Rubric: "01.01.00 Mathematics"
519.612.2 Performance analysis of iterative methods of combined linear algebraic equations solution
Marchevsky I. K. (Bauman Moscow State Technical University), Puzikova V. V. (Bauman Moscow State Technical University)
doi: 10.18698/2309-3684-2014-4-3752
When sampling partial differential equations one has to solve a system of linear algebraic equations. To select the optimal in the sense of the computational efficiency of iterative method for solving such equations, in addition to the rate of convergence we should take into account such characteristics of the system and method, as the condition number, the smoothing factor, the indicator "costs on." The last two characteristics are calculated by the coefficients of harmonics amplification that give evidence of the smoothing properties of the iterative method and its "costs on", i. e. how worse the method suppresses frequency components of the error as compared with the highfrequency ones. The suggested method of determining harmonic gain factors is based on of the discrete Fourier transform. As an example, an analysis of the effectiveness of the BiCGStab method with ILU and multigrid preconditioning when solving difference analogues of the Helmholtz and Poisson equations is described.
Marchevsky I., Puzikova V. Performance analysis of iterative methods of combined linear algebraic equations solution. Маthematical Modeling and Coтputational Methods, 2014, №4 (4), pp. 37-52
Dimitrienko Y. I. (Bauman Moscow State Technical University), Dimitrienko O. Y. (Bauman Moscow State Technical University)
doi: 10.18698/2309-3684-2016-1-105122
The article considers the concept of applying the multidimensional continuum model to one of the main problems emerging in the theory of large scale data array treatment i.e. forecasting the dynamics of data cluster change. The concept is based on the model of multidimensional continua in spaces of high dimensionality (more than three) earlier developed by the authors. The model includes the integral conservation laws, which are reformulated for informational data clusters, as well as the model of motion kinematics and cluster deformation. The model of deformable multidimensional cluster is developed. The movement of the cluster in multidimensional data space includes translational and rotational motion and uniform tension-compression strain. The system of differential tensor equations describing the dynamics of the deformable multivariate cluster motion over time is formulated. A numerical algorithm for solving the system of differential equations for the ellipsoidal model of multidimensional cluster is worked out. An example of the developed model application for predicting the dynamics of economic data (data on goods purchases in a large supermarket) is considered. The results of forecasting the data on purchases of different consumer groups are shown.
Dimitrienko Y., Dimitrienko O. A model of multidimensional deformable continuum for forecasting the dynamics of large scale array of individual data. Маthematical Modeling and Coтputational Methods, 2016, №1 (9), pp. 105-122
519.63 Development and testing for methods of solving stiff ordinary differential equations
Galanin M. P. (Bauman Moscow State Technical University/Keldysh Institute of Applied Mathematics of the Russian Academy of Scienсes), Khodzhaeva S. R. (Bauman Moscow State Technical University)
doi: 10.18698/2309-3684-2014-4-95119
The paper is aimed at research of the (m,k)-method, CROS, finite superelement method and 4-stage explicit Runge–Kutta method for solving stiff systems of ordinary differential equations. Analysis of tests results showed that the best choice for systems with high stiffness is CROS. The finite superelement method is the «precise» method for solving linear systems of ordinary differential equations, the best supporting method for its implementation is (4,2)-method. The variation of the finite superelement method has been built and tested for solving nonlinear problems, this method proved to be unsuitable for problems with high stiffness.
Galanin M., Khodzhaeva S. Development and testing for methods of solving stiff ordinary differential equations. Маthematical Modeling and Coтputational Methods, 2014, №4 (4), pp. 95-119
Bazilevsky M. P. (Irkutsk State Transport University)
doi: 10.18698/2309-3684-2016-2-104116
The article deals with the problem of building regression models, in which all variables have stochastic nature. To solve it, we propose to use the determination coefficient. We obtain analytical dependencies of the determination coefficients from the ratio of error variances of the test items. We set the optimization problem, assuming the maximization of the determination coefficients sum for each Deming regression equation. We give a model example of the numerical processing of Deming regression with its parameters and sign errors which are known.
Bazilevsky M. Analytical dependences between the determination coefficients and the ratio of error variances of the test items in Deming regression model. Маthematical Modeling and Coтputational Methods, 2016, №2 (10), pp. 104-116
Polyanin A. D. (Ishlinsky Institute for Problems in Mechanics), Sorokin V. G. (Ishlinsky Institute for Problems in Mechanics), Vyazmin A. V. (Moscow State University of Mechanical Engineering)
doi: 10.18698/2309-3684-2014-4-5373
In the article we explored nonlinear hyperbolic delay reaction-diffusion equations with varying transfer coefficients. A number of generalized separable solutions were obtained. Most of the equations considered contain arbitrary functions. Global nonlinear instability conditions of solutions of hyperbolic delay reaction-diffusion systems were determined. The generalized Stokes problem for a linear delay diffusion equation with periodic boundary conditions was solved.
