**
Number 1, pp. 1-95
Number 2, pp. 97-199
Number 3, pp. 201-301
Number 4, pp. 303-404**

**
Antyufeev V.S.**

**
Theorem of training for a competition algorithm**

**
**

This paper is an extension of [1], where a new decision algorithm was proposed.
In its operation, the unit resembles artificial neural networks. However the
functioning of the algorithm proposed is based on the different concepts. It
does not use the concept of a net, a neuron. The theorem of training for the new
competition algorithm is proved**.**

**
Key words:**
theorem of training, probabilistic convergence, artificial neural network.

**
Zadorin A.I., Tikhovskaya S.V.**

**
Solution of second order nonlinear singular perturbation ordinary
differential equation based on the Samarskii scheme**

**
**

A boundary value problem for a second order nonlinear singular perturbation
ordinary differential equation is considered. We propose the method based on the
Newton and the Picard linearizations using known modified Samarskii scheme on
the Shishkin mesh in the case of a linear problem. It is proved that the
constructed difference schemes are of second order and uniformly convergent. To
decrease the number of the arithmetical operations, we propose a two-grid
method. The results of some numerical experiments are discussed.

**
Key words:**
second
order nonlinear ordinary differential equation, singular perturbation, Newton
method, Picard method, Samarskii scheme, Shishkin mesh, uniform convergence,
two-grid algorithm.

**
Zorkaltsev V.I., Perzhabinsky S.M.**

**
Theoretical justification of interior point algorithms for solving
optimization problems with nonlinear constraints**

**
**

A family of interior point algorithms is considered. These algorithms can be
used for solving mathematical programming problems with nonlinear inequality
constraints. The weighted Euclidean rates are applied to find a descent
direction for improving a solution. These rates are varying in iterations.
Theoretical justification of the algorithms with some assumptions (such as
non-degeneracy of a problem) is presented.

**
Key words:**
interior point method, weighted Euclidean rate, linearization.

**
Kabanikhin S.I., Krivorotko O.I., and Shishlenin M.A.**

**
A numerical method for solving inverse thermoacoustic problem
**

**
**

**
**In this paper, we consider the inverse problem of determining the initial
condition of the initial boundary value problem for the wave equation with
additional information about solving the direct initial boundary value problem
that is measured at the boundary of the domain. The main objective of our
research is to construct a numerical algorithm for solving the inverse problem
based on the method of simple iteration (MSI) and to study the resolution of the
inverse problem and its dependence on the number and location of measurement
points. We consider three two-dimensional inverse problems. The results of
numerical calculations are presented. We show that the MSI for each iteration
step reduces the value of the object functional. However, due to the ill-posedness
of an inverse problem the difference between the exact and the approximate
solutions of the inverse problem decreases up to some fixed number
*k*_{min} and then monotonically
increases. This reflects the regularizing properties of the MSI, in which the
iteration number is a regularization parameter.

**
Key words:**
thermoacoustic problem, inverse and ill-posed problems, wave equation, method of
simple iteration.
**
**

**
Kostin V.I., Lisitsa V.V., Reshetova G.V., Tcheverda V.A.**

**
Finite difference simulation of elastic waves propagation through 3D
heterogeneous
multiscale media based on locally refined grids
**

**
**

In order to simulate the interaction of seismic waves with
microheterogeneities (like cavernous/fractured reservoirs), a finite difference
technique based on locally refined in time and in space grids is used. The need
to use these grids is due to essentially different scales of heterogeneities in
the reference medium and in the reservoir. Parallel computations are based on
Domain Decomposition of the target area into elementary subdomains in both the
reference medium (a coarse grid) and the reservoir (a fine grid). Each subdomain
is assigned to its specific Processor Unit which forms two groups: for the
reference medium and for the reservoir. The data exchange between PU within the
group is performed by non-blocking iSend/iReceive MPI commands. The data
exchange between the two groups is done simultaneously with coupling a coarse
and a fine grids and is controlled by a specially designated PU. The results of
numerical simulation for a realistic model of fracture corridors are presented
and discussed.

