**
Number 1, pp. 1-117
Number 2, pp. 119-233
Number 3, pp. 235-344
Number 4, pp. 345-447**

**
Aleksandrov V.M.**

**
Forming an approximating construction for calculation and implementation
of optimal control in real time**
(

**
**

A new
approach to realization of the time-optimal control in real time for linear
systems under control with a constraint is proposed. It is based on dividing the
computer costs into those made in advance of the control process and those
carried out as it proceeds. The preliminary computations do not depend on
certain initial condition and rely on approximation of attainability sets in
different periods of time by a union of hyperplanes. Methods of their
construction and singling out the support hyperplane are given. Methods of
approximate finding and subsequent correction of the normalized vector of the
initial conditions of the conjugate system as well as switching times and
instants of switching of time-optimal control are proposed. Results of modeling
and numerical calculations are presented.

**
Key words:
**
optimal
control, attainability sets, hyperplane, real time, adjoint system, edge point,
first approximation, approximating construction

====================================================================

**
Amelkin V.A.**

**
Enumeration problems solutions for serial sequences with a permanent
difference in adjacent series heights**
(

**
**

In this
paper, the sets of *n*-value serial
sequences are considered. The structure of such series is defined by constraints
on the number of series, the length of series, and the height of series.
The
problem of recalculation, numeration, and generation has been solved for the
sets of ascending, descending, and one-transitive sequences with permanent
differences in the adjacent
series
heights.

**
Key words:
**
series
length, series height, constraints, numerical coding.

=====================================================================

**
Artemiev S.S., Ivanov A.A., Korneev V.D.**

**
Numerical analysis of stochastic oscillators on supercomputers****
**
(

**
**

In this
paper we investigate the numerical analysis problem of stochastic differential
equations
(SDEs)
with oscillating solutions. The dependence of mathematical expectation and
dispersion of the SDE numerical solution on the mesh size of integrating the
generalized Euler method is determined. The results of numerical experiments
with simulation of linear and nonlinear stochastic oscillators on the
supercomputer of the Siberian Supercomputer Center are presented.

**
Key words:
**
stochastic
differential equations, statistical algorithms, parallelization, supercomputer,
cluster, van der Pol equation, phase trajectory, stochastic oscillators.

=====================================================================

**
Dmitriev M.N., Lisitsa V.V.**

**
Application of absorbing boundary conditions M-PML for numerical
simulation of wave
propagation in anisotropic media. Part II: Stability****
**
(

**
**

This paper deals with studies of the detailed properties of absorbing boundary conditions M-PML (Multiaxial Perfectly Matched Layer) that arise when a computational domain is limited. These conditions are stable for any type of anisotropy with a correct choice of a stabilization parameter. In the first part of this paper [3], the authors show a linear dependence of the reflectivity on the stabilization parameter. Based on this study, the problem of finding the optimal stabilizing parameter, which provides stability and minimal reflection has been formulated. In this paper, we provide a necessary stability condition of M-PML, which allows limiting the lower value of the stabilizing parameter. It is shown that this criterion is not sufficient.

**
Key words:
**
anisotropy, reflectionless boundary conditions, perfectly matched layer, elastic
wave equa

=====================================================================

**
Kalgin K.V.**

**
Parallel implementation of asynchronous cellular automata on 32-core
computer**
(

**
**

In this
paper we investigate in what way and how efficiently different parallel
algorithms of asynchronous cellular automata simulation can be mapped onto the
architecture of modern 32-core computer (4×Intel Xeon X7560). As an example, a
model of CO+O=CO_{2} reaction on the surface of palladium particle is
used.

**
Key words:
**
parallel
implementation, cellular automata, parallel algorithm, multicores.

=====================================================================

**
Karchevsky A.L.**

**
Reconstruction of pressure and shear velocities and boundaries of thin
layers in a thinly stratified layer**
(

**
**

In this
paper, a result of reconstruction of velocities of elastic waves and boundaries
of thin layers in a thinly stratified layer is presented. For this purpose, the
method of residual functional minimization was used. Differentiation of the
residual functional with respect to coordinates of gap points of a medium was
proved and the appropriate derivative was obtained.