Polyanin A., Sorokin V., Vyazmin A. Nonlinear delay reaction-diffusion equations of hyperbolic type: Exact solutions and global instability. Маthematical Modeling and Coтputational Methods, 2014, №4 (4), pp. 53-73
517.9:519.6 Analysis of bifurcations in double-mode approximation for Kuramoto — Tsuzuki system
Malinetsky G. G. (Keldysh Institute of Applied Mathematics of the Russian Academy of Scienсes), Faller D. S. (Keldysh Institute of Applied Mathematics of the Russian Academy of Scienсes)
doi: 10.18698/2309-3684-2014-3-111125
The article discusses emergence of chaotic attractors in the system of three ordinary differential equations arising in the theory of reaction–diffusion models. We studied the dynamics of the corresponding one- and two-dimensional maps and Lyapunov exponents of such attractors. We have shown that chaos is emerging in an unconventional pattern with chaotic regimes emerging and disappearing repeatedly. We had already studied this unconventional pattern for one-dimensional maps with a sharp apex and a quadratic minimum. We applied numerical analysis to study characteristic properties of the system, such as bistability and hyperbolicity zones, crisis of chaotic attractors.
Malinetsky G., Faller D. Analysis of bifurcations in double-mode approximation for Kuramoto — Tsuzuki system. Маthematical Modeling and Coтputational Methods, 2014, №3 (3), pp. 111-125
Andrianov I. K. (Комсомольский-на-Амуре государственный технический университет), Grinkrug M. S. (Комсомольский-на-Амуре государственный технический университет)
doi: 10.18698/2309-3684-2016-2-2438
The paper considers a mathematical model of the heat exchange process occurring in the thin-walled curved shells of turbomachines. We propose an algorithm for calculating the thermal state on the boundary surfaces of the shell and of the coating according to the desired thermal condition. We present the calculation results of the temperature distribution in the given temperature field for the most terminally loaded shell surface caused by the heat effect.
Andrianov I., Grinkrug M. Parametric identification of a mathematical model of heat exchange process for thin-walled curved shells of turbomachines. Маthematical Modeling and Coтputational Methods, 2016, №2 (10), pp. 24-38
519.8 “Mixed” probabilistic models of bilateral military operations of numerous groups
Chuev V. U. (Bauman Moscow State Technical University), Dubogray I. V. (Bauman Moscow State Technical University), Dyakova L. N. (Bauman Moscow State Technical University)
doi: 10.18698/2309-3684-2017-1-91101
The purpose of this work was to develop "mixed" probabilistic models of bilateral military operations according to the theory of continuous Markov processes. In our research we obtained calculation formulas for estimating the main combat indices of groups small in number. Moreover, we developed a numerical algorithm to calculate the main combat indices of numerous groupings and made a comparison with the results of combat simulation using a deterministic model of two-way combat operations, the model being developed according to themean-value method dynamics. Findings of the research show that the correlation of the forces of the opposing sides, rather than their initial numbers, affects the errors in the mean-value method dynamics.
Chuev V., Dubogray I., Dyakova L. “Mixed” probabilistic models of bilateral military operations of numerous groups. Маthematical Modeling and Coтputational Methods, 2017, №1 (13), pp. 91-101
519.8 Stochastic models of the two unit duel fight
Chuev V. U. (Bauman Moscow State Technical University), Dubogray I. V. (Bauman Moscow State Technical University)
doi: 10.18698/2309-3684-2016-2-6984
On the basis of the theory of continuous Markov processes we developed models of the two unit duel fight. We obtained computing formulas for calculating the basic fight indicators. Moreover, we found that the pre-emptive strike of one of the units participating in the fight has a significant impact on the fight outcome of the units which are similar in forces. The strike has a negligible impact, if one of the units has a significant advantage. The findings of the research show that the use of model with constant effective firing rates can lead to significant errors in the evaluation of its results. Finally, we found that the pre-emptive strike, coupled with a high degree of effective firing rate growth, can sometimes compensate for more than the double initial superiority of the opponent. We show the possibility of using approximations of the effective firing rate of the fighting units by the different functions of the fight time.
Chuev V., Dubogray I. Stochastic models of the two unit duel fight. Маthematical Modeling and Coтputational Methods, 2016, №2 (10), pp. 69-84