Number 1, pp. 1-105
Number 2, pp. 107-234
Number 3, pp. 237-347
Number 4, pp. 349-467
Aleksandrov Vladimir Mikhailovich
Computing of optimal inertial control with a linear system (in Russian), pp.1-13
Computing of time-optimal inertial control amounts to solving the three
problems: 1) computing of optimal control on the assumption that the control is
without inertia; 2) finding the optimal switching time of the control; 3)
calculating of the error induced by the time lag of the control followed by
correcting the control time and switching moments. Characteristics of the
problems are considered and methods of their solution are given. A way of
assignment of initial
approximation is presented. A computational algorithm, results of modeling and
numerical computations are performed.
Key words: optimal control, speed, switching moment, inertial switching, non-inertial switching, phase trajectory.
Artemiev Sergey Semenovich, Ivanov Aleksandr Aleksandrovich, Smirnov Dmitriy Dmitrievich
New frequency characteristics of the numerical solution to stochastic
The problems of the numerical analysis of stochastic differential equations with
oscillatory solutions trajectories are studied. For the analysis of the
numerical solution it is proposed to use the frequency response of generalizing
the integral curve and the phase portrait. The results of numerical experiments
carried out on a cluster of NCC-30T Siberian Supercomputer Center at the ICM\&MG
SB RAS using a set of programs PARMONC are presented.
Borovko IrinaVladimirovna, Krupchatnikov Vladimir Nikolaevich
Simulation of the response of the Hadley cell and extratropical
on the climate changes using a general atmosphere circulation of medium
In this paper, the response of the general atmosphere circulation to the climate
changes is simulated with use of a medium complexity model. It is shown that
with a temperature gradient decrease, the Hadley circulation weakens and its
boundary moves to the poles. The troposphere height dynamics depending on
atmosphere temperature is investigated. A relation between characteristics
determining the atmosphere baroclinic instability is obtained.
Key words: Hadley cell, atmosphere stratification, climate changes.
Gervich Lev Romanovich, Kravchenko Evgeniy Nikolaevich, Steinberg Boris Yakovlevich, Yurushkin Mikhail Viktorovich
Automatic program parallelization with block data distribution
This paper discusses several automated methods of acceleration programs. The
acceleration is achieved by parallelization and optimization of memory access.
Optimization of accesses to RAM is achieved by switching to a block code and
block placements arrays. When using a distributed memory, the automated
distribution of arrays and array distribution with overlapping are employed.
Automation is implemented using the C language with pragmas in Open
Parallelizing System. This paper presents the numerical results for linear
algebra and mathematical physics. Some features of this demonstration converter
have a remote access to the Internet.
Key words: automatic parallelization, tiling, memory, distributed memory, block distribution of arrays, optimization of memory, distribution with overlapping.
Kozak Anatoliy Vsevolodovich., Khanin Dmitriy Igorevich
Approximate solution of large systems of equations with multi-dimensional
The conditions using the inverse operator and its form in the truncated
two-dimensional convolution operators on sets with smooth boundaries are known.
The presence of the corner points adds complexity to the task. The equations
with multi-dimensional convolution operators on polyhedra is considered. The
approximate method for them is proposed, and estimates for the errors are
obtained. The possibility of approximation solutions of these equations with
multi-dimensional cyclic matrices is also investigated.
Key words: approximate solution, Toeplitz matrices, multi-dimensional cyclic matrices, multi-dimensional convolution operators on polyhedral.
Mohanty R.K., Talwar Jyoti
A new coupled reduced alternating group explicit method for non-linear
singular two point boundary value problems on a variable mesh
In this paper, we discuss a new coupled reduced alternating group explicit
(CRAGE) and Newton-CRAGE iteration methods to solve non-linear singular two
point boundary value problems u'' =
f(r,u,u'), 0< r< 1, subject
to given natural boundary conditions u(0)
u(1)= A2, where
A2 are finite constants,
along with a third order numerical method on a geometric mesh. The proposed
method is applicable to singular and non-singular problems. We discuss the
convergence of the CRAGE iteration method in detail. The results obtained from
the proposed CRAGE iteration method are compared with the results of the
corresponding two parameter alternating group explicit (TAGE) iteration methods
to demonstrate computationally the efficiency of the proposed method.
Key words: two point singular boundary value problems, geometric mesh, third order method, singular equation, CRAGE method, Newton-CRAGE method, Burgers equation, RMS errors.
Tarakanov Victor Ivanovich, Dubovik Alexey Olegovich.