**
Key words:**
seismic
waves, finite difference techniques, domain decomposition, interpolation, groups
of processor elements.

**
Popov A.S.**

**
The cubature formulas on a sphere invariant with respect to a dihedral
group of rotations with inversion D_{6h}**

**
**

An algorithm of searching for the best (in a sense) cubature formulas on
a sphere that are invariant with respect to a dihedral group of rotations with
inversion *D*_{6h} has been veloped. This algorithm was applied to find
parameters of all the best cubature formulas of this group of symmetry up to the
23rd order of accuracy *n*. In the
course of the study carried out, the exact values of parameters of the
corresponding cubature formulas were found for
*n ≤* 11, and the approximate ones were obtained by the numerical
solution of systems of nonlinear algebraic equations by a Newton-type method for
the other *n*. For the first time, the
ways of obtaining the best cubature formulas for the sphere were systematically
investigated for the case of the group which is not a subgroup of the groups of
symmetry of the regular polyhedrons.

**
Key words:**

numerical integration, invariant cubature formulas, invariant polynomials, dihedral group of rotations.

**
Pchelintsev A.N.**

**
On constructing the generally periodical solutions of a complicated
structure of a non-autonomous system of differential equations
**

In this paper, a numerical scheme of constructing approximate generally
periodical solutions of a complicatedtructure of a non-autonomous system of
ordinary differential equations with the periodical right-hand sides on the
surface of a torus is considered. The existence of such solutions as well as
convergence of the method of successive approximations are shown. There are
given results of the computational experiment.

**
Key words:**
generally-periodical solution, system of ordinary differential equations,
Fourier series, almost periodical solution, irrational winding of torus.

**
Saberi Najafi H., Edalatpanah S.A.**

**
Comparison analysis for improving preconditioned SOR-type iterative
method
**

In this article, on the basis of nonnegative matrices, some preconditioners from
class of (*I+S*)-type based on the SOR
method have been studied. Moreover, we prove the monotonicity of spectral
radiuses of iterative matrices with respect to the parameters in [12]. Also,
some splittings and preconditioners are compared and derived by comparisons. A
numerical example is also given to illustrate our results.

**
Key words:
**
preconditioning, comparison theorems, spectral radius, SOR,

**
Chistyakov V.F., Chistyakova E.V.**

**
Application of the least square method to the solving linear
differential-algebraic equations**

**
**

We consider application of the least square method to the numerical solution of
a linear system of ordinary differential equations (ODEs) with an identically
singular matrix multiplied a higher derivative by the desired vector-function.
We discuss the behavior of the gradient method for minimizing the functional of
the residual square in the Sobolev space and some other issues. The results of
the numerical experiments are given.

**
Key words:
**
differential-algebraic equations, index, least square method, gradient methods.

An algorithm for statistical simulation of random-structure systems with distributed transitions has been constructed. The proposed algorithm is based on numerical methods for solving stochastic differential equations, and uses a modified maximum cross-section method when the transition intensity depends on the vector of state.

**Akimova E.N., Belousov D.V., Misilov V.E.**

**Algorithms for solving inverse geophysical problems on parallel computing
systems**** ** **(****in Russian****)****, pp.
**
**107-121**

For solving inverse gravimetry problems, efficient stable parallel algorithms
based on iterative gradient methods are proposed. For solving systems of linear
algebraic equations with block-tridiagonal matrices arising in geoelectrics
problems, a parallel matrix sweep algorithm, a square root method, and a
conjugate gradient method with preconditioner are proposed. The algorithms are
implemented numerically on the MVS-IMM parallel computing system, NVIDIA
graphics processors, and the Intel multi-core CPU with the use of new computing
technologies. The parallel algorithms are incorporated into a developed system
of remote computations «Specialized Web-Portal for Solving Geophysical Problems
on Multiprocessor Computers». Problems with «quasi-model» and real data are
solved.

**Key words:**
inverse gravimetry problems, parallel algorithms, direct
and iterative methods, parallel computing systems.