**
Key words:
**
inverse
problem, pressure velocity, shear velocity, gap point of medium, horizontaly
stratified medium, thinly stratified layer, residual functional, gradient of
residual functional, layer stripping method.

====================================================================

**Leonov A.S.**

**
A posteriori accuracy estimations of solutions of ill-posed inverse
problems and extra-optimal regularizing
algorithms for their solution****
**
(

**
**

A new scheme of a posteriori accuracy estimation for approximate solutions of ill-posed inverse problems is presented along with an algorithm of calculating this estimation. A new notion of extra-optimal regularizing algorithm is introduced as a method for solving ill-posed inverse problems having optimal in order a posteriori accuracy estimation. Sufficient conditions of extra-optimality are formulated and an example of extra-optimal regularizing algorithm is given. The developed theory is illustrated by numerical experiments.

**
Key words:**
ill-posed
problems, regularizing algorithms, a posteriori accuracy estimation,
extra-optimal algorithm.

=====================================================================

**
Tarakanov V.I., Lysenkova S.A.**

**
Iterative algorithm of defining the stability of oscillations equation
with damping**
(

The problem of studying parametric oscillations with damping is reduced to a
spectral problem for a linear bunch of operators in the Hilbert space. Such
spectral problem has an efficient algorithm of its solution. The boundaries of
the first stability domain for different values of the damping factor and a
special form of the periodic function being a part of the equation have been
calculated.

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

**
Asnaashari A., Brossier R., Castellanos C., Dupuy B., Etienne V., Gholami
Y., Hu G., Métivier L., Operto S., Pageot
D., Prieux V., Ribodetti A., Roques A., Virieux J.**

**
Hierarchical Approach to Seismic Full Waveform Inversion**
(

**
**

Full waveform inversion (FWI) of seismic traces recorded at the free surface
allows the reconstruction of the physical parameters structure on the underlying
medium. For such reconstruction, an optimization problem is defined where
synthetic traces, obtained through numerical techniques as finite-difference or
finite-element methods in a given model of the subsurface, should match the
observed traces. The number of data samples is routinely around 1 billion for 2D
problems and 1 trillion for 3D problems, while the number of parameters ranges
from 1 million to 10 million degrees of freedom. Moreover, if one defines the
mismatch as the

standard least-squares norm between values sampled in time/frequency and space,
the misfit function has a significant number of secondary minima related to the
ill-posedness and non-linearity of the inversion problem linked to the so-called
cycle skipping.

Taking into account the size of the problem, we consider a local linearized
method where the gradient is computed using the adjoint formulation of the
seismic wave propagation problem. Starting for an initial model, we consider a
quasi-Newton method which allows us to formulate the reconstruction of various
parameters, such as P and S wave velocities, density, or attenuation factors. A
hierarchical strategy is based on an incremental increase in the data complexity
starting from low-frequency content to high-frequency content, from initial
wavelets to later phases in the data space, from narrow azimuths to wide
azimuths, and from simple observables to more complex ones. Different synthetic
examples of realistic structures illustrate the efficiency of this strategy
based on data manipulation.

This strategy is related to the data space, and has to be inserted into a more
global framework, where we could improve significantly the probability of
convergence to the global minimum. When considering the model space, we may rely
on the construction of the initial model or add constraints, such as smoothness
of the searched model and/or prior information collected by other means. An
alternative strategy concerns building the objective function, and various
possibilities must be considered which may increase the linearity of the
inversion procedure.

**
Key words:**
seismic traces, optimization problem, cycle skipping, quasi-Newton method.

=====================================================================

**
Barucq H., Dupouy St-Guirons A.-G., and Tordeux S.**

**
Non-reflecting boundary condition on ellipsoidal boundary
**(

**
**

Modeling of wave propagation problems using finite element methods usually
requires the truncation of the computation domain around the scatterer of
interest. Absorbing boundary conditions are classically considered in order to
avoid spurious reflections. In this paper, we investigate some properties of the
Dirichlet to Neumann map posed on a spheroidal boundary in the context of the
Helmholtz equation.

**
Key words: **
Helmholtz equation, Boundary value problem for second-order elliptic equation,
Wave propagation, Scattering problems.

