Siberian Journal of Numerical Mathematics

Volume 1, 1998

Contents

Number 1, pp. 1-98
Number 2, pp. 99-196
Number 3, pp. 197-297
Number 4, pp. 295-394


Number 1, pp. 1-98

A.S. Alekseev
On the scientific and educational potential of Siberia
in the field of computational and applied mathematics (foreword of the Editor-in-Chief)
(in Russian), pp. 1-2.

A.S. Alekseev
On the scientific and educational potential of Siberia
in the field of computational and applied mathematics (foreword of the Editor-in-Chief)
(in English), pp. 3-4.

Godunov S.K., Gordienko V.M.
The Krylov space and the Kalman equation
(in Russian), pp. 5-10.

An optimal, in some respects, representation of vectors in the Krylov space is built with the help of a variational problem. The extremum of the variational problem is the solution of the Kalman matrix equation and the 2-norm of the solution is suggested to use as a characteristic of the Krylov space. This characteristic can also be used as the measure of controllability in stationary discreet problems of optimal control.

Gorban' A.N.
Generalized approximation theorem and computational capabilities of neural networks
(in Russian), pp. 11-24.  

Computational capabilities of artificial neural networks are studied. In this connection comes up the classical problem on representation of function of several variables by means of superpositions and sums of functions of one variable, and appears a new edition of this problem (using only one arbitrarily chosen nonlinear function of one variable). 

It has been shown that it is possible to obtain arbitrarily exact approximation of any continuous function of several variables using operations of summation and multiplication by number, superposition of functions, linear functions and one arbitrary continuous nonlinear function of one variable. For polynomials an algebraic variant of the theorem is proved. 

For neural networks the obtained results mean that the only requirement for activation function of neuron is nonlinearity - and nothing else.

Konovalov A.N.
Conjugate-factorized models in mathematical physics problems
(in Russian), pp. 25-58. 

Linear  mathematical models are studied, which are based on a certain law (laws) of conservation. It  is shown that in this case the basic operators of a continuous model have initially a conjugate-factorized  structure. This property allows one  to simplify essentially the transfer to adequate grid models and to construct efficient algorithms to determine parameters of a model in different statements. The results  obtained can be considered as further development of the theory of support operators for difference   schemes of the divergent form.

Menikhes L.D., Tanana V.P.
The finite-dimensional approximation for the Lavrent'ev method
(in Russian), pp. 59-66.  

The generalization of the Lavrent'ev method for a solution of ill-posed problems is considered. The convergence criterion for the finite-dimensional approximation in terms of duality of the Banach spaces has been obtained.

Mikhailov G.A.
New Monte Carlo methods for solving boundary value problems (and related topics)
(in English), pp. 67-76. 

The article contains a survey of new Monte Carlo methods presented in recently published papers [2-9]. They are related to solving the Dirichlet problem with complex parameters, the mixed problem to a parabolic equation, a main eigenvalue estimation problem and similar problems with stochastic parameters. Besides, the effective method of improving random number generators by the modulo one summation is presented. There are used references only to papers [1-9], in which the detailed bibliography is considered.

Ostapenko V.V.
Approximation of Hugoniot's conditions by explicit conservative difference schemes for non-stationar shock waves
(in Russian), pp. 77-87. 

Introducted   here,  is  the  concept  of (ε, δ)-Hugoniot's condition  being  the  relatioship  which   links   generalised solution magnitudes in points (t-δ, x(t) + ε) and  (t+δ, x(t) - ε)  for both sides of non-stationary  shock wave front line x=x(t). It is showed here, that the explicit bi-layer with respect  to  time  conservative difference   schemes   for δ 0  approximate (ε, δ)-Hugoniot's conditions only with the first  order,  independent  of  their accuracy  for  smooth solutions. At the same time, if the front lines are quite smooth, then for δ = 0 these schemes approximate (ε, 0)-Hugoniot's conditions with the same order they have for smooth solutions.

Sorokin S.B.
Step-by-step inversion method for elasticity problems
(in English), pp. 89-98.

The paper presents a new efficient method for solving the difference problem in the domains of a  standard shape for the elasticity problems (step-by-step inversion method) for two-dimensional case. N^{3/2} arithmetic operations are required for obtaining a solution to the problem by this method,  N is the number of unknowns. It is  greater than N\ln(N) - the number of operations necessary to realize the conventional efficient direct methods for equations of the elliptic kind with separated variables (the fast Fourier transform, cyclic reduction technique). But it is considerable less than N^3 - the number of operations necessary to realize  the Gauss type method for this problem.


