** Number 1, pp.1-95 Number 2, pp.97-185 Number 3, pp.187-282 Number 4, pp.283-376**

**
Bakushinskii A.B., Kokurin M.Yu.**
(

Continuous methods for stable approximation of solutions to nonlinear equations in the Banach space based on the regularized Newton-Kantarovich scheme

We propose and study a class of methods for approximation of solutions to
nonlinear equations with smooth operators in a Banach space, when the
operators are approximately given and their derivatives are not regular.
The construction of the presented methods is connected with the operator
differential equation obtained by linearization of the original equation
using the Newton-Kantorovich scheme and various ways of its regularization.
When the initial discrepancy possesses a sourcewise representation, we
establish estimates for the approximation errors.

**
Key words:** nonlinear equation, irregular equation, Banach space,
operator differential equation, regularization, stopping rule.

**
Klimenko O.A.**
(

A method for solution of elasto-electrodynamics problem

The problem of elasto-electrodynamics is under investigation in this paper. The theory of elasto-electrodynamics deals with the mutual influence of a deformation field and an electric field in the elastic solid. An elastic conducting medium on a three-dimensional half-space is under consideration (for example, the Earth). A specific instantaneous point source of deformation is created on the boundary of the medium. This deformation involves the motion of charged particles in a conducting medium. It is required to find a coefficient that defines this current as a function of depth. One of intensity components of electric field on the boundary of the medium is known.

Very complicated elasto-electrodynamics model is simplified and for a new
model, the algorithms for direct and inverse problems are
constructed.

**
Key words:** inverse problem, elasto-electrodynamics.

**Larin M.R. ** **
On a multigrid method for solving partial eigenproblems** (

Recently the direct application of a multigrid technique for computing
the smallest eigenvalue and its corresponding eigenvector of a large
symmetric positive definite matrix *A* has been
investigated in \cite{L2000}. This method solves the eigenvalue problems
on a sequence of nested grids using an interpolant of solution on
each grid as initial guess for the next one and improving it by the full
approximation scheme applied as an inner nonlinear
multigrid method.
In the present paper, the generalization of the method for computing
a few smallest eigenvalues and their corresponding eigenvectors of the
elliptic self adjoint operator is presented. Moreover, the
quality of the method is improved by using the nonlinear Gauss-Seidel
iteration instead of its linearized version as pre- and post-smoothing
steps. Finally, we give some advice for a good choice of
multigrid-related parameters.

**
Key words:** multigrid methods, eigenvalue problems, sparse matrices.

**
Marchenko M.A.
The program package MONC for distributed computations by Monte Carlo method **
(

In this paper, the technique of distributed computations for Monte
Carlo methods within personal computers using the program system
MONC is considered. The following items are discussed: parallel
modification of congruential pseudorandom numbers generator;
functional capabilities of the MONC; demands for the user's program
to execute using the MONC; the estimate of computational costs of
distributed computations using the MONC. Advantages of the MONC are
shown when solving diffusion problems.

**
Key words: ** parallel computations, Monte Carlo method,
pseudorandom number generators, computing system.

**
Nechepurenko M.I.
On some characteristics of the multigraph arc coherence **
(

Papers [1-3] present values of the greatest arc coherence* **λ*(*p,q*)
and the smallest number *B*(*p,q*) of the power cuts * **λ*(*p,q*) for
(*p,q*)-multigraphs. The present paper states the complete corrected proof
of the results from [3], which brings about obtaining the asymptotic values
of probabilities of coherence of one class of random multigraphs.

**Key words:** multi-graph, minimum edge's connectivity, maximum
edge's connectivity cuts.

**
Shurina E.P., Gelber M.A.
On vector finite element method for solution to electromagnetic problems **
(

The vector finite element method is a comparatively new approach,
therefore for this method neither a general theory no computational scheme
has been developed. The aim of the present
paper is to analyze application of the method to solution of
electromagnetic problems. Special vector variational formulations
have been constructed depending on the problem. Interpolation
properties of different types of elements are investigated both
theoretically and experimentally.

