and Computational Methods

Rubric: "01.01.00 Mathematics"

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

doi: 10.18698/2309-3684-2017-1-102117

The article proposes a nonparametric criterion of the Kiefer — Gihman type to test the Cox model validity for several progressively censored samples. As estimates of the reliability function for each sample we are using the Kaplan — Meyer ones. The paper proves that if the hypothesis is valid, the Kiefer — Gihman distribution can be used as an approximation of the asymptotic distribution of the criterionstatistics. Based on the particle random walk model over a multidimensional cells array, the paper has developed the method for calculating the exact statistics distributions. The article presents obtained probability values tables of the proposed statistics exact distributions for a wide range of samples possible values. Statistical modeling methods show Cox parameters estimating method consistency, based on the statistics minimization. We present the obtained estimates histograms for the developments exponential distribution to failure. The research results are used when analyzing the redundant technical systems of different multiplicity tests results operating in different operating conditions.

Analyzed systems find applications in all industries — from machine building to radio electronic.

Timonin V., Tyannikova N. Cox model validity checking for several progressively censored samples. Маthematical Modeling and Coтputational Methods, 2017, №1 (13), pp. 102-117

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

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

doi: 10.18698/2309-3684-2017-1-321

The article introduces the methods and techniques of simulating various characteristics (such as cross-section, reaction rate etc.) of atomic and molecular physics elementary processes based on the quantum scattering theory within the system of several particles. We have analyzed the results of simulating the electrons and atoms processes of scattering by the diatomic and polyatomic molecules being in specific excited rovibrational states. The article considers different approximations necessary for constructing adequate models of the real physical systems consisting of several bodies which are applicable for both forward reactions and the reactions accompanied by the formation of the intermediate transition complex. We have compared the results of simulating the cross sections of the collisions between the electrons, atoms and molecules as well as between the molecules to the existing experimental data and calculations results of other researchers.

Pozdneev S. Simulating atomic and molecular physics processes based on the quantum scattering theory. Маthematical Modeling and Coтputational Methods, 2017, №1 (13), pp. 3-21

doi: 10.18698/2309-3684-2016-4-3446

For modeling piping systems we made a transition from the mass balance equations, based on 1m and 2m Kirchhoff laws, to the mathematical description of a hydraulic network using the continuity equation discretization. For this purpose we applied a controlvolume method. This paper introduces an extension of the developed control-volume method for extended period simulations in hydraulic networks. This extension is developed for slow time-varying conditions in the hydraulic networks and is not intended to calculate rapidly occurring local phenomena such as waterhammer. The control-volume method was successfully applied to test tasks.

Volkov V., Golibrodo L., Zorina I., Kudryavtsev O., Krutikov A., Skibin A. Applying the control−volume method to extended period simulations in pipe network hydraulics. Маthematical Modeling and Coтputational Methods, 2016, №4 (12), pp. 34-46

doi: 10.18698/2309-3684-2016-4-84102

The study examines extremum problems for eigen spectra components of Lagrangian dynamical systems. Mathematical models of the systems studied are described by the matrices depending on the parameters. The eigenproblems defined for such systems, in general, are characterized by a spectrum, which can contain multiple eigenvalues. Subtests in extremum problems are assumed to be continuous, Lipschitzian, multiextremum and maybe not everywhere differentiable functions. The search for global solutions is conducted using new hybrid algorithms that combine a stochastic algorithm for scanning the variables space and deterministic local search methods. The study gives numerical examples of solving the problems of global nondifferentiable minimization of the maximum systems eigenvalues.

Sulimov V., Shkapov P., Goncharov D. Use of hybrid algorithms in extremum eigenproblems of Lagrangian dynamical systems. Маthematical Modeling and Coтputational Methods, 2016, №4 (12), pp. 84-102

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

doi: 10.18698/2309-3684-2016-2-85103

The paper deals with the factorial modeling of the initial stage arterial hypertension. The modeling was carried out by the factorization method based on the neural network and the back propagation of error algorithm. This factorization method is an alternative to the classical factor analysis. We implemented an algorithm for constructing the factorial structure based on the neural network in software. This method has been improved for the factor rotation and obtaining an interpretable solution. The hypertension factorial structure obtained by this factorization method is in accordance with the results of the factorial modeling by other methods.

Chauvigny V., Goltiapin V. Factorial modeling using neural network. Маthematical Modeling and Coтputational Methods, 2016, №2 (10), pp. 85-103