Number 2, pp. 99-196

Bykova E.G. and Shaidurov V.V.
A nonuniform difference scheme with fourth order of accuracy in a domain with  smooth boundary
(in Russian), pp. 99-118.  

The paper is devoted to construction and justification of a nonuniform difference scheme with fourth order of accuracy for a two-dimensional boundary-value problem for an elliptic equation of second order in a domain with  smooth curvilinear boundary. This scheme is called nonuniform because a stencil of difference operator alternates from node to node. In interior nodes a nine-point and standard five-point stencils are used. The special type of stencil is used near the boundary. The paper contains a description for constructing the scheme and for the proof accuracy. Numerical tests confirm the theoretical conclusions about the fourth order of accuracy for the approximate solution.

A.V. Voytishek
On the permissible class of interpolations for discrete-stochastic procedures of global estimation of functions (in Russian), pp. 119-134

Problems are considered of construction and convergence of numerical discrete--stochastic procedures for approximation of the functions presented in integral form. The procedures include introduction of a discrete grid, estimation of the function at grid nodes using Monte Carlo methods and interpolation of the function using the values at the grid nodes. The results obtained formerly for the multilinear approximation are generalized to the case of interpolation which is based on expanding  the function under study in a basis of positive functions forming a partition of unity. The Strang-Fix approximation is considered as an example of such interpolation. The statements are formulated  on the rate of convergence of discrete-stochastic procedures for approximation of an integral depending  on a parameter.

Malbackov V.M.
On the statistical properties of the hydrodynamic models based on solutions to the Boussinesq equations
(in Russian), pp. 135-142.  

Using simplified Boussinesq equations as an example, we show that their solutions describing vortex  structures (convective cells) are unstable perturbations of finite-amplitude. This property of the  solutions makes it possible to advance a hypothesis concerning the mechanism of formation of the spectrum of the ensemble of convective cells and go from the hydrodynamic model to a statistical model. The results obtained earlier for the adiabatic atmosphere are generalized to a more general case of a polytropic atmosphere. This case includes the spontaneous convection that leads to formation of mesoscale  convective ensembles. Such ensembles consisting of thermals and convective clouds play an important role in formation of the weather and climate of the planet.

Namm R.V.
On  characterization of limit point in the iterative prox-regularization method
(in Russian), pp. 143-152.  

The article studies the dependence of a solution on the  points generated in the iterative prox-regularization method as applied to semicoercive variational inequalities.

Rukavishnikov V.A. and Bespalov A.Yu.
On the h-p version of the finite element method for one-dimensional boundary value problem with singularity of a solution
(in English),  pp. 153-170.  

The paper analyzes the h-p version of the finite element method for a one-dimensional model boundary value problem with coordinated degeneration of initial data and with strong singularity of a solution. The scheme of the finite element method is constructed on the basis of the definition of  R_\nu-generalized solution to the problem, and the finite element space contains singular power functions. By using meshes with concentration at a singular point and by constructing the linear degree vector of approximating functions in a special way, a nearly optimal two-sided exponential estimate is obtained for the residual of the finite element method.

Skurin L.I.
An iterative-marching method for solving problems of fluid and gas mechanics problems
(in Russian), pp. 171-182. 

A possibility is studied of applying the idea of global iterations with aspect to the pressure for the complete Navie--Stokes equations for a fluid as well as for a gas, for stationary and unstationary problems, for two and three dimensional problems. We give a generalization for the results published earlier and present new results concerning stability and convergence of the iterative-marching method, and its testing on the problems of motion of a fluid (twist fluid flows with bubbles; internal and surface waves generation by an eddy pair) and gas motions (flows in a nozzle with a bubble; shock wave formation resulting from viscous effects). 

The main conclusion is as follows. The proposed method allows us to develop numerical algorithms for various problems of fluid and gas mechanics based on a common principle. These algorithms are simple because their basic element is a marching procedure. The above implies the possibility of developing rather universal programs.

T.A. Sushkevich, S.A. Strelkov, A.K. Kulikov, and S.V. Maksakova
A model of the polarized radiation transfer in a planar layer with interface of two media (in Russian), pp. 183-194.

