Siberian Journal of Numerical Mathematics

Volume 15, 2012

Contents

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


Number 1, pp. 1-117 

 

Aleksandrov V.M.

Forming an approximating construction for calculation and implementation of optimal control in real time (in Russian), pp. 1-19

 

         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.

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Amelkin V.A. 

Enumeration problems solutions for serial sequences with a permanent difference in adjacent series heights (in Russian), pp. 21-29 

 

          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.

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

Numerical analysis of stochastic oscillators on supercomputers (in Russian), pp. 31-43

 

          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.

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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 (in Russian), pp. 45-54

 

          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 equations.

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Kalgin K.V.  

Parallel implementation of asynchronous cellular automata on 32-core computer (in Russian), pp. 55-65

 

          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=CO2 reaction on the surface of palladium particle is used. 

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

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Karchevsky A.L.  

Reconstruction of pressure and shear velocities and boundaries of thin layers in a thinly stratified layer (in Russian), pp. 67-82 

 

            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.

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 Leonov A.S. 

A posteriori accuracy estimations of solutions of ill-posed inverse problems and extra-optimal regularizing algorithms for their solution (in Russian), pp. 83-100  

 

         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.

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Tarakanov V.I., Lysenkova S.A.

Iterative algorithm of defining the stability of oscillations equation with damping (in Russian), pp. 101-117

 

           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.


Number 2, pp. 119-233

 

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 (in Russian), pp. 119-130 

 

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.

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

Non-reflecting boundary condition on ellipsoidal boundary (in Russian), pp. 131-139

 

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.

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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 Russian), pp. 141-149

 

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.

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Bogulskii I.O., Volchkov Yu.M.

Numerical solution of dynamic problems of elastoplastic deformation of solids (in Russian), pp. 151-156

 

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.

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

Artificial boundary conditions to compute correctors in linear elasticity (in Russian), pp. 157-164 

 

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.

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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 Russian), pp. 165-174

 

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.

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

Efficient finite difference multi-scheme approach for simulation of seismic waves in anisotropic media (in Russian), pp. 175-181 

 

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.

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Voronin K.V., Laevsky Yu.M. 

On splitting schemes in the mixed finite element method (in Russian), pp. 183-189

 

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.

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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 Russian), pp. 191-196

 

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.

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Jaffré J., Roberts J.E.

Modeling flow in porous media with