Rubric: "01.01.00 Mathematics"
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
519.63 Parallel multigrid algorithms
Martynenko S. I. (Baranov Central Institute of Aviation Motor Development)
doi: 10.18698/2309-3684-2015-2-105120
The paper represents the main directions of development of the parallel classic multigrid algorithms and discusses their disadvantages. The possibility of efficient parallelization of smoothing iterations at the levels of coarse grids is shown using the Robust Multigrid Technique. Then multigrid structure is used for developing hybrid multigrid method. The paper describes estimations of speed-up and efficiency of different parallel multigrid algorithms as well as the results of numerical experiments.
Martynenko S. Parallel multigrid algorithms. Маthematical Modeling and Coтputational Methods, 2015, №2 (6), pp. 105-120
681.513.5 Stabilization of an unstable limit cycle of relay chaotic system
Krasnoschechenko V. I. (Bauman Moscow State Technical University)
doi: 10.18698/2309-3684-2015-2-87104
The article presents an algorithm of synthesis for stabilization of an unstable limit cycle of relay chaotic system. One-dimensional discrete Poincare map is used in algorithm for finding fixed points of the period one (limit cycles of initial continuous system). It is shown, that classical OGY method of dead beat regulator synthesis does not solve the problem as it takes into account only speed of the target coordinate what is not sufficient for stabilizing. The proposed algorithm is based on search of the necessary regulator factor by solving an inverse problem: at first some factor is assigned and then two-step procedure of system transition to the following switching point (with correction) is carried out. The task of correction is performed in a complete neighborhood of target coordinate position and speed, and it provides stabilization of a limit cycle by adjusting small amplitude pulses in the chosen area of entry conditions (area of stabilization) as evidenced by the simulation results.
Krasnoschechenko V. Stabilization of an unstable limit cycle of relay chaotic system. Маthematical Modeling and Coтputational Methods, 2015, №2 (6), pp. 87-104
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
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
Polyanin A. D. (Ishlinsky Institute for Problems in Mechanics), Zhurov A. I. (Cardiff University/Ishlinsky Institute for Problems in Mechanics)
doi: 10.18698/2309-3684-2015-4-337
We present a number of new simple separable, generalized separable, and functional separable solutions to one-dimensional nonlinear delay reaction-diffusion equations with varying transfer coefficients of the formut = [G(u)ux ]x F(u,w),where w = u(x,t) and w = u(x,t ), with denoting the delay time. All of the equations considered contain one, two, or three arbitrary functions of a single argument. The generalized separable solutions are sought in the form =1 = () () N
n n n u x t , withn (x) and n (t) to be determined in the analysis using a new modification of the functional constraints method. Some of the results are extended to nonlinear delay reaction-diffusion equations with time-varying delay = (t). We also present exact solutions to more complex, three-dimensional delay reactiondiffusion equations of the formut = div[G(u)u] F(u,w).Most of the solutions obtained involve free parameters, so they may be suitable for solving certain problems as well as testing approximate analytical and numerical methods for non-linear delay PDEs.
Polyanin A., Zhurov A. Nonlinear delay reaction-diffusion equations with varying transfer coefficients: generalized and functional separable solutions. Маthematical Modeling and Coтputational Methods, 2015, №4 (8), pp. 3-37
519.237.07 Factorial modeling using neural network
Chauvigny V. A. (Sobolev Institute of Mathematics, Omsk branch, Siberian Branch of the Russian Academy of Sciences), Goltiapin V. V. (Sobolev Institute of Mathematics, Omsk branch, Siberian Branch of the Russian Academy of Sciences)
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
519.6 Use of hybrid algorithms in extremum eigenproblems of Lagrangian dynamical systems
Sulimov V. D. (Bauman Moscow State Technical University), Shkapov P. M. (Bauman Moscow State Technical University), Goncharov D. A. (Bauman Moscow State Technical University)
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
Parkhomenko V. P. (ФИЦ ИУ РАН/Bauman Moscow State Technical University)
doi: 10.18698/2309-3684-2016-3-93109
The paper analyzes some factors affecting the parallel implementation performance of the atmospheric general circulation model designed on a cluster type multiprocessor computer. It considers several modifications of the initial parallel code of this model in order to improve both its computational efficiency and processor load balancing. The numerical scheme is modified according to the time of the atmospheric general circulation model for parallel computing of dynamics and physics blocks. The proposed procedure is used along with the procedures of paralleling the dynamics and physics blocks based on decomposition of the computational domain. It allows both optimizing the processor load balancing and increasing the paralleling efficiency. The data obtained while using the scheme for the physics block load balancing allow for complication of the physics block without increasing the total computational time. The results of numerical experiments are given.
Parkhomenko V. Algorithm for computational performance improvement and processor load balancing to simulate the general atmosphere circulation. Маthematical Modeling and Coтputational Methods, 2016, №3 (11), pp. 93-109