**
Key words:** vector finite element method,
modeling of 3D electromagnetic fields.

**
Smelov V.V.
On efficient approximation of piecewise smooth functions with
their presentation by rapidly converging piecewise polynomial
series **
(

A variant of expansion of piecewise smooth functions in
rapidly converging series about specific
piecewise polynomial functions is proposed. These specific
functions are constructed on the basis of the Legendre
polynomials. This paper is the sequel of the author's previous
publications [1] and forms the basis of the efficient
approximations of the above-mentioned functions.

**
Key words:** piecewise smooth functions, rapidly converging
series, piecewise polynomial basis, approximation.

**Kamenskii G.A., Varfolomeyev E.M.**

**Approximate solution of variational problemsfor the mixed type nonlocal
functionals **(*in English*), pp.115-123

There are considered variational problems for the mixed type nonlocal functionals. The application of the Ritz method and the
method of least square for the quadratic functionals of the above-mentioned type are investigated.

**Key words:** calculus of variations, nonlocal functionals, approximate solutions, methods of least squares and the Ritz
method.

**Kargin B.A., Sabelfeld K.K., Artemiev S.S., Voytishek A.V.**

**On the anniversary of Gennady Alekseevich Mikhailov **(*in
Russian*), pp.97-101

**Noskov M.V., Osipov N.N.**

**Minimal and almost minimal rank 1 lattice rules, exact on trigonometric polynomials in two
variables** (*in Russian*), pp.125-134

Two-dimensional rank 1 lattice rules of trigonometric degree *d* (*d* ≥
1) are characterized. The number of nodes of these cubature formulas is minimal or
differs from minimal by one for even *d*, or by two for odd *d*.

**Key words: ** minimal cubature formula, lattice rule of trigonometric degree *d*.

**Omelayeva O.S.**

** A version of the commutative alternating direction method**
(*in Russian*), pp.135-141

In this paper, we consider a version of the iterative adaptive commutative alternating
direction method. For the optimization of the method we need not require a priori
spectrum information. The convergence rate estimate is kept the same as in the case with a
priori information.

**Key words:** optimization, two-level iterative methods.

**Palymskiy I.B.**

**Linear and nonlinear analysis of the numerical method for the calculation of convective
flows** (*in Russian*), pp.143-163

Spectral characteristics of the numerical method for calculation of convective flows are investigated. These characteristics are
compared to spectral characteristics of the original differential problem. Nonlinear analysis of the numerical method is made on a model system of equations. The results of
calculation of turbulent convection with the Rayleigh number up to 1350 critical values are presented. These results are
compared to experimental data and numerical results obtained by other authors.

**Key words:** convection, spectral characteristics, Rayleigh number, Prandtl number, turbulence, chaotic mode, super-criticality.

**Prigarin S.M., Martin A., Winkler G.**

** Numerical models of binary random fields on the basis of thresholds of Gaussian
functions ** (*in English*), pp.165-175

We present a few numerical models of binary stochastic fields based on the thresholds of Gaussian functions and discuss the results of
numerical experiments on estimating the models' parameters and simulation of the observed data. The considered models can be used, in
particular, for texture analysis and synthesis, for simulation of stochastic structure of clouds in the atmosphere, as well as for
solving other problems when statistical analysis and construction of binary random fields are a part of research.

**Key words:** stochastic simulation, numerical models of random fields, binary fields, texture analysis and synthesis.

**Rukavishnikov V.A., Rukavishnikova E.I.**

**On the error estimation of the finite element method for the third boundary value problem with singularity
in the space** *L ^{*}*

The paper analyzes the finite element method for the third boundary value problem for non-self-conjugate second order elliptic equation with coordinated degeneration of initial data and with strong singularity of solution. The scheme of the finite element method is constructed on the basis of the definition of

A boundary value problem for a linear system of ordinary second order differential equations with a small parameter at higher derivatives on a semi-infinite interval is considered. Systems of reaction-diffusion and convection-diffusion equations are considered. The method of reduction of a problem to a finite interval problem, based on the extraction of a set of solutions satisfying the limit conditions on infinity, is investigated. Auxiliary singular Cauchy problems for the differential matrix Riccati equations are solved with the use of a series in powers of a small parameter and an independent variable. Accuracy of the method proposed is estimated. The Shishkin mesh is proposed for solving a problem after its reduction to a finite interval. The results of numerical experiments are presented.