=====================================================================

**
Bendali A., Cocquet P.-H., and Tordeux S.**

**
Scattering of a scalar time-harmonic wave by
N small spheres by the method of matched asymptotic expansions
**(

**
**

In this paper, we construct an asymptotic expansion of a time-harmonic wave
scattered by *N* small spheres. This
construction is based on the method of matched asymptotic expansions. Error
estimates give a theoretical background to the approach.

**
Key words:**
Helmholtz equations, matched asymptotic expansions, homogenization.

=====================================================================

**
Bogulskii I.O., Volchkov Yu.M.**

**
Numerical solution of dynamic problems of elastoplastic deformation of
solids
**(

Numerical algorithms for solving two-dimensional dynamic problems of elasticity
theory were developed based upon several local approximations for each of the
required functions. The schemes contain free parameters (constants of
dissipation). The explicit form for formulas of the artificial dissipation of
solutions allows us to control its size and to build effective both explicit and
implicit schemes. As an example, the principle of constructing such schemes is
presented for a plane dynamic problem of elasticity theory. We describe a class
of problems, for which numerical algorithms are constructed using several local
approximations for each of the required functions. Examples of solving applied
problems are given.

**
Key words: **
dynamic problem of elasticity theory, local approximation of required functions,
implicit and explicit finite difference schemes.

=====================================================================

**
Bonnaillie-Noël V., Brancherie D., Dambrine M., and Vial G.**

**
Artificial boundary conditions to compute correctors in linear elasticity
**(

**
**

We present the derivation of a transparent boundary condition of order two to
solve the equations of linear elasticity in a half plane. The resolution of the
boundary value problem leads to a noncoercive variational formulation. We also
present some numerical examples.

**
Key words: **
linear
elasticity equations, transparent boundary conditions.

=====================================================================

**
Burel A., Impériale S., and Joly P.**

**
Solving the homogeneous isotropic linear elastodynamics equations using
potentials and finite elements. The case of the rigid boundary condition
**(

**
**

In this article, elastic wave propagation in a homogeneous isotropic elastic
medium with a rigid boundary is considered. A method based on the decoupling of
pressure and shear waves via the use of scalar potentials is proposed. This
method is adapted to a finite element discretization, which is discussed. A
stable, energy preserving numerical scheme is presented, as well as 2D numerical
results.

**
Key words:**
elastic wave propagation, vector potentials, finite elements, clamped boundary
condition.

====================================================================

**
Vishnevsky D.M., Lisitsa V.V., Tcheverda V.A.**

**
Efficient finite difference multi-scheme approach for simulation of
seismic waves in
anisotropic media
**(

This paper presents an original multi-scheme approach to numerical simulation of
seismic wave propagation in models with anisotropic formations. In order to
simulate wave propagation in anisotropic parts of the model, the Lebedev scheme
is used. This scheme is anisotropy-oriented but highly intense in terms of
computation. In the main part of the model, a highly efficient standard
staggered grids scheme is proposed for use. The two schemes are coupled to
ensure the reflection/transmission coefficients to converge with a prescribed
order. The algorithm presented combines the universality of the Lebedev scheme
and the efficiency of the standard staggered grid scheme.

**
Key words:**
finite difference schemes, differential approach, elastic wave equation,
anisotropy.

=====================================================================

**
Voronin K.V., Laevsky Yu.M.**

**
On splitting schemes in the mixed finite element method
**(

Within the research into some geothermal modes, a 3D heat transfer process was
described by the first order system of differential equations (in terms of
«temperature - heat-flow»). This system was solved by an explicit scheme for the
mixed finite element spatial approximations based on the Raviart-Thomas degrees
of freedom. In this paper, a few algorithms based on the splitting technique for
the vector heat-flow equation are proposed. Some comparison results of accuracy
of the algorithms proposed are presented.