Akysh A.Sh.

Convergence of splitting method for the nonlinear Boltzmann equation**(**in Russian**)**, pp.

The question of convergence of the splitting method scheme for the nonlinear
Boltzmann equation is considered. On the basis of the splitting method scheme,
boundedness of positive solutions in the space of continuous functions is
obtained. By means of the solution boundedness and found a priori estimates,
convergence of the splitting method scheme and uniqueness of the limiting
element are proved. The found limiting element satisfies the equivalent integral
Boltzmann equation. Thereby global solvability of the nonlinear Boltzmann
equation in time is shown.

**Key words:**
splitting method, convergence
of the splitting method scheme, nonlinear Boltzmann equation, global solvability
of the nonlinear Boltzmann equation in time, existence and uniqueness of a
solution to the Boltzmann equation, a priori estimates.
**
**

Aleksandrov V.M.

Transferring a system with unknown disturbance under optimal control to a state
of dynamic balance and to ε-vicinity of a final state**
**

**
**

The problem of transferring a linear system to a state of dynamic balance under
simultaneous action of an unknown disturbance and time-optimal control is
considered. Optimal control is calculated along the phase trajectory, and it is
periodically updated for discrete phase coordinate values. It is proved that the
phase trajectory comes to the dynamic equilibrium point and makes undamped
periodic motions (a stable limit cycle). The location of the dynamic equilibrium
point and the limit cycle form are considered as functions of different
parameters. With the disturbance calculated in the process of control, the
accuracy of transferring to the required final state increases. A method for
estimating attainable accuracy is presented. Results of simulation and numerical
calculations are given.

**Key words:****
optimal
control, speed, computing time, disturbance, phase trajectory, dynamic balance,
limit cycle, transferring accuracy, linear system.
**

**Babkina L.A, Garmai Yu.P., Lebedev D.V., Pantina, R.A., Filatov M.V.,
Isaev-Ivanov V.V.**

**Using Zernike moments for analysis of images****
** **( in Russian**

A method for analyzing AFM images of the cell nuclei of higher organisms by
expanding these images by Zernike moments is proposed. This method allows for
expanding the pilot image by Zernike moments whose spatial harmonics are Zernike
polynomials. It is shown that the reverse procedure of image reconstruction
using Zernike polynomials converges to the experimental image and the expansion
amplitude is a quantitative spectral characteristic when comparing the
morphological features of different images. It is shown that expansion
amplitudes can be used as input vectors for cluster analysis of images by PCA.**
**

**Key words:****
image analysis, Zernike moments, atomic force microscopy, cell nuclei of higher
organisms, PCA.
**

**
Matsokin A.M.**

**
Preconditioner for a Laplace grid operator on a condensed grid**

**
In this paper, it is proved that a Laplace grid operator approximating a Dirichlet boundary value problem for the Poisson equation by the finite element method with piecewise-linear functions on an evenly condensed grid that is topologically equivalent to a rectangular grid (i.e. obtained by
**

**
Fadeev S.I., Kogai V.V., Mironova V.V., Omelyanchuk N.A., and Likhoshvai V.A.**

**
Mathematical modeling of matter distribution in cells assembling into a ring**

**
In this paper, a mathematical model describing substance transport in a circular
cell ensemble is considered. The model is represented by an autonomous system of
equations. With a model of continuation with respect to a parameter, it is shown
that stationary solutions may have different symmetry representing closed
curves. Periodic solutions have the same property, whereas the component plots
repeat each other by a simple shift.**

**
Key words:
cell ensemble, gene networks, autonomous system, circular model, stationary
solution, auto-oscillations, model for continuation with respect to parameters.**

**
Hou T.****
**

**
Superconvergence and a posteriori error estimates of RT1 mixed methods for
elliptic control problems with an integral constraint**

