Siberian Journal of Numerical Mathematics

Volume 2, 1999

Contents

Number 1, pp.1-98
Number 2, pp.99-196
Number 3, pp.197-294
Number 4, pp.295-394


Number 1, pp.1-98

S.S. Artemiev and M.A. Yakunin
Estimates of the parameters in system of stochastic differential equations with linear inclusion of parameters (in Russian), pp.1-11

In this paper we give the method of calculating the maximum-likelihood estimates of the parameters in nonlinear system of stochastic differential equations, when unknown parameters are linearly included in the right-hand side of system. Maximum-likelihood function is constructed on the basis of the Euler scheme, that describes the discrete observations of the solution to equations system. We set the conditions to attain the maximum by the likelihood function. Estimates of the parameters are calculated by the iterative method. We consider the special cases of equations and give examples of calculations.

Yu.A. Bogan
Stationary problems of the theory of elasticity with a small viscosity
(in Russian), pp.13-20

Some singularly perturbed boundary value problems in the theory of elasticity, arising in averaging of laminated media with nonstandard contact conditions on the interface boundary, have been studied. The studied problems have been formulated in the variational set-ups, and theorems on convergence to solutions of the limiting ones have been proved.

A.I. Zadorin
The transfer of the boundary condition from the infinity for the numerical solution to the second order equations with a small parameter (in Russian), pp.21-36

The ordinary second order differential equations with a small parameter effecting the higher derivative on the semi-infinite interval are considered. The method of the transition of that problem to the finite interval is proposed. To solve the auxiliary Cauchy problem the asymptotic method is used.

K.I. Kutchinsky
An error estimation in -approximation via statistical methods
(in English), pp.37-46

This paper offers a statistical approach to obtain a numerical estimate of
-approximation algorithm efficiency on a fixed functional class. This approach consists of two steps. The first one is finding the distribution of -approximation coefficients. The second one is the simulation of a random vector with obtained probability density and calculation the integer (number of summands in -series) that provides given accuracy with given probability.

V.V. Ostapenko
Finite difference scheme of high order of convergence at a non-stationary shock wave (in Russian), pp.47-56

The finite difference scheme is constructed for the hyperbolic system of two conservation laws of the “shallow water” theory. It has not less than the second order of weak convergence when calculating the non-stationar shock wave. This scheme is not monotone, however, as differentiated from all other known now “high accuracy” schemes (considering monotone ones), it reproduces the Hugoniot conditions with high accuracy and correspondingly conserves the high order of the strong local convergence in the area of the non-stationary shock influence.

S.M. Prigarin
On convergence and optimization of functional Monte Carlo estimators in Sobolev spaces (in Russian), pp.57-67

The paper deals with the study of convergence and optimization of unbiased functional Monte Carlo estimators. We have obtained the estimators, that are optimal in Sobolev's Hilbert spaces, for calculating the integrals dependent on a parameter and for calculating the families of functionals of the solution to the integral equation of the second kind. The results have been obtained in the context of the new concept proposed by the author in order to compare the efficiency of functional estimators in Monte Carlo methods. New conditions of estimators convergence in the space of continuous functions have been proved.

V.I. Sabinin
The solution of a problem of groundwater salt-heat transport by an incomplete factorization technique (in Russian), pp.69-80

Mathematical model of the groundwater salt-heat transport on which freezers affect as protective screens is described. A numerical method of the solution, realizing this model with the help of iterative incomplete factorization techniques is offered. Outcomes of calculations of a groundwater contaminant filtration to two rivers and a water intake, and also ones of dynamics of an icing of a horizontal series of underground freezers in a salt flow are resulted, which demonstrate efficiency and economy of the method.

S.B. Sorokin
Conjugate-factorized models in plate theory (in English) pp.81-88

The conjugate-factorized model of the problem of sagging of thin plate is formulated in the work. The statements in “saggings” and “moments” are described. It is shown that in both cases the problem operator has the conjugate-factorized structure , where is a differential matrix-operator of the second order, and is a numerical matrix. The presented results are analogous to those obtained for the problems of elasticity theory and give the posibility to use the method of support operator to build the difference scheme, and to apply the iteration methods in subspace or the direct method of step by step conversion for the numerical solution to the difference problem.