**
Key words: **
heat transfer, mixed formulation, finite element method, splitting scheme.

=====================================================================

**
Demidov G.V., Martynov V.N., and Mikhailenko B.G.**

**
A method of solving evolutionary problems based on the Laguerre
step-by-step transform
**(

In his previous publications, B.G. Mikhailenko proposed a method of solving
dynamic problems of elasticity theory based on the Laguerre transform with
respect to time. In this paper, we offer a modification of the given approach,
which is in that the Laguerre transform is applied to a sequence of finite
temporal intervals. The solution obtained at the end of one temporal interval is
used as initial data for solving the problem at the next temporal interval. When
implementing the approach in question, there arises a necessity of selecting the
four parameters: the scale factor needed for approximating a solution by the
Laguerre functions, the exponential coefficient of the weight function that is
used for finding a solution on a finite temporal interval, the duration of this
interval and the number of projections of the Laguerre transform. The way of
selecting the above parameters for the stability of calculations is proposed.
The effect of the applied method parameters on the accuracy of calculations when
using difference schemes of second and fourth orders of approximation has been
studied. It is shown that the use of such an approach makes possible to obtain a
solution with a high accuracy on large temporal intervals.

**
Key words:**
dynamic problems, Laguerre transformation, step-by-step method, difference
approximation, accuracy, stability.

=====================================================================

**
Jaffré J., Roberts J.E.**

**
Modeling flow in porous media with fractures; discrete fracture models
with matrix-fracture exchange
**(

This article is concerned with a numerical model for flow in a porous medium
containing fractures. The fractures are modeled as (*d*-1)-dimensional surfaces inside the
*d*-dimensional matrix domain, and a mixed finite element method
containing both *d* and (*d*-1)
dimensional elements is used. The method allows for fluid exchange between the
fractures and the matrix. The method is defined for single-phase Darcy flow
throughout the domain and for Forchheimer flow in the fractures. We also
consider the case of two-phase flow in a domain in which the fractures and the
matrix are of different rock type.

**
Key words:**
flow in porous media, fractures, multiscale modeling.

=====================================================================

**
Kabanikhin S.I., Krivorotko O.I.**

**
Singular value decomposition in the source problem
**(

Inverse source problem for the wave equation is considered. The additional
information is measured on different parts of the boundary. The degree of ill-posedness
of the inverse problem is investigated. Numerical algorithm which is based on
the SVD of the discrete inverse problem is constructed and tested.

**Key words:**
inverse source problem, singular value decomposition, degree of ill-posedness.

=====================================================================

**
Calandra H., Gratton S., Lago R., Pinel X., and Vasseur X.**

**
Two-level preconditioned Krylov subspace methods for the solution of
three-dimensional heterogeneous Helmholtz problems in seismics
**(

**
**

In this paper we address the solution of three-dimensional heterogeneous
Helmholtz

problems discretized with compact fourth-order finite difference methods with application to acoustic waveform inversion in geophysics. In this setting, the numerical simulation of wave propagation phenomena requires the approximate solution of possibly very large linear systems of equations. We propose an iterative two-grid method where the coarse grid problem is solved inexactly. A single cycle of this method is used as a variable preconditioner for a flexible Krylov subspace method. Numerical results demonstrate the usefulness of the algorithm on a realistic three-dimensional application. The proposed numerical method allows us to solve wave propagation problems with single or multiple sources even at high frequencies on a reasonable number of cores of a distributed memory cluster.

**
Key words:**
flexible Krylov subspace methods, Helmholtz equation, inexact preconditioning,
inhomogeneous media.

=====================================================================

**
Kalinkin A.A., Laevsky Yu.M.**

**
Iterative solver for systems of linear equations with a sparse stiffness
matrix for clusters
**(

In this paper, a package of programs for solving systems of linear equations
with a sparse matrix for computers with distributed memory is proposed. This
package is based on the iterative algorithm for solving the initial system of
equations with preconditioner constructed using the algebraic domain
decomposition. Such an approach makes possible to multiply by the preconditioner
and a stiffness matrix on cluster. Also, to improve the efficiency of
computation, PARDISO and SparseBlas functionalities from Intel®MKL library are
used on each process. In addition to parallelization among processes, this
package uses OpenMP parallelization on each of these processes as well as
Intel$\textregistered$MKL internal functional parallelization.