**
In this paper, we investigate the superconvergence property and a posteriori
error estimates of mixed finite element methods for a linear elliptic control
problem with an integral constraint. The state and co-state are approximated by
order k=1 Raviart-Thomas mixed finite
element spaces, and the control variable is approximated by piecewise constant
functions. Approximations of the optimal control of the continuous optimal
control problem will be constructed by a projection of the discrete adjoint
state. It is proved that these approximations have convergence order
h^{2}. Moreover, we derive a
posteriori error estimates both for the control variable and the state
variables. Finally, a numerical example is given to demonstrate the theoretical
results.**

**
Key words:****
elliptic equations, optimal control
problems, superconvergence, a posteriori error estimates, mixed
finite element methods, postprocessing.**

**
Amelkin V.A.**

**
Enumeration problems of sets of increasing and decreasing
n-valued serial sequences with
double-ended constraints on series heights **

Enumeration problems for *n*-valued
serial sequences are considered. Sets of increasing and decreasing sequences
whose structure is specified by constraints on lengths of series and on a
difference in heights of the neighboring series in the case when this difference
lies between
δ_{1 }
and
δ_{2 }
are examined.

**
Formulas for powers of these sets and algorithms for the
direct and reverse numerations (assigning smaller numbers to the
lexicographically lower-order sequences or smaller numbers
to the lexicographically higher-order sequences) are obtained.**

**
Key words:
serial sequences, series length, series height, constraints.
**

**
Antyufeev V.S.**

**
Finding the most probable non-negative solution of systems of linear
algebraic equations by the likelihood method
**

A probabilistic method for regularization is proposed. This method
enables one to obtain a non-negative solution to systems of linear algebraic
equations. A theorem of existence of the best possible solution is proved. A
numerical example of the method application is given.

**
Key words:** system
of linear equations, non-negative regularization, probability distribution,
stochastic ensemble.**
**

**
Zabinyako G.I., Kotel'nikov E.A.**

**
Minimization of nonlinear functions with linear constraints
**

**
**

In this paper, some aspects of numerical realization of algorithms from
the software package for solving problems of minimization of nonlinear functions
including non-smooth functions with allowance for the linear constraints set by
sparse matrices are considered. Examples of the solution of test problems are
presented.

**
Key words:**
nonlinear programming, reduced gradient, method of conjugate gradients,
quasi-Newton method, subgradient method, basis, superbasis.**
**

**
Penenko V.V., Tsvetova E.A.**

**
Variational methods for constructing of monotone approximations for
atmospheric
chemistry models
**

**
**

A new method for constructing efficient monotone numerical schemes for
solving direct, adjoint, and inverse atmospheric chemistry problems is
presented. It is a systhesis of a variational principles combined with splitting
and decomposition methods and a constructive realization of the Eulerian
integrating factors (EIM) by means of the local adjoint problem technique. To
provide the efficiency of calculations, a method to decompose the multi
component substances transformation operators in terms of mechanisms of
reactions is also proposed. With the analytical EIMs, the decomposed systems of
stiff ODEs are reduced to the equivalent systems of integral equations. To solve
them, non-iterative multistage algorithms of given order of accuracy are
developed. An original variational method for constructing of mutually
consistent algorithms for direct and adjoint problems, and sensitivity studies
for complex models with constraints is developed.

**
Key words:**
variational principle, stiff systems ODE, integrating multipliers,
discrete-analytical approximations, atmospheric chemistry, algorithms for
sensitivity studies.**
**

**
Savelyev L.J.**

**
A minimum of the centered discrete random variables dispersion
**

**
**

The problem of isolation of discrete random values and vectors with
discrete distributions having a given average value and a minimum dispersion is
solved. The vector model is associated with statistical methods of calculation
of multiple integrals and solutions to systems of the integral equations.

**
Key words:
**
discrete
distribution, random variable, random vector, average value, dispersion.**
**

**
Sorokin S.B.**

**
Analytical solution of generalized spectral problem in the method of
recalculating boundary conditions for a biharmonic equation
**

An iterative algorithm with an efficient preconditioner for the numerical solution of an elastic problem in approximation of plate theory with mixed boundary conditions is proposed and substantiated. Exact constants of energy equivalence for optimization of iteration method are obtained. Inversion of the preconditioner is equivalent to the double inversion of a discrete analog of the Laplace operator with the Dirichlet boundary conditions.