T.A. Sushkevich and S.A. Strelkov
Model of the polarized radiation transfer in atmosphere-earth surface system (in Russian), pp.89-98

The general vectorial boundary-value problem for kinetic equation, describing the polarized radiation transfer in a planar layer with a horizontally nonhomogeneous anisotropically reflecting underlying surface, is not solved by the finite differences methods. Mathematical model, giving an asymptotically accurate solution to this problem in the slow growth functions class, has been proposed and justified. The new model has been stated by the influence functions and the spatial frequence characteristics method. This model is effective for the parallel computing.


Number 2, pp.99-196

I.I. Belinskaya
The solution to direct and inverse problems for oscillation's equation with Monte Carlo methods (in Russian), pp.99-110

There are the well-known probabilistic methods for solving the elliptic and parabolic problems of mathematical physics. As for hyperbolic equations the probabilistic methods are used very seldom. In this work the Monte Carlo algorithm for solving the inverse and direct problems of the oscillation's equation is presented.

A.V. Voytishek
Using the Strang-Fix approximation for Monte Carlo calculating of multiple integrals (in Russian), pp.111-122

In this paper the possibility of using the Strang-Fix approximation for constructing of optimal density in the standard Monte Carlo method for calculating multiple integrals is investigated. Optimal parameters of the constructed density are obtained. The comparison between the constructed important sampling method and the separation of main part method is provided. The combined method, which contains advantages of both algorithms, is suggested.

B.S. Jovanovic, P.P. Matus, and V.S. Shchehlik
The rates of convergence of the finite difference schemes on nonuniform meshes for parabolic equation with variable coefficients and weak solutions (in Russian), pp.123-136

The convergence of the difference schemes of the second order of local approximation on space for a one-dimensional heat conduction equation with variable factors on an arbitrary nonuniform grid is investigated. For the schemes with averaged coefficient of a thermal conduction and averaged right part the evaluations of a rate of convergence in a grid norm , agreed with a smoothness of a solution of a boundary value problem are obtained.

A.L. Karchevsky
Behavior of the misfit functional for a one-dimensional hyperbolic inverse problem (in Russian), pp.137-160

In this paper we investigate the behavior of the misfit functional for a one-dimensional hyperbolic inverse problem when an unknown coefficient stands by a lowest term of a differential equation. Assuming an existence of an inverse problem solution we prove a uniqueness of a stationary point of the functional. If the minimization sequence belongs to a bounded set, we show that the following estimates of the convergence rate for the suggested method of the descent takes place.

Yu.I. Kuznetsov
Complement of Jacobi matrix (in Russian), pp.161-170

The new properties of Jacobi matrix are discussed. The matrix analogs for the differential operator of the hypergeometric type and of the Pearson one are derived on the base of Jacobi matrix properties exclusivly: the conjugate Sturm system is derived from the properties of symmetric Jacobi matrix , as well as the complementary Jacobi matrix , for which this conjugate Sturm system is an ordinary Sturm system.

V.A. Rukavishnikov, E.V. Kashuba
On the properties of an orthonormalized singular polynomials set
(in English), pp.171-184

The paper introduces a concept of singular polynomials, and their orthonormalized set is constructed. The recurrence formula for three neighbouring orthonormalized singular polynomials is given and an analogue of the Rodrigues formula is found. It is established that the derivatives of polynomials from the built set have a property, close to the property of orthogonality. A differential equation and a family of relations for the derivatives of orthonormalized singular polynomials have been obtained. The results are applicable to the construction and investigation of the finite element method for mathematical physics problems with a strong singularity of solution.

Kh.M. Shadimetov
Weight optimal cubature formulas in Sobolev's periodic space
(in Russian), pp.185-196

In the present paper the weight lattice optimal cubature formulas in the periodic Sobolev's space are constructed. Under writing out of the algorithm of the construction the extremal function is found, and by means of the function the norm of the error functional of the cubature formula is calculated. Minimizing the norms, periodic Winier-Hopf's systems are obtained. Then the uniqueness solution to the system has been proved.