**
Key words: **
sparse solver, domain decomposition, parallelization, MPI and OpenMP.

=====================================================================

**
Fatyanov A.G.**

**
A wave method for multiple waves suppression for any complex subsurface
geometries
**(

A wave method for suppression of multiple waves that does not require knowledge
of a depth-velocity medium model has been developed. The method is constructed
so as to completely suppress multiple waves in the case of a layer in a
half-space. This is the principle distinction from the existing methods. In
particular, this leads to the fact that no a priori data about the medium
structure is required and the depth-velocity medium model is considered unknown.
The efficiency of the method for arbitrary 3D plane-layered media is
demonstrated both theoretically and numerically. Examples of the method
application to real media showing a substantial decrease in the multiple wave
amplitudes without distortion of the dynamics of useful reflections are given.

**
Key words:**
mathematical modeling, analytical solution, multiple waves suppression, any
complex subsurface geometries.

**
Andreev A.B., Racheva M.R.**

**
Lower bounds for eigenvalues and postprocessing by an integral type
nonconforming FEM
**
(

**
**

In this paper, we analyze some approximation properties of a nonconforming
piecewise linear finite element with integral degrees of freedom. A
nonconforming finite element method (FEM) is applied to second-order eigenvalue
problems (EVPs). We prove that the eigenvalues computed by means of this element
are smaller than the exact ones if the mesh size is small enough. The case when
an EVP is defined on a nonconvex domain is considered.

A superconvergent rate is established to a second-order elliptic problem by the
introduction of nonstandard interpolated elements based on the integral type
linear element. A simple postprocessing method applied to second-order EVPs is
also proposed and analyzed.

Finally, computational aspects are discussed and numerical examples are
presented.

**
Key words: **eigenvalues,
lower bounds, Crouzeix-Raviart element, postprocessing, superconvergence.

=====================================================================

Voevodin A.F.

The method of conjugate operators for solving boundary value problems for ordinary second order differential equations

In this paper, for a linear boundary value problem, we propose a method that
reduces a solution to a differential problem to a discrete (difference) problem.
Difference equations, which are an exact analog of differential equation, are
constructed by the conjugate operator method. Conjugate equations are solved by
the factorization method.

**
Key words: **
boundary value problem, conjugate operator, difference equations, condition
numbers.

=====================================================================
*
*

**
Gasenko V.G., Demidov G.V., Il'in V.P., Shmakov I.A.**

**
Modeling of wave processes in a vapor-liquid medium
**
(

**
**

Numerical methods for modeling nonlinear wave processes in a vapor-liquid medium for a model two-phase spherical symmetric cell, with an applied pressure jump on its external boundary are considered. The viscosity and compressibility of liquid are neglected as well as the space variation of vapor in the bubble. The problem is described by the heat equations in vapor and liquid, and by the system of ODEs for velocity, pressure and a radius at the bubble boundary. The space discretization of equations is made by an implicit finite-volume scheme on the dynamic adaptive grid with the geometrical refinement near the bubble boundary. The «nonlinear» iterations are implemented at each time step to provide a necessary high accuracy. The results of numerical experiments are presented and discussed for critical thermodynamic parameters of water, for different initial values of the bubble radius and pressure jumps.

**
Key words:**
nonlinear wave oscillation, vapor-liquid cell, implicit scheme, dynamic adaptive
grid, inverse characteristics method, numerical experiments.

=====================================================================

**
Laevsky Yu.M., Kandryukova T.A.**

**
On approximation of discontinuous solutions to the Buckley-Leverett
equation
**
(

In this paper, the Lax-Wendroff and «cabaret» schemes for the Buckley-Leverett
equation are studied. It is shown that these schemes represent unstable
solutions. The choice of an unstable solution depends on the Courant number,
only. The finite element version of the «cabaret» scheme is given equation are
studied. It is shown that these schemes represent unstable solutions. The choice
of an unstable solution depends on the Courant number, only. The finite element
version of the «cabaret» scheme is given.

**
Key words:
**

Buckley-Leverett equation, Lax-Wendroff scheme, «cabaret» scheme, unstable
solutions.

=====================================================================

**
Okuonghae R.I., Ikhile M.N.O.**

**
On the construction of high order $A(\alpha)$-stable hybrid linear
multistep methods
for stiff IVPs in ODEs
**
(

**
**

In this paper, we present a class of *A*(α)-stable
hybrid linear multistep methods for the numerical solution of stiff initial
value problems (IVPs) in ordinary differential equations (ODEs). The method
considered uses a second derivative like the Enright's second derivative linear
multistep methods for stiff IVPs in ODEs.