**
Key words:**
biharmonic equation, boundary conditions, iterative method, Poisson's
equation, plate, Dirichlet problem.**
**

Cherepennikov V.B.

Numerical analytical method of studying
some linear functional differential equations

This paper presents the results of studies of the scalar linear
functional-differential equation of a delay type*
ẋ*(*t*)=*a*(*t*)*x*(*t*-1)+*b*(*t*)*x*(*t/q*)+*f*(*t*),
*q*>1. The main attention is being given to the original
problem with the initial point, when the initial condition is specified at the
initial point, and the classical solution, whose substitution into the original
equation transforms it into the identity, is sought for. The method of
polynomial quasi-solution, based on representation of an unknown function
*x*(*t*)
as polynomial of degree
*N*
is applied as the method of investigation. Substitution of this function in the
original equation results in the residual
δ(*t*)=*O*(*t*^{N}),
for which an accurate analytical representation is obtained. In this case, the
polynomial quasi-solution is understood as exact solution in the form of
polynomial of degree
*N*,
disturbed because of the residual of the original initial problem. The theorems
of existence of polynomial quasi-solutions for the considered linear
functional-differential equation and exact polynomial solutions have been
proved. The results of the numerical experiment are presented.

**
Key words: ** functional
differential equations, initial value problem, polynomial quasi-solutions, exact
solutions.

**
Shumilov B.M.**

**
Cubic multiwavelets orthogonal to polynomials and a splitting algorithm
**

In this paper, an implicit method of decomposition of hermit cubic
splines using the new type multiwavelets with supercompact supports is
investigated. The splitting algorithm of wavelet-transformations on the parallel
solution of two three-diagonal systems of the linear equations with strict
diagonal domination is reasonable. The results of numerical experiments are
presented.

**
Key words: **hermit
splines, multiwavelets, implicit relations of decomposition, parallelization.

Artemiev S.S., Korneev V.D., Yakunin M.A.

**
Numerical solution to stochastic differential equations with a random
structure on supercomputers**
(in Russian)

**
**
**
**
In this paper we investigate the precision of estimate of the expectation
of solutionsto
stochastic differential equations with a random structure. The dependence of the
precision of estimate on the size of the integration step of the generalized
Euler method and on the volume of the simulated trajectories is shown. A strong
loss of the precision of estimate at deterministic or random times of changing
the SDE structure is shown on an example of a simple equation. This requires the
use of supercomputers for the statistical modeling. The results of the numerical
experiments carried out in the Siberian SuperСomputer Center are presented.

**
Key words:
**
stochastic differential equations, parallelization, supercomputer, the
methods of statistical modeling, the generalized Euler method.

Zadorin A.I., Zadorin N.A.

An analogue of Newton-Cotes formula with four nodes for a function with a
boundary-layer component**
**

The construction of the Newton-Cotes formulas is based on approximating
an integrand by the Lagrange polynomial. The error of such quadrature formulas
can be serious for a function with a boundary-layer component. In this paper, an
analogue to the Newton-Cotes rule with four nodes is constructed. The
construction is based on using non-polynomial interpolation that is accurate for
a boundary layer component. Estimates of the accuracy of the quadrature rule,
uniform on gradients of the boundary layer component, are obtained. Numerical
experiments have been performed.

**
Key words:
**
one-variable function, boundary-layer component, high gradients, definite
integral, non-polynomial interpolation, quadrature rule, error estimate.

Mastryukov A.F.

The numerical solution of the inverse problem for Maxwell's equations
based on the Laguerre functions**
**

The inverse problem is solved by an optimization method using the
Laguerre functions. Numerical simulations are carried out for the
one-dimensional Maxwell's equations in the wave and diffusion approximations.
Spatial distributions of permittivity and conductivity of the medium are
determined from a known solution at a certain point. The Laguerre harmonics
function is minimized. The minimization is performed by the conjugate gradient
method. Results of determining permittivity and conductivity are presented. The
influence of shape and spectrum of a source of electromagnetic waves on the
accuracy of solution of the inverse problem is investigated. The accuracies of
the solutions with a broadband and a harmonic sources of electromagnetic waves
are compared.