Number 3, pp.197-294

A.Yu. Bezhaev
Slope's choice in plane curve interpolation (in Russian), pp.197-206

The problem of plane curve construction, given by supporting points, is considered. The methods, based on the piece-wise Hermit interpolation by the polynomials of third degree (like the Fergusson and Bezier approaches), additionally require slope vectors in supporting points. Here two methods for slopes are suggested. In some sense they present extremal interpolation variants. Their convex combinations give multiparameter set of interpolating curves. There we select one-parameter set, whose parameter influences to the same extent on visual smoothness and curvature on all curve patches. On the base of numerical experiments the parameter limits have been found out and which provide the simplicity and sufficient smoothness of the interactively managed curve with the help of supporting points.

V.G. Belyakov, N.A. Miroshnichenko
Use of queueing network methods at analysis of the realistic models of the telecommunication systems (in Russian), pp.207-222

In this paper are the developed methods of analysis of the closed multiclass multiplicative queueing network models provided that service rate in each node is a strictly positive function of two variables: of the total number of customers of different classes in that node and of the total number of customers in fixed set of service nodes. The basic analytic expressions are derived to convolution, coalesce computation, mean-value analysis, asymptotic expansions and decomposition methods.

V.E. Gheit
On the polynomials, the least deviating from zero in metrics
(in Russian), pp.223-238

For polynomials, the least deviating from zero in metrics with given number of leading coefficients, it is obtained the representation through the so-called extremal polynomials of the least deviation, whose number of sign changes is equal to their degree. The instruments of effective calculating of these polynomials are represented.

Katya O. Gorbunova
Formal Kinetic Model of Structureless Small-grained Parallelism (in Russian), pp.239-256

A new abstract model of parallel calculations – the Kirdin kinetic machine – is suggested. It is expected that this model will play the same role for parallel calculations, as Markov normal algorithms, Kolmogorov and Turing machine or the Post schemes for consecutive calculations. The basic ways of realization of calculations are described in the article, correctness of the elementary programs for the Kirdin kinetic machine is investigated, it is proved that the determined Kirdin kinetic machine is the effective calculator.

A.S. Leonov
Application of function of several variables with bounded variation to numerical solution of two-dimensional ill-posed problems
(in Russian), pp.257-272

The problem of numerical piece-uniform regularization of two-dimensional ill-posed problems with bounded discontinuous solutions are under consideration. The functions of two variables with bounded variations of several kinds (total variation, variation of Arzela) are applied to solve the problem by use of regularizing algorithms. In finite-dimensional form, these algorithms are reduced to solution of mathematical programming mith non-smooth target functions or with non-smooth restrictions. After smooth approximation, the algorithms are effectively implemented numerically and ensure piece-uniform convergence of approximate solutions to exact solution to be found. The numerical experiments in problems of distored image reconstruction illustrate the influence of different kinds of variations on the quality of obtained solution.

V.A. Leus
Strong-linear independence for differential images of Gauss potentials (in Russian), pp.273-280

J.C. Mairhuber theorem concerning the Chebyshev approximation problem provides a resolution uniqueness only for the case of one-dimensional compacts. In present paper an attempt to overcome the above restriction by means of stochastic interpretation applied to solvability is taken. In the context of such a position the multiparametric system of Gauss potentials is studied. The strong linear independence for potentials and its images resulting from the constant factors linear differential operator is proved. The differentially conditioned analytic function generating based on a linear combination of Gauss potentials is examined.

I.A.R. Moghrabi, A. Obeid Samir
Curvature-based multistep quasi-Newton method for unconstrained optimization (in English), pp.281-294

Multi-step methods derived in [1-3] have proven to be serious contenders in practice by outperforming traditional quasi-Newton methods based on the linear Secant Equation. Minimum curvature methods that aim at tuning the interpolation process in the construction of the new Hessian approximation of the multi-step type are among the most successful so far [3]. In this work, we develop new methods of this type that derive from a general framework based on a parameterized nonlinear model. One of the main concerns of this paper is to conduct practical investigation and experimentation of the newly developed methods and we use the methods in [1-7] as a benchmark for the comparison. The results of the numerical experiments made indicate that these methods substantially improve the performance of quasi-Newton methods.