A new mathematical model is stated for the polarized radiation transfer in the two-media planar layer with internal reflecting and refracting interface. Solution of the general vectorial boundary-value problem for the kinetic equation is reduced to computing the vectorial optical transfer operator (VOTO). The VOTO kernels are tensors of the influence functions (TIF) of both media. The basic models of the vectorial influence functions (VIF's) are distinguished. The T-matrix method is developed and generalized to a multiple scattering theory with the mechanism of the radiation polarization and depolarization in two-media system taken into account.

Information on the International Conference on Inverse Problems of Mathematical Physics (in Russian and English), pp. 195-196.


Number 3, pp. 197-297

I.A. Blatov
On incomplete factorization for fast Fourier transform method for the discrete Poisson equation in a curvilinear boundary domain (in Russian), pp. 197-216.

For the discrete Laplasian on the rectangular grid the spectral equivalent preconditioner of the type of the incomplete block-factorization is constructed. The inversion of this preconditioner with accuracy ε = O(N^-1) is realized with the help of the fast Fourier transform with O(N\ln N \ln(ε^-1)) arithmetical operations.

L.V.Gilyova
A cascadic multigrid algorithm in the finite element method for the three-dimensional Dirichlet problem (in Russian), pp. 217-226.

A standard scheme of the finite element method with the use of piecewise-linear elements on tetrahedrons is considered as applied to the three-dimensional elliptic second order Dirichlet problem. In order to solve this scheme, a cascadic arrangement of two iterative algorithms is used on a sequence of embedded three-dimensional triangulations that gives a simple version of the multigrid methods without preconditioning and restriction to a coarser grid. The cascadic algorithm starts on the coarsest grid where the grid problem is solved by direct methods. In order to obtain approximate solutions on finer grids, the iteration method is used; the initial guess is taken by interpolation of the approximate solution from the preceeding coarser grid. It has been proved that the convergence rate of this algorithm does not depend on the number of unknowns and the number of grids.

V.A.Debelov, A.M.Matsokin, and S.A.Upol'nikov
Subdivision of a plane and set operations on domains (in Russian), pp. 227-247.

The paper is devoted to the description of an algorithm for subdivision of a plane into non-intersecting domains by a finite set of the simple Jordan arcs. Each resulting domain is defined via a set of its boundary arcs and its indicator (a bounded or an unbounded domain), which determines the characteristic domain function. Also, the algorithm for implementation of a regularized set operations on domains without cut-offs is proved. It is based on the subdivision of a plane by common boundaries on sub-domains and construction from the latter of a result of operation. To compute intersection points of the boundary arcs the Newton method is applied whose square convergence is proved for the case of convex and monotone curves.

A.I. Zadorin
Numerical solution of the equation with a small parameter and a point source on the infinite interval (in Russian), pp. 249-260.

The second order equation with a small parameter effecting a higher derivative and a point source on the infinite interval is considered. The question of the transformation of the boundary conditions to the finite interval is investigated. The difference scheme for the problem on the finite interval is constructed. The uniform convergence of the difference scheme is proved.

B.G. Mikhailenko and O.N. Soboleva
Absorbing boundary conditions for the elastic theory equations (in Russian), pp. 261-269.

High order absorbing conditions for the artificial boundary equations are studied in the paper. The system of equations with the boundary conditions is solved by a spectral-finite difference algorithm that is based on the finite cosine and sine Fourier transforms. For the obtained ordinary differential equations a numerical finite difference scheme is proposed. The efficiency of the method is illustrated on the numerical experiments.

V.F. Raputa, A.I. Krylova, and G.A. Platov
Inverse problem of estimating the total emission for the nonstationary boundary layer of the Atmosphere (in Russian), pp. 271-279.

Opimization problems for estimating the upper and the lower bounds of pollutant emission from multiple point sources according to concentration measurement data are considered on the basis of nonstationary models of the Atmosphere boundary layer and air pollutants transfer. The summary strength of pollution sources is used as objective function. The results of numerical experiments on estimation of the daily dynamics of bounds of the summary emission are presented for different variants of the observation network.

G.I. Shishkin
Grid approximations of singularly perturbed systems for parabolic convection-diffusion equations with counterflow (in English), pp. 281-297.