**On the anniversary of Sergey Konstantinovich Godunov **(

(

Monotone operators provide a basis for pointwise bounds of the solution and discretization errors. We apply this technique for convection-diffusion problems, including an anisotropic diffusion term and show that the discretization error has a higher order of accuracy near Dirichlet boundaries or, alternatively, the second order of the global error remains even if we use a lower order of approximation near the Dirichlet boundary.

(

A standard scheme of the finite element method with the use of bicubic elements on a rectangular quasi-uniform grid is considered as applied to the two-dimensional Dirichlet problem for the biharmonic equation in a rectangle. To solve this scheme, two multigrid algorithms are treated on a sequence of embedded rectangular grids: a full multigrid with

(

The flutter of viscoelastic cylindrical shells streamlined by a gas current are investigated. The basic direction of the present work consists in taking into account of viscoelastic material properties at supersonic speeds. The vibration equations relative to deflections are described by integro partial differential equations. By the Bubnov-Galerkin methods, the problems are reduced to investigation of a system of ordinary integro-differential equations (IDE). The IDEs are solved by a numerical method which is based on using the quadrature formulas. Critical speeds for the shell flutter are defined.

(

Implicit multi-step quasi-Newton methods, introduced in [1], use the existing Hessian approximation to compute, at each iteration, the parameters required in the interpolation. To avoid the burden of computing the needed matrix-vector products, required by this approach, approximations based on the Secant Equation were proposed. Based on [2], a different approach to dealing with this difficulty was suggested, in which standard single-step quasi-Newton updates were replaced by successive iterations, by two-step updates, so that approximations were no longer necessary. The recent research has shown that the quantities required to compute the parameters referred to the above may be exactly computed by means of recurrence, so that the technique of alternation is no longer the only alternative. In this paper, we consider the derivation of new recurrences for the implicit update methods based on the well-known Symmetric Rank One (SRI) update formula. We present the results of a range of numerical experiments to compare and evaluate the methods developed here.

Refinement of convergence conditions of the Chebyshev method.

(

The iterative Chebyshev method of an approximate solution of equations of the form

(

A non-stationary multiresolution analysis

The space *V _{k}* is the same as the space of discrete splines

The paper deals with determination of spatial distribution of oscillation sources by remote measurements on a finite set of points. This problem is assumed to be a problem of reconstruction of the original tsunami waveform from the measurement of the arrived wave on a finite set of coastal receivers. The propagation of the wave is described by linearized shallow-water equations when the depth depends on two variables. The direct problem is approximated by the explicit-implicit finite difference scheme. The ill-posed inverse problem of reconstruction is regularized by means of singular value decomposition, so

(

The subject of consideration is the integral equations with

(

It is shown that the algorithms proposed in [1, 2] for solving problems of rigid body dynamic deformation are efficient when a time iterative procedure is applied to the case of essentially inhomogeneous domains with not extensive but very rigid impurities. In multidimensional case this, in particular, solves the problems of constructing a mesh adaptive to an inhomogeneous medium and solution matching at the boundaries of subdomains.

**
Bubyakin A.A., Laevsky Yu.M. **

**
On one approach to constructing schemes of increased
order of accuracy in the finite element method.**

(*in Russian*), pp.287-300

The paper considers schemes of increased order of accuracy in the
finite element method with the same number of degrees of freedom
as in the schemes constructed by the Galerkin method with the
use of piecewise-polynomial functions. The approach proposed is
based on a special choice of the grid scalar products and the
right-hand side linear functionals limited on a set of grid
functions. The fourth order of accuracy is established in the
grid energy norm.