Number 4, pp.295-394

A.B. Andreev, T.D. Todorov
Lumped mass approximation for an isoparametric finite element eigenvalue problem (in English), pp.295-308

We consider isoparametric variant of the lumped mass modification of the standard Galerkin method for second-order elliptic eigenvalue problems. The lumping of the mass matrix results from the use of an appropriate isoparametric quadrature formula for the integrals over triangular Lagrange finite elements. The analysis of the 7-node finite element transformations is made. The convergence of the eigenvalue approximations is proved.

B.M. Bagaev, H.-G. Roos
The finite element method on adapted meshes for the two-dimensional convection diffusion problem (in English), pp.309-320

We consider a singularly perturbed elliptic boundary value problem which models a special channel flow. Basing on the decomposition of the exact solution we obtain a priori bounds for derivatives of the exact solution. For the numerical solution, we used linear finite elements on an adapted meshes, further we discuss the additive extraction method. We obtain uniform estimates for the approximate solution in some energetic norm.

A.F. Voevodin, T.V. Yushkova
Numerical solution to initial value problems for the Navier-Stokes equations in closed regions based on splitting method (in Russian), pp.321-332

On the basis of splitting method with respect to physical processes, a numerical method is proposed for solving initial value problems for the Navier-Stokes equations written in the terms of “eddy - current” variables. For the solution to the implicit difference systems a modified method of “two-field” (separate) calculation of the stream function and vorticite is suggested. For the first time in solving the Poisson equation for stream function two boundary conditions are simultaneously used on the boundaries (). A stability analysis in the linear approach is made. Some test numerical examples are given.

A.V. Kel'manov, S.A. Khamidullin
Optimal detection of given number of identcal subsequences in quasiperiodic sequence (in Russian), pp.333-349

The problem of the detection of given number of identical subsequences in quasiperiodic sequence distorted by the uncorrelated Gaussian interference with a known dispersion is studied. The aposteriori computing algorithm for the solution to this problem is justified. The case is considered, when the boundaries of an interval of the beginning and ending of the observations above the distorted sequence do not break the first and the last subsequence of hidden quasiperiodic sequence into two parts, and the instants of the begining of the subsequences are the determined values. It is established that the given problem is a particular problem of the test of the hypothesis about the mean of the Gaussian random vector. The recursion formulas of step by step discrete optimization ensuring maximum likelihood decsion are obtained. The estimation of temporary and capacitive of the algorithm is given. The outcomes of the numerical modeling are adduced.

B.P. Kolobov, Yu.I. Molorodov
Calculation of the optimal collocation points for solving parabolic equations using high precision schemes (in Russian), pp.351-360

The quadrilateral finite element's collocation points increasing a precision of the numerical collocation (NC) alhorithm for solving the initial-boundary problem for one-dimensional parabolic equations of the second order are obtained.

Yu.M. Laevsky, A.M. Matsokin
Decomposition methods for the solution to elliptic and parabolic boundary value problems (in Russian), pp.361-372

In the paper, a short survey of investigations on the decompositiom methods for elliptic and parabolic problems that where carried out in the ICM\&MG SB RAS during 20 years is presented.

V.I. Paasonen
The improved boundary conditions at the singular points of coordinate systems for non-stationary boundary value problems
(in Russian), pp.373-384

In the article, two types of non-stationary boundary value problems in polar, cylindrical and spherical coordinates are considered. These are symmetric problems whose initial data do not depend on angle variables and whatever problems. The difference fourth-order accuracy boundary conditions in poles of coordinate systems are constructed for their application in high-order schemes. The conditions represent special difference analogues of the differential equation in the Cartesian coordinates noted in poles. The modes of realization of boundary conditions in high-order schemes based on a method of an approximate factorization are developed.

V.V. Smelov
On representation of piecewise-smooth functions by rapidly convergent trigonometric series (in Russian), pp.385-394

A specific variant of expansions of smooth and piecewise-smooth functions to rapidly convergent series with respect to trigonometric functions is suggested. This result is the basis for the effective approximations of the above-mentioned functions.