The first boundary value problem is considered on a strip for a system of two singularly perturbed parabolic equations. The perturbation parameters multiplying the highest derivatives of each of the equations are mutually independent and can take arbitrary values from the interval (0,1]. When these parameters equal zero, the system of parabolic equations degenerates into a system of hyperbolic first order equations coupled by the reaction terms. The convective terms (i.e., their components orthogonal to the boundaries of the strip) that are involved in the different equations have the opposite directions (convection with counterflow). This case brings us to the appearance of boundary layers in the neighbourhood of both boundaries of the strip. For this boundary value problem, the difference schemes that converge uniformly with respect to the parameters are constructed here using the condensing mesh method. We also consider the construction of parameter uniform convergent difference schemes for a system of singularly perturbed elliptic equations that degenerate into first order equations if the parameter equals zero.


Number 4, pp. 299-400

On the anniversary of Anatoly Semenovich Alekseev (in Russian), pp. 299-300.

V.A. Vasilenko and A.V. Elyseev
Abstract splines with the tension as the functions of parameters in energy operator (in Russian), pp. 301-311.

In the framework of general optimization of energy functional in variational spline interpolation approach we consider the multiparametric family of energy operators linear depending on parameters (it corresponds to abstract splines with the tension). Our main result consists in the representation of this kind of spline as the function of parameters in energy operator. It provides the explicit calculation of various criteria functions for forthcoming choice of optimal parameters.

A.V. Gavrilov
On best quadrature formulas in the reproducing kernel Hilbert space (in Russian), pp. 313-320.

The best quadrature formulas with free nodes in some Hilbert space with a reproducing kernel are considered. It is shown that they are at the same time optimal among the formulas with the same nodes and having directional derivatives. To prove this, the error norm gradient of the formula with the optimal weights is determined as a function of nodes.

V.P. Il'in and K.Yu. Laevsky
On incomplete factorization methods with generalized compensation (in Russian), pp. 321-336.

The iterative incomplete factorization methods are described on the base of definition of preconditioning $B$ matrix from generalized compensation principle By_k = Ay_k, k = 1,...,m, where A is the matrix of original system of linear algebraic equations and {y_k} is the set of so called probe vectors. The correctness of such algorithms and conditions of positive definiteness of preconditioning matrices are investigated for solution to the Stieltjes type block-tridiagonal systems. The estimates of condition number of matrix product B^{-1}A, that define the iterative convergence rate, are derived in the terms of the properties of original matrices.

O.A. Klimenko
Stability of an inverse problem for transport equation with discrete data (in English), pp. 337-345.

A validity of numerical algorithms is under investigation in this article. Namely, stability for two-dimensional problem of restoring the right-hand side in the transport equation by discrete data is considered. It is suggested that the radiation incident on the boundary is omitted. The outgoing radiation is known at discrete numbers on the boundary. Stability theorems for two problem formulations are proved. In the former case an interparticle collision is not considered. In the latter case the transport equation is considered with the integral term and absorption, and scattering properties of the medium are taking into account.

Yu.M. Laevsky and O.V. Rudenko
On the locally one-dimensional schemes for solving the third boundary value parabolic problems in nonrectangular domains (in Russian), pp. 347-362.

The paper deals with studying some modifications of the local one-dimensional schemes for solving the mixed and the third boundary value parabolic problems in nonrectangular domains. Contrary to the usual schemes with the error estimate O(h+τ / sqrt{h}), these modifications have unconditional convergence with the error estimate O(h+τ) for the problem with the mixed boundary conditions of special type and O(h+τ^5/6) for the third boundary value problem.

Leus V.A.
On the differentially conditioned function generating based on degree potentials (in Russian), pp. 363-371.

The notion of the strong linear independence for the functions of many arguments is introduced. Multiparametric family containing degree potentials (of arbitrary indices) and all their partial derivatives is studied. Strong linear independence is proved for it. The function generating problem with differential conditions referred to scattered data is formulated. Geometric aspect of this problem is examined and technology based on degree potentials for the problem solving is argued.

Rozhenko A.I.
Spline approximation in tensor product spaces (in Russian), pp. 373-390.

The normal solvability of the energy operator for spline approximation problem in tensor product of abstract Hilbert spaces is proved. Hence, the correctness of the problem (the existence of a solution) is derived. The general method for the regularization of a semi-Hilbert space reproducing map is proposed. It allows to construct the reproducing map in tensor case. The general theory is illustrated by examples.

Smelov V.V.
On completeness of hemispherical harmonics system (in Russian), pp. 391-395.

Completeness of the hemispherical harmonics system is proved, and the exact formula for the square of their norm is found (simultaneously, the recursion formula for the Jacobi adjoined function is derived).