** Number 1, pp.1-88 Number 2, pp.89-176 Number 3, pp.177-271 Number 4, pp.273-362**

** Andramonov M.Yu.**
(

Solving systems of nonlinear equations by the parametric approach with an arbitrary initial point

We propose a number of algorithms for solving systems of nonlinear equations,
when a good approximation to solution is unknown, and the Newton method is not
efficient. These methods are based on the choice of weights for an auxiliary function and on the descent in the space of weights. The
convergence depends on relations between the measures of regions of attraction
of the solutions. In order to improve the performance, we consider perturbation
methods.**
Key words: **nonlinear equations, arbitrary initial point,

**Andreev A.B., Maximov J.T., Racheva M.R. Finite element modelling for a beam on the Winckler type basis with variable
rigidity **
(

We study constructing a beam on the Winckler basis that is under the
influence of a cross-force. This force rotates around the axis of the beam. The
rigidity of this basis depends on the time variable. A general mathematical
model is deduced for this type of constructions. Variational formulations of the
boundary value problems in question are obtained. The finite element method is
used to determine the stresses of a beam. We discuss the corresponding
eigenvalue problems in order to apply the method of normal shapes. Finally, a
numerical result with practical application is presented. ** Key words:** finite element method, the Winckler type basis, dynamic
stresses.

**Averina T.A., Artemiev S.S. Numerical solution to stochastic differential equations with growing variance **(

The paper considers a new method for transition from
an initial unstable in the mean square SDE system to
the SDE system with a solution close to a stationary process.
The SDE systems for a stochastic component are obtained with the use of the
Ito formula both in the case of linear and nonlinear
initial SDE systems. ** Key words:** stochastic differential equations (SDEs),
unstable SDEs, numerical methods for solution of SDEs, Monte Carlo methods.

**Kashuba E.V., Rukavishnikov V.A. On the p-version of the finite element method for the boundary
value problem with singularity** (

The one-dimensional first-type boundary value problem for the second order
differential equation with strong singularity of a solution caused by
coordinated degeneration of input data at the origin is considered. For this
problem we define the solution as*
-*generalized
one. It has been proved that solution belongs to the weighted Sobolev space
under
proper assumptions for coefficients and the right-hand side of the differential
equation. The scheme of the finite element method is constructed on a fixed mesh
using polynomials of an arbitrary degree *p* (the *p*-version of the
finite element method). The finite element space contains singular polynomials.
Using the regularity of*
*-generalized
solution, the estimate for the rate of
convergence of the second order with respect to the degree *p* of
polynomials is proved in the norm of the weighted Sobolev space. **
Key words:** the

**Kretinin A.V. Forming a neuronet database for perceptrons structure optimizatio** (

Results of the solution of a variety of neuronet approximations of different topology functions are used for formation of a training set on which the neuronet database is constructed for the perceptrons structure optimization.**
Key words:** perceptron, structure optimization.

**Nemirovskii Yu.V., Yankovskii A.P. Generalization of the Runge-Kutta methods and their application to integration of initial-boundary value problems of mathematical physics **
(

An idea is proposed and tested to generalize the Runge-Kutta methods to a
bidimensional case for the approximate integration of the initial-boundary value
problems corresponding to the partial differential equations. It is shown that some classical finite difference schemes of integration of
the equation of transport and non-stationary one-dimensional heat conductivity
can be obtained as consequence of such generalization. New schemes of high
orders of accuracy for various problems of mathematical physics are obtained.
Stability of these schemes is proved, and results of calculations for problems
with large gradients of the solution are presented. On concrete examples it is
shown that classical schemes of low orders of accuracy unsatisfactorily describe
solutions of such problems, and the schemes of high orders constructed by means
of the generalized Runge-Kutta methods presented, give a good approximation to
exact solutions.**
Key words:** numerical integration, initial-boundary value problems, generalization of the Runge-Kutta methods, large gradients of solution, stability of numerical schemes.

**Shevaldin V.T. Approximation by local parabolic splines with arbitrary knots **
(

For the class of the functions
with almost bounded second
derivatives, a new linear local method of parabolic spline approximation on an
arbitrary grid is constructed. This method has some smoothing properties and
inherits the monotonicity and the convexity of the initial data (values of a
function at the grid points). On this class the error of approximation by the
splines constructed is exactly calculated.**
Key words:** local method, parabolic spline approximation, the error of
approximation.

**Andreev A.B., Todorov T.D.**

**Superconvergence of the gradient for cubic triangular finite elements **(*in English*), pp.89-100

Superconvergence of the gradient of approximate solutions to
second order elliptic equations is analyzed and justified for the
10-node cubic triangular elements. The existence of superconvergent
points is proved. A recovery gradient technique in a subdomain is presented.
The superclose property is proved. A rigorous proof of
the superconvergent error estimate in a recovered gradient function is
obtained. Numerical experiments supporting the theory under study are presented. **
Key words:** finite element method, superconvergence,
recovered gradient.