Iterative algorithm for calculation of spectral parameters of a quadratic bunch of operators in the Hilbert space (in Russian), pp. 79-93
A new iterative algorithm is suggested for calculating spectral parameters of a
quadratic bunch of partially symmetrical compact operators in the Hilbert space.
Key words: operator, spectrum, iterative algorithm.
Khatuntseva Olga Nikolaevna
Method for description of heat transfer processes in fractal systems
using scale variable
A method is offered for the description of heat transfer (diffusion) processes
in fractal systems based on the heat conductivity equation enhanced by an
additional variable specifying a scale of the consideration of the fractal.
Key words:fractal, fractional dimension, scaling, heat conduction, diffusion.
(in Russian), pp. 107-120
In this paper, we propose a regular iterative method of identifying a numerical parameter in the kernel of the integral equation of the first kind of the convolution type. It is shown that an unambiguous identification of the parameter is possible when an exact solution has discontinuities of the first kind. The convergence theorem is proved, and an example of the equation with a parameter, for which the method constructed is applicable, is given.
Artemiev Sergey Semenovich, Ivanov Aleksandr Aleksandrovich
Analysis of the effect of random noise on the strange attractors of Monte
Carlo on a supercomputer
Keywords: stochastic differential equations, cumulative frequency curve, frequency phase portrait, generalized Euler's method, strange attractors.
Voronin Kirill Vladislavovich, Laevsky Yuri Mironovich
On the stability of some flux splitting schemes
In this paper, we investigate the stability of some splitting schemes
approximating the equations for a heat flux, obtained by a mixed finite element
method. For the two-dimensional problem, the splitting scheme is based on the
alternating direction method, and for the three-dimensional problem the
splitting scheme is based on the Douglas-Gunn scheme.
heat transfer, mixed formulation, finite element method, splitting
Gusev Sergey Anatol’evich
Application of SDE's to estimating the solution of heat equations with discontinuous coefficients (in Russian), pp. 147-161
This paper proposes the use of the numerical solution to stochastic differential equations (SDE's) to find estimates of the solutions to boundary value problems for linear parabolic equations with discontinuous coefficients. The solution of the problem with smoothed coefficients is taken as an approximation of the generalized solution to the considered boundary value problem. The results of calculations for a thermal barrier coating comprising a composite
material are presented.
Keywords: heat equation, discontinuous coefficients, integral averaging, diffusion process, stochastic differential equations, Euler method.
Kotel'nikov Evginii Alekseevich
Non-convex minimization of a quadratic function on a sphere (in Russian), pp. 163-176
The minimization of convex functions on a sphere reduces to a sequence of
problems minimizing its convex majorants on a sphere. To build majorants, the
representation of the target function as a difference of convex quadratic
functions and the solutions of the problem at the previous step is used.
Representation of the target function in the form of a difference of convex
quadratic functions is based on a modified procedure of decomposition of the
Cholesky symmetric alternating-sign matrices.
Nikolaev Aleksey Georgievich, Tanchik Evgeniy Andreevich
The first boundary value problem of elasticity theory for a cylinder with
N cylindrical cavities
An efficient method for the analytical-numerical solution to the
non-axyally symmetric boundary value problem of elasticity theory for a
multiconnected body in the form of a cylinder with
N cylindrical cavities is proposed. The solution is constructed as
superposition of the exact basis solutions of the Lame equation for a cylinder
in the coordinate systems assigned to the centers of the boundary surfaces of
the body. The boundary conditions are exactly satisfied with the help of the
apparatus of the generalized Fourier method. As a result, the original problem
reduces to an infinite system of linear algebraic equations, which has a
Fredholm operator in the Hilbert space l2.
The resolving system is numerically solved by the reduction. The rate of
convergence of the reduction is investigated. The numerical analysis of stresses
in the areas of their greatest concentration is carried out. The reliability of
the results obtained is confirmed by comparing them for the two cases: a
cylinder with sixteen and a cylinder with four cylindrical cavities.
Keywords: boundary value problem, multiconnected body, generalized Fourier method, resolving system, cylindrical boundary, addition theorems.
Saveliev Lev Yakovlevich
Calculation of the number of states in binary Markov stochastic models
This paper derives exact and approximate formulas for the distribution,
average values and variances of the number of units on the segments of binary
Markov sequences. Various ways to calculate these formulas are proposed.
Estimates of the errors are given. An example of the calculation for a binary
Markov model of the precipitation process is presented.