**
Key words: **
numerical algorithm, Maxwell's equations, electromagnetic wave, conductivity,
inverse problem, the Laguerre method, finite difference, linear equations,
accuracy.

Nikolaev V.E., Ivanov G.I., Rozhin I.I.

Numerical modeling of the influence of heat exchange of reservoir beds
with enclosing rocks on gas production from a single well**
(in Russian), pp.
**

In the computational experiment, the influence of heat exchange through
top and bottom of the gas-bearing reservoir on the dynamics of temperature and
pressure fields in the process of real gas production from a single well is
investigated. The experiment was carried out with a modified mathematical model
of non-isothermal gas filtration, obtained from the energy and mass conservation
laws and the Darcy law. The physical and caloric equations of state together
with the Newton-Rihman law of heat exchange of a gas reservoir with surrounding
enclosing rocks are used as closing relations. It is shown that the influence of
the heat exchange with environment on the temperature field of a gas-bearing
reservoir is localized in a narrow zone near its top and bottom, though the size
of this zone increases with time.

**Key words: **
mathematical modeling, non-isothermal filtration, real gas, finite difference
methods.
**
**

**
Okuonghae R.I.**

A class of *A*(α)-stable numerical methods for stiff problems in
ordinary differential equations**
(in Russian), pp.
**

The *A*(α)-stable numerical methods (ANM) for the number of
steps *k *≤ 7 for stiff initial value
problems (IVPs) in ordinary differential equations (ODEs) are proposed. The
discrete schemes proposed from their equivalent continuous schemes are obtained.
The scaled time variable *t* in a
continuous method, which determines the discrete coefficients of the discrete
method is chosen in such a way as to ensure that the discrete scheme attains a
high order and *A*(α)-stability. We select the value of α for which the
schemes proposed are absolutely stable. The new algorithms are found to have a
comparable accuracy with that of the backward differentiation formula (BDF)
discussed in [12] which implements the Ode15s in the Matlab suite.

**
Key words:** stiff
IVPs, continuous LMM, collocation and interpolation approach, boundary locus.

Rozhenko A.I., Shaidorov T.S.

On spline approximation with a reproducing kernel method**
(in Russian), pp.
**

Spline approximation with a reproducing kernel of a semi-Hilbert space is
studied. Conditions are formulated that uniquely identify the natural Hilbert
space by a reproducing kernel, a trend of spline, and the approximation domain.
The construction of *spline with external
drift* is proposed. It allows one to approximate functions having areas of
big gradients or first-kind breaks. The conditional positive definiteness of
some known radial basis functions is proved.

**
Key words:
**
spline, reproducing kernel, trend, radial basis function,
external drift.

**
**

Rybakov K.A.

An approximate solution of the optimal nonlinear filtering problem for
stochastic differential systems by statistical modeling**
(in Russian), pp.
**

An algorithm for solving the optimal nonlinear filtering problem by
statistical modeling is proposed. It is based on reducing the filtration problem
to the analysis of stochastic systems with terminating and branching paths,
using a structure similarity of the Duncan-Mortensen—Zakai equations and the
generalized Fokker-Planck-Kolmogorov equation. The solution of such problem of
analysis can be approximately found by using numerical methods for solving
stochastic differential equations and methods for modeling inhomogeneous Poisson
flows.

**Key words:**

Tarakanov V.I., Lysenkova S.A., Nesterenko M.V.

The precession of a parametric oscillation pendulum with the Cardano
suspension**
(in Russian), pp.
**

The probability of the precession of a pendulum with the Cardano
suspension in conditions of an oscillation point of suspension based on the
mathematical proof is investigated.

**
Key words:** operator,
spectrum, iterative algorithm, parametric oscillation, stability.