**Artemiev S.S., Korsun A.E., Yakunin M.A. **

**Investigation of probability characteristics for a particular trade algorithm**
(*in Russian*), pp. 101-108

We investigate probability characteristics of a random sequence which
forms the total profitability for the trade algorithm based on a
simple model of a price series. In the case of the model of a
price series with a Gaussian distribution of profitabilities, the
formulas for calculation of some probability characteristics are obtained. **
Key words:** trade algorithm, profitability, probability
density.

**Kamenskii A.G., Kamenskii G.A.**

** On convergence of a finite difference scheme to solution of the
third boundary value problem for a system of abstract elliptic equations**
(*in Russian*), pp. 109-126

There is considered the third boundary value problem for abstract
elliptic equations. The problem of stability of solutions to a system of
elliptic equations on a restricted domain by non-smooth perturbations of the
boundary of this domain is studied. There are proposed a difference scheme
for an approximate solution of the considered problem and the conditions for
the convergence of solutions of this scheme to the exact solution of the
problem. **
Key words:** elliptic type equations, difference
schemes, boundary value problems.

**Larin M.
Using a compensation principle in the algebraic multilevel
iteration method for finite element matrices** (

In the present paper, an improved version of the algebraic multilevel iteration
(AMLI) method for finite element matrices, which was offered in [7],
is proposed. To improve the quality of the AMLI-preconditioned, or
(which is the same) to speed up the
rate of convergence, a family of iterative parameters defined on an error
compensation principle is proposed and analyzed.
The performance results on standard test problems are presented and discussed. **
Key words:** algebraic multilevel iteration method,
preconditioned conjugate gradient method, finite element approximation.

**Popov A.S. The search for the best cubature formulae invariant
under the octahedral group of rotations with inversion for a sphere ** (

The definition of the best cubature formula invariant under the octahedral
group of rotations with inversion for a sphere is given. The process of
searching for the best cubature formulae of the given symmetry type is
described. The table which contains the main characteristics of all the best
today
cubature formulae of the octahedral group of rotations with inversion
up to the 53rd algebraic order of accuracy is given. The weights and the
coordinates of the new cubature formulae of the 21st, 25th, 27th, 31st and
33rd orders of accuracy are given to 16 significant digits.

**Key words:** numerical integration, cubature formulae, octahedral
group of rotations.

**Shapeev A.V.
Investigation of mixed spectral and finite difference approximation
on the basis of a viscous flow problem in a diffusor**
(

A general approach to derivation of efficient numerical methods
based on a mixed spectral and finite difference approximation
for problems, whose solutions are smooth in some variables
and non-smooth in other variables is considered.
The approach considered is applied to the problem of
a viscous liquid flow in a plane diffusor.
Properties of the numerical method are investigated on the basis of computational
experiments.

**Key words:** non-stationary self-similar, fluid flow in a diffusor,
confusor; mathematical simulation, numerical method,
mixed spectral and finite difference approximation, discretization.

**Shkarupa E.V.
A functional random walk-on-grid algorithm for the biharmonic equation.
The error estimation and optimization ** (

We consider a functional algorithm of random walk-on-grid
as applied to the global solution of the Dirichlet
problem for the biharmonic equation. In the metric space *C*, a certain upper error bound is constructed,
and optimal values (in the sense of
the upper error bound) of the algorithm parameters, i.e.,
the number of grid nodes and the sample size are obtained.
We carry out numerical comparison of efficiency of the algorithm in question
and the global random walk on spheres algorithm,
based on the use of the fundamental solution to the biharmonic equation
for the problem of a bending of a thin elastic plate
with a simply supported boundary.

**Key words:** Monte Carlo methods, functional algorithms, random walks, biharmonic
equation, error estimation, optimization.

**Alekseev A.S., Glinski B.M.,Kotelevski S.P., Kuchin N.V.,
Malyskin V.E., Selikhov A.V. **

**
History of creation of the Siberian Super Computer Center,
state-of-the-arts and prospects for its development**
(*in Russian*), pp. 179-187

The history of creation of computational resources of Siberian Super
Computer Center (SSCC) and organization of the large-scale problem
solution are considered. The results of the project on the development
of multicomputer system SIBERIA in the 80-s, the current structure
of the SSCC and prospects of its development are considered and analyzed.

**Key words:** supercomputing, supercomputer center, multicomputer,
numerical modelling.

**Andreev A.B., Petrov M.S., Todorov T.D.**

**General results for lumped mass approximation of
isoparametric eigenvalue problem on triangular meshes** (*in English), *pp. 189-205

This paper deals with a FE-numerical quadrature method giving a diagonalization of the mass matrix (lumped mass matrix). The method is applied for a class of second order selfadjoint elliptic operators defined on a bounded domain in the plane. The isoparametric finite element transformations and triangular Lagrange finite elements are used.

The paper concludes with the investigation started by the authors
in \cite{and1,and2} for the isoparametric variant of the lumped mass
modification for second order planar eigenvalue problems.
Thus the relationship between the possible quadrature formulas
and the precision of the method is proved. The effect of these
numerical integrations on the error in eigenvalues and eigenfunctions is
estimated. At the end of the paper, the numerical results confirming the theory
are presented.

