurements of a refractive index dependence on the wavelength in 400−1000 nm band for a transparent polymer. The convergence of the proposed criterium to the known analytical solution for the case of linear regression is also studied.

Heat transfer modeling of an electroconductive liquid in a spherical layer

Sergey Victorovich Solovjov
Pacific National University, Tihookeanskaya St. 136, Khabarovsk, 680035, Russia
Keywords: математическое моделирование, конвективный теплообмен, джоулева диссипация, магнитная гидродинамика, сферический слой, mathematical modeling, convective heat transfer, Joule dissipation, magnetic hydrodynamics, spherical layer

Abstract >>
In this paper, based on mathematical modeling, the convective heat transfer of an electroconductive liquid with regard to the internal sources of heat and the Joule dissipation in a spherical layer with heat from below is investigated. The structure of a flow, temperature field, magnetic field distribution and the Nusselt numbers are investigated.

A splitting algorithm for wavelet transforms of the Hermite splines of the seventh degree

Boris Mikhailovich Shumilov
Tomsk State University of Architecture and Building, Solyanaya sq., 2, 634003, Tomsk, Russia
Keywords: эрмитовы сплайны, В«ленивыеВ» вейвлеты, неявные соотношения разложения, распараллеливание, Hermite splines, В«lazyВ» wavelets, implicit relations of decomposition, parallelization

Abstract >>
In this paper, an implicit method of decomposition of 7-th degree Hermite splines to a series of «lazy» wavelets with displaced supports is investigated. A splitting algorithm for wavelet transforms of solving four five-diagonal s stems of linear equations with a strict diagonal dominance in parallel is justified. Results of numerical experiments on exactness for polynomials and on compression of spline-wavelet decomposition are presented.