Stepanova Larisa Valentinovna, Igonin Sergej Aleksandrovich
Asymptotics of the near crack-tip stress field of a fatigue growing crack
in damaged materials: numerical experiment and analytical solution
In this paper, the asymptotic analysis of the near fatigue growing
crack-tip fields in a damaged material is done. The integrity parameter
describing the damage accumulation process in the vicinity of a crack tip is
incorporated into the constitutive law of the isotropic linear elastic material.
The asymptotic solution based on the eigenfunction expansion method is obtained.
It is shown that the problem is reduced to the nonlinear eigenvalue problem. The
analytical solution of the nonlinear eigenvalue problem is found by the
artificial small parameter method. The perturbation theory approach allows us to
derive the analytical presentation of the stress and integrity distributions
near the crack tip. The technique proposed permits us to find the higher-order
terms of the asymptotic expansions of the stress components and the integrity
Keywords: fatigue crack growth, cyclic loading, asymptotic analysis, nonlinear eigenvalue problem, analytical solution.
Shkarupa Elena Valer'evna
Functional algorithms of the statistical modeling are designed to
construct an approximation of the problem solution as function on a required
domain. The approaches to construction of the upper error bound in the metrics
of the space C with allowance for the
degree of dependence of the estimates were devised for functional algorithms
with different types of stochastic estimates in the nodes. Furthermore, there
exists a universal approach applicable at any degree of dependence. The
constructed upper error bound of the functional algorithm is used for choosing
an optimal value of parameters, such as the number of grid nodes and the sample
size. Optimality of the chosen parameters directly depends on the accuracy of
the used upper error bound. The primary intent of the present paper is a
comparison of universal approaches and those with allowance for the degree of
dependence of the estimates.
Keywords: functional algorithms of statistical modeling, biharmonic equation, error estimation, optimization.
Balandin Alexander Leonidovich
Tomography of force-free fields
(in Russian), pp. 237–253
In order to investigate the force-free fields it is proposed to use the computerized tomography methods. For the inversion of the ray transformation, the method of multipole fields expansion has been developed. This method is based on the expansion of a vector field and the ray transformation over the special basis of vector-functions. Analytical expressions for the ray transform of the basis vector-functions and the results of computer simulation are given.
Bandman Olga Leonidovna, Kireeva Anastasiyfa Evgenevna
Stochastic cellular automata simulation of
oscillations and autowaves in reaction-diffusion systems
In this paper, experience in the conducted investigation of the stochastic cellular automata models of forming stable oscillations and autowaves in active media is generalized. As a result, the concept of stochastic cellular automaton (CA), corresponding to the asynchronous CA with probabilistic transition rules, is formulated. The formal notions of a stochastic CA and a stochastic CA model are given. Properties of the CA models and methods of their synthesis, using a specified set of elementary physical and chemical transformations, are described. The possibility of the autowave and oscillatory processes simulation is shown on an example of the carbon monoxide oxidation reaction on the platinum catalyst with reconstructing its surface structure. The CA-simulation enabled to reveal the range of reaction parameters values, at which stable oscillations of the reagents concentration occur, and to observe autowaves over the platinum surface. Considerable attention has been given to a high efficiency of the stochastic CA parallel implementation, which demands preliminary transformation of the asynchronous mode to the block-synchronous one with validation of its equivalence to the asynchronous mode. The latter is done for the investigated reaction CA model by means of the comparative statistical analysis of the simulation results.
Bychkov Igor Vicheslavovich, Zorkaltsev Valery lvanovich, Kazazaeva Anna Vasilevna
The weight coefficients in the weighted least squares method
the problem of estimating parameters of linear mathematical models. It is proved
that due to the choice of weights in the least squares method it is possible to
obtain solutions by minimizing any penalty functions of a wide class, including
those of the Holder norms. A limitation on a set of solutions resulting from the
variation of the weights in the least squares method has been determined. The
possibility of the practical use of the established theoretical facts is
illustrated by the ecology-mathematical models.
Keywords: mathematical model, agreement of parameters, the least squares method, weight coefficients.
Zadorin Alexander Ivanovich
The Lagrange interpolation and the Newton-Cotes formulas for functions with a
boundary layer component on piecewise-uniform meshes
The interpolation problem of a one-variable function, which can be considered as a solution of a boundary value problem for an equation with a small parameter ε with a higherderivative is investigated. The application of the Lagrange interpolation for such a function on a uniform grid can result in serious errors. In the case of the Shishkin mesh, ε-uniform error estimates of the Lagrange interpolation are obtained. The Shishkin mesh is modified to increase the interpolation accuracy. The ε-uniform error estimates of the Newton-Cotes formulas on such meshes are obtained. Numerical experiments have been carried out. The results obtained confirm the theoretical estimates.