**
Key words:** mixed derivative, finite element, compact scheme,
bilinear form, accuracy.

**
Gorunescu F., Gorunescu M., Gorunescu R. **

**
A metaheuristic GAs method as a decision support for the choice
of cancer treatment.**

(*in English*), pp.301-307

This paper focuses on a metaheuristic method that
helps in evaluating the cancer treatment complexity.
We show how to help find a (near) optimal treatment
formula by using a genetic algorithms approach.
When the diagnosis problem has been solved, attention
is given to designing the treatment procedure.
The goal of this paper is to explore a GA-based approach
to determine the (near) optimum treatment formula depending
on some features of the patient. An application to breast and
uterus cancers is presented as well.

**
Key words:** cancer treatment, genetic
algorithm, evolution program, Java implementation.

**
Kotel'nikov E.A. **

**
Searching for the global maximum of a quadratic function with linear
constraints.**

(*in Russian*), pp.327-334

The global maximum of a quadratic function is localized with
the help of a decreasing sequence of linear or quadratic majorants
of the objective function. The majorants are constructed on subsets
of the set of admissible solutions.

**
Key words:** global optimum of quadratic function.

**
Larin M., Padiy A. **

**
On the theory of the generalized augmented matrix preconditioning method.**

(*in English*), pp.335-343

This paper is devoted to an improvement of the theory of
the recently proposed generalized augmented matrix preconditioning
method*. Namely, we compute a sharp lower bound on
the eigenvalues of a preconditioned matrix based on the
properties of a projector involved in its definition.

(* Padiy A., Axelsson O., Polman B. Generalized augmented matrix preconditioning approach and its
application to iterative solution of ill-conditioned algebraic systems//
SIAM J. Matrix Anal. Appl.- 2000.- N 22.- P.793-818.)**
**

**
Prigarin S.M., Fedchenko N.V. **

**
Solution of boundary value problems for linear
systems of stochastic differential equations.**

(*in Russian*), pp.345-361

The paper deals with methods to solve boundary value problems
for linear systems of stochastic differential equations. We
investigate numerical algorithms, the problem of existence and
uniqueness of solutions, and other more specific problems
(including steady-state boundary value problems, reduction of a
boundary value problem to a Cauchy problem, extended boundary
value problems, active and passive boundary conditions, etc.).

**
Key words:** stochastic differential equations, boundary
value problems, numerical algorithms, linear systems, existence and
uniqueness of solutions, active and passive boundary conditions.

**
Shary S.P. **

**
Solving tied interval linear systems.**

(*in Russian*), pp.363-376

This paper presents a survey of modern techniques for enclosing
the solution sets to interval linear systems whose parameters are
subject to additional ties. For optimal (exact) component-wise
estimation of the solution sets to interval linear systems with
symmetric, persymmetric, Hankel and Toeplitz matrices, we develop
so-called *parameter partitioning methods* (PPS-methods)
based on adaptive partitioning of the interval initial data of
the problem under consideration.

**
Key words:** interval linear systems, tied parameters, adaptive
partitioning, PPS-methods.

**
Zhelezovskii S.E.**

**On error estimates for schemes of the projection-difference method for
hyperbolic equations **(*in Russian*), pp.309-325

We study the convergence of a three-level scheme of the
projection-difference method for an abstract quasi-linear
hyperbolic equation. We establish asymptotic energy estimates
for the error. The order of these estimates is unimprovable. A
preliminary result on the conditional stability of the scheme
(*W*-stability in the sense of the definition formulated in the
paper) forms the basis of our derivation of the estimates. We
illustrate the use of our general results by an example of a
scheme with finite element space discretization applied to the
first initial boundary-value problem for a second-order
hyperbolic equation. We also note the possibility of application
of our general results in the case when the space discretization
is realized by the Galerkin method in the form of Mikhlin.

**
Key words:** quasi-linear hyperbolic equation,
projection-difference method, asymptotic error estimates.