Siberian Journal of Numerical Mathematics

Volume 18, 2015

Contents

Number 1, pp. 1-105
Number 2, pp. 107-234
Number 3, pp. 237-347
Number 4, pp. 349-467


Number 1, pp. 1-105  

 

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 differential equations (in Russian), pp. 15-26 

 

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. 

Key words: stochastic differential equations, cumulative frequency curve, frequency phase portrait, generalized Euler's method. 

 

Borovko IrinaVladimirovna, Krupchatnikov Vladimir Nikolaevich

Simulation of the response of the Hadley cell and extratropical troposphere stratification on the climate changes using a general atmosphere circulation of medium complexity (in Russian), pp. 27-40 

 

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 (in Russian), pp. 41-53 

 

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 Toeplitz matrices (in Russian), pp. 55-64 

 

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 Russian), pp. 65-78 

 

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) = A1, u(1)= A2, where A1 and 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 (in Russian), pp. 95-105 

 

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.

 


Number 2, pp. 107-234

 

Antonova Tatiana Vladimirovna

Methods of identifying a parameter in the kernel of the first kind equation of the convolution type on the class of functions with discontinuities (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.

Keywords: ill-posed problems, localization of singularities, equation of the first kind, parameter identification.

 

Artemiev Sergey Semenovich, Ivanov Aleksandr Aleksandrovich

Analysis of the effect of random noise on the strange attractors of Monte Carlo on a supercomputer (in Russian), pp. 121-134

 

In this paper, we numerically investigate the influence of random noise on the behavior of the trajectories of strange attractors defined by a system of ordinary differential equations. The resulting stochastic differential equations are solved by the generalized Euler method. The results of numerical experiments conducted on a cluster of NKS-30T Siberian Supercomputer Center at ICMMG using the program package PARMONC. For the analysis of the numerical solutions, the frequency characteristics of generalizing the integral curve and the phase portrait are used. 

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 Russian), pp.135-145

 

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. 

Keywords: heat transfer, mixed formulation, finite element method, splitting scheme. 

 

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

cellular 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. 

Keywords: quadratic optimization on sphere, collinearity gradients, convex majorant, Cholesky decomposition. 

 

Nikolaev Aleksey Georgievich, Tanchik Evgeniy Andreevich

The first boundary value problem of elasticity theory for a cylinder with N cylindrical cavities (in Russian), pp. 177-189

 

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 (in Russian), pp. 191-198

 

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. 

Keywords: stochastic model, binary Markov chain, distribution, generating function, mean, variance. 

 

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 Russian), pp. 201-217

 

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 parameter. 

Keywords: fatigue crack growth, cyclic loading, asymptotic analysis, nonlinear eigenvalue problem, analytical solution. 

Shkarupa Elena Valer'evna

 Comparison of approaches to optimization of functional statistical modeling algorithms in the metric of the space C (in Russian), pp. 219-234

 

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.

 


Number 3, pp. 237-347  

 

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. 

Keywords: computerized tomography, spherical harmonics, inverse problems.

Bandman Olga Leonidovna, Kireeva Anastasiyfa Evgenevna

Stochastic cellular automata simulation of oscillations and autowaves in reaction-diffusion systems  (in Russian), pp. 255–274 

 

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.

Keywords: stochastic cellular automata, computer simulation, asynchronous cellular automata, parallel calculations, catalytic reactions, autowaves.

Bychkov Igor Vicheslavovich, Zorkaltsev Valery lvanovich, Kazazaeva Anna Vasilevna

The weight coefficients in the weighted least squares method   (in Russian), pp. 275–288

 

We consider 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   (in Russian), pp. 289–303

 

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 higher  derivative 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.

Keywords: one-variable function, boundary layer, high gradients, Shishkin mesh, Lagrange interpolation, Newton-Cotes formula, error estimate.

Okuonghae R.I., Ikhile M.N.O.

Stiffly stable second derivative linear multistep methods with two hybrid points   (in Russian), pp. 305–317

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 k≤7. 

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  (in Russian), pp. 319–326

 

The method 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  (in Russian), pp. 327–335

 

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  (in Russian), pp. 337–347

 

A new 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 examples. 

Keywords: traveling salesman problem, recurrent neural Hopfield network, 2-opt, CUDA technology, LKH algorithm.

 


Number 4, pp. 349-467

 

Zabinyako Gerard Idelfonovich

An algorithm of the simplex method using a dual basis (in Russian), pp. 349–359

 

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. 

Keywords: LU-decomposition, decomposition updating, sparse matrices, simplex method, linear programming.

Zorkaltsev Valery Ivanovich, Kiseleva Marina Alexandrovna

Oligopolistic interacting markets (in Russian), pp.361–368

 

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. 

Keywords: Cournot model, Nash equilibria, potential game.

 

Ikramov Khakim Dododjanovich 

Normality conditions for semilinear matrix operators of the Stein type (in Russian), pp. 369–375

 

Normality 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 = C. 

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

 

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. 

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 Russian), pp. 389–405

 

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 (in Russian), pp. 407–424

 

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. 

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 models (in Russian), pp. 425–434

 

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. 

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 layer (in Russian), pp. 435–451

 

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. 

Keywords: mathematical modeling, convective heat transfer, Joule dissipation, magnetic hydrodynamics, spherical layer.

Shumilov Boris Mikhailovich

A splitting algorithm for wavelet transforms of the Hermite splines of the seventh degree   (in Russian), pp. 453–467

 

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. 

Keywords: Hermite splines, «lazy» wavelets, implicit relations of decomposition, parallelization.