Okuonghae R.I., Ikhile M.N.O.
Stiffly stable second derivative linear
multistep methods with two hybrid points
This paper presents a family of hybrid linear multistep
methods (LMM) with a second derivative term for the numerical solution of stiff
initial value problems (IVPs) for ordinary differential equations (ODEs). The
methods are stiffly stable for the step number
Keywords: continuous linear multistep methods, stiff problem, stiff stability, boundary locus, hybrid LMM.
Perepelkin Evgenii Aleksandrjvich
An inverse eigenvalue problem for a class of matrices of second and third orders
for solving the inverse eigenvalue problem for the product of matrices of second
and third orders is proposed. The necessary and sufficient conditions for the
existence of the problem solution have been obtained.
Keywords: eigenvalues, inverse problem, product of matrices.
Solodusha Svetlana V., Yaparova Natalia M.
A numerical solution of an inverse boundary value problem of heat conduction
using the Volterra equations of the first kind
We consider an inverse boundary value problem of heat conduction. To solve it, we propose a new approach based on the Laplace transform. This approach allows us to confine the original problem to solving the Volterra equations of the first kind. We have developed algorithms of the numerical solution to the resulting integral equations. The algorithms developed are based on the application of the product integration method and the quadrature of middle rectangles. A series of test calculations were performed to test the efficiency of the numerical methods.
Keywords: Volterra integral equations, numerical solution, product integration method.
Tarkov Mikhail Sergeevich
Solving the traveling salesman problem using a recurrent neural network
algorithm (NWTA-algorithm) for solving the traveling salesman problem (TSP) is
proposed. The algorithm is based on the use of the Hopfield recurrent neural
network, the WTA method (“Winner takes all”) for the cycle formation, and 2-opt
optimization method. A special feature of the algorithm proposed is in the use
of the method of partial (prefix) sums to accelerate the solution of the system
of the Hopfield network equations. For attaining additional acceleration, the
parallelization of the algorithm proposed is performed on GPU with the CUDA
technology. Several examples from the TSPLIB library with the number of cities
from 51 to 2,392 show that the algorithm proposed finds approximate solutions of
the TSP (a relative increase in the length of the route with respect to the
optimum is 0.03 ÷ 0.14). With a large number of cities (130 and more) the
NWTA-algorithm running duration on the CPU is in 4 ÷ 24 times less than the
duration of the heuristic LKH algorithm giving optimal solutions for all TSPLIB
Zabinyako Gerard Idelfonovich
An algorithm of the simplex method using a dual
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
Keywords: LU-decomposition, decomposition updating, sparse matrices, simplex method, linear programming.
Zorkaltsev Valery Ivanovich, Kiseleva Marina Alexandrovna
Oligopolistic interacting markets
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
conditions are found for the operators associated with the semilinear analogs of
the Stein matrix equation, namely, with the equations
X - A\overline X B = C and
X – AX*B =
Keywords: Stein matrix equation, semilinear operator, adjoint operator, self-adjointness, normality, simultaneous singular value decomposition, conjugate-normal matrix.
Marchuk Andrey Gurevich
The assessment of tsunami heights above the bottom slope within the wave-ray approach (in Russian), pp. 377–388
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.
Keywords: tsunami propagation, shallow-water equations, wave ray, wave front kinematics.
Mohanty Ranjan Kumar, Talwar Jyoti
A new compact alternating group explicit
iteration method for the solution of nonlinear time-dependent viscous Burgers' equation
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.
Keywords: non-linear parabolic equation, viscous flow, Compact AGE Method, Burgers' equation, Reynolds number.
Novikov Ivan Sergeevich
Solving the optimization problem of economic damage from environmental
pollution by local sources
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.
Keywords: adjoint equations, optimal control, Tikhonov regularization, economic damage, numerical modeling of pollution.
Rudoy Georgy Igorevich
On applying Monte Carlo methods to analysis of nonlinear regression
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.
Keywords: symbolic regression, nonlinear models, solution stability, transparent medium dispersion, Monte Carlo methods.
Solovjov Sergey Victorovich
Heat transfer modeling of an electroconductive liquid in a spherical
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.
Shumilov Boris Mikhailovich
A splitting algorithm for wavelet transforms of the Hermite splines of
the seventh degree
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.
Keywords: Hermite splines, «lazy» wavelets, implicit relations of decomposition, parallelization.