**
Key words: **
hybrid
methods, continuous methods, collocation, interpolation, boundary locus,
*A*(α)-stability.

=====================================================================

**
Ostapenko V.V.**

**
On compact approximations of divergent differential equations
**
(

The method for construction of compact difference schemes approximating the
divergent differential equations is proposed. The schemes have an arbitrary
order of approximation on a stencil of a common type. It is shown that
construction of such schemes for partial differential equations is based on
spatial compact schemes approximating ordinary differential equations depending
on several independent functions. Necessary and sufficient conditions on factors
of these schemes, at which they have a high order of approximation, are
obtained. Examples of restoration with these schemes of compact difference
schemes for partial differential equations are given. It is shown that such
compact difference schemes have the same order as classical approximation on
smooth solutions and weak approximations on discontinuous solutions.

**
Key words:**
divergent differential equations, compact difference schemes, high order of
approximation.

=====================================================================

**
Perevarukha A.U.**

**
The cyclic and unstable chaotic dynamics in models of two populations of
sturgeon fish
**
(

This paper considers nonlinear effects in the dynamics of biological models. We
describe two dynamic systems, which are elaborated for the simulation of
populations (Russian sturgeon and stellate sturgeon) and are based on the
formalization of the relationship between the spawning stock and recruitment
according to the analysis of observational data. For the numerical study of
differential equations with structurally changing right-hand side, we use the
method of representing models, based on maps of states with conditional
transitions. For the dynamic systems, the presence of the qualitatively
different modes of behavior of trajectories is revealed: stable periodic
oscillations (model sturgeon), unstable chaotic (model stellate), realized in a
limited time interval due to the presence in the phase space of a chaotic subset
not being an attractor.

**
Key words:**
modeling of population dynamics, hybrid automata, chaotic modes.

=====================================================================

**
Savchenko A.O., Savchenko O.Ya.**

**
Calculation of charges screening an external coaxial electric field
on the surface of a conducting axial symmetric body
**
(

A one-dimensional integral equation for finding charges on the surface of a
conducting axially symmetric body is given. For the case of an ellipsoid of
rotation in the electric field with polynomial values on the axis of symmetry an
exact solution is obtained. The axis of symmetry of the body and the axis of the
external field coincide with each other. A numerical algorithm, based on a
combination of the projective method and the method of iterative regularization
for solving the first kind Fredholm equations, is proposed. The projectors were
chosen as B-splines. The charges calculated for the ellipsoid of rotation are
close to analytical ones.

**
Key words:**
charges, electric field, potential, conductor, axially symmetric, screening,
first kind Fredholm equations, B-splines.

=====================================================================

**
Svetov I.E.**

**
Reconstruction of solenoidal part of a three-dimensional vector field by
its ray transforms along straight lines, parallel to the coordinate planes
**
(

The numerical solution of a vector field reconstruction problem is offered. It
is assumed that a field is given in a unit ball. The approximation of the
solenoidal part of the vector field is constructed from ray transforms known
over all the straight lines parallel to one of the coordinate planes. Good
results of reconstruction of solenoidal vector fields by the numerical
simulations are proposed. **
**

**
Key words: **vector
tomography, solenoidal field, approximation, inversion formula, ray transform,
fast Fourier transform.

**
Antonova T.V.**

**
Localization method for lines of discontinuity of approximately defined
in the small function of two variables**
(

**
**

A function of two variables with a line of discontinuity is considered, which
has a
discontinuity of the first kind. It is assumed that outside discontinuity lines
the function to be measured is smooth and has a limited partial derivative.
Instead of the accurate function its approximation in
*L*_{2 }and perturbation level
are known. The problem in question belongs to the class of nonlinear ill-posed
problems, for whose solution it is required to construct regularizing
algorithms. We propose a reduced theoretical approach to solving the problem of
localizing the discontinuity line of the function that is noisy in the space
*L*_{2}. This is done in the
case when conditions of an exact function are imposed «in the small». Methods of
averaging have been constructed, the estimations of localizing the line (in the
small) have been obtained.