**Key words:** eigenvalue problem, isoparametric FEM,
lumped mass, numerical integration.

**Kwak Do Y., Lee Jun S. **

**The V-cycle multigrid convergence of some finite difference scheme for
the Helmholtz equation **
(

In this paper, we analyze the *V*-cycle multigrid algorithm for a positive
definite Helmholtz equation on a hexagonal grid.
Specifically, we apply the *V*-cycle multigrid algorithm to the numerical
scheme based on the mean value solutions for the Helmholtz equation on
hexagonal grids introduced in \cite{An-Do}, and show its convergence.
The theory for the *V*-cycle multigrid convergence is carried out in
the framework in \cite{Br-Xu91} by estimating the energy norm of the
prolongation operator and proving the approximation and regularity
conditions. In numerical experiments, we report the eigenvalues, condition
number and contraction number.**
Key words:** multigrid method, mean value solution,
finite difference methods.

**Lisitsa V.V.
Optimal grids for solution to the wave equation with variable coefficients**
(

This paper represents investigation of the method of constructing optimal
grids. Extension of this method to the wave equation with variable coefficients
and estimation of numerical solution obtained on optimal grids are also
considered. The experiments presented illustrate a decrease of the time required
for calculations. **
Key words:** optimal grids, Pade-Chebyshev approximation, Gaussian
quadrature rules, Lanczos method.

**Lugovkin S.E. ***
*

We study the problem of finding the necessary and sufficient conditions
of the fact that a given set of numbers could be a set of probabilities
of certain events and probabilities of the pairwise interselection
of such events. The algorithm of building a system of inequalities for
solving this problem is obtained. The application of this algorithm
to more general problems is discussed. **
Key words:** event, probability, the exception method, reliability,
compatibility of parameters.

**Marchenko M.A. **

**
Monte Carlo simulation of spatially inhomogeneous coagulation of particles
altogether with their diffusion**
(*in Russian*), pp. 245-258

Monte Carlo algorithm for simulation of coagulation of particles
altogether with their diffusion is developed. The problem to solve is
the boundary-value problem for the 1D Smoluchowski
equation containing convection and diffusion terms.
The stochastic particles method is underlying the algorithm.
The principal features of the algorithm are the use of special Markov
process and a splitting scheme according to physical processes.
A special technique to reduce the estimator variance is developed.
The method of tentative estimation of the algorithm parameters is given.

**Key words:** Monte Carlo, Smoluchowski's equation, coagulation, diffusion,
nucleation.

**Tabarintseva E.V.
On error estimation for the quasi-inversion method for solving a semi-linear
ill-posed problem **
(

In this paper, the approximate solutions error estimates are obtained
for an ill-posed semi-linear Cauchy problem. The continuity module of the
inverse operator is used as a standard estimator for obtaining the error estimates.
The value of the continuity module is calculated for two classes of uniform
regularization of the original problem. The quasi-inversion method is used
to construct stable approximate solutions.

**Key words:** differential-operator equation, Cauchy problem, ill-posed
problem, method of approximate solution, error estimate.

**Akhysh A.Sh.**

**The \ell_{p} stability of some difference schemes for one system of
nonlinear parabolic equations **
(

In the present paper, the stability in the space \ell

explicit scheme, implicit scheme, splitting scheme.

A parametric analysis of fundamental characteristics of profitability and risk of the two trade algorithms is realized by

Monte Carlo method. Numerical experiments are executed on the model prices of stocks, which are a discrete analogue to stochastic differential equations. The description of a modeling program is presented.

Dynamic problem of linear viscoelasticity in velocity-stress statement

A conjugate-operator viscoelastic model in the velocity-stress statement is studied. To implement it numerically, a class of implicit difference schemes based on the spatial variables splitting is constructed.

Monte Carlo estimates of derivatives with respect to parameters of the solution of the parabolic equation based on numerical SDE solution

In this paper, a statistical method of estimation of the solution of the parabolic equation and its derivatives with respect to parameters is proposed. This method is based on the numerical solution of stochastic differential equations (SDE's) by the Euler method. The order of convergence of using functionals of the SDE's is determined. Some numerical results are given.

On extrapolation with respect to a parameter in the perturbed mixed variational problem

In this paper, the extrapolation with respect to a regularization parameter in the mixed variational problem is investigated. The estimates obtained are applied to a few examples of boundary value problems. The results of numerical experiments are given.

In this paper, some features of the polar vortex dynamics are investigated. We use a mathematical model, in which a stream with a linear shift with overlapped stationary waves is taken as basic state. The interaction between the basic stream and the

non-stationary Rossby waves is examined. The stability of trajectories is studied. The numerical estimation of some characteristic parameters is made, and the phenomenon of chaotic advection is discussed.

equation from the Ulyanov

The paper dealt with a problem of numerical integration, and approximate restoration of functions and solutions to the heat conductivity equation with functions of distribution of starting temperatures from the classes

Average discrepancy for periodic integrands

In the numerical integration of periodic integrands over the