**
Key
words:**
ill-posed problems,
localization of singularities, line of discontinuity, regularization

=====================================================================

**
Vabishchevich P.N., Vasil'eva M.V.**

**
Explicit-implicit schemes for convection-diffusion-reaction problems**
(

**
**

The basic models of problems in continuum mechanics are boundary value problems
for the time-dependent convection-diffusion-reaction equations. For their study,
various numerical methods are involved. After applying the finite difference,
finite element or finite volume
approximation in space, we arrive at the Cauchy problem for systems of ordinary
differential equations whose main features are associated with the asymmetry of
the operator and its indefinite. The explicit-implicit approximation time is
conventionally used in constructing

**
Key words: **
convection-diffusion-reaction problems, explicit-implicit scheme, stability of
difference schemes.

=====================================================================

**
Egorshin A.O.**

**
On counter orthogonalization processes**
(

**
**

The deduction the counter orthogonalization equation for homogeneous, i.e.,
generated by isometric operators, vector systems in the Hilbert space is given.
The application of this theory deals, in particular, with solving the Toeplitz
algebraic and integral equations, some problems of the signals estimation,
inverse problems of mathematical modeling, and identification.

**
Key words:
**
homogeneous vectors systems, orthogonal projection, orthogonal projector, the
Grama-Shmidt
orthogonalization, counter orthogonalization.

=====================================================================

**
Moskalensky E.D.**

**
On the evolution of wavefront of a plane wave passing through an area
with heterogeneities**
(

A two-dimensional eikonal equation with the right-hand side tending to unity as
the distance from the origin increases is considered. Formulas describing a
wavefront in such a medium have been obtained.

**
Key words: **
wave propagation, eikonal equation.

=====================================================================

**
Penenko A.V.**

**
Discrete-analytic schemes for solving an inverse coefficient
heatconduction problem in a layered medium with gradient methods****
**
(

**
**

A method for constructing numerical schemes for inverse coefficient inverse
heatconduction problem with boundary measurement data and piecewise-constant
coefficients is considered. A set of numerical schemes for a gradient
optimization algorithm is presented. The method is based on the combined use of
locally-adjoint problems along with approximation methods in the Hilbert spaces.

**
Key words: **
inverse problem, gradient algorithm, numerical schemes, locally-adjoint
problems.

=====================================================================

**
Potapov D.K.**

**
On solutions of the Gol'dshtik problem****
**
(

The Gol'dshtik model for separated flows of incompressible fluid is considered.
A solution of the given two-dimensional problem in mathematical physics for a
finite domain is found with the finite element method. Estimations of the
differential operator are obtained. A result on the number of solutions of the
Gol'dshtik problem is obtained using the variational method.

**
Key words:
**
Gol'dshtik problem, nonlinear differential equation, discontinuous nonlinearity,
finite element method, variational method, estimations of differential operator,
number of solutions.

=====================================================================

**
Savelev L.Ya., Balakin S.V.**

**
A stochastic model of a digit transfer by computing**
(

This paper describes a stochastic model of the digit transfer. The main
characteristics of the transfer process are the number of transfers, a number of
groups of consecutive transfers and a maximum number of consecutive transfers.
Two binary numbers with a digit transfer form a triplet, and a sequence of these
triplets generates a Markov chain. In our model the above-mentioned
characteristics can be described by functionals on trajectories of this chain.
They are: the number of events, the number of runs of these events and a maximum
run length. These characteristics can be efficiently used for estimation of a
computation speed.

**
Key words:
**
summator, summation, digit, transfer, stochastic model, random sequence, Markov
chain, run, functional, expectation, variance.

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Hou T.**

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Error estimates and superconvergence of semidiscrete mixed methods for
optimal control problems governed by hyperbolic equations****
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In this paper, we investigate *L*^{∞}(*L*^{2})
-error estimates and superconvergence of semidiscrete mixed finite element
methods for quadratic optimal control problems governed by linear hyperbolic
equations. The state and the co-state are discretized by order
*k* Raviart--Thomas mixed finite element
spaces and the control is approximated by piecewise polynomials of order
*k* (*k*≥0).
We derive error estimates for both the state and the control approximation.
Moreover, we present superconvergence analysis for mixed finite element
approximation of the optimal control problems.

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Key words:
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a priori error estimates, superconvergence, optimal control problems, hyperbolic
equations, semidiscrete mixed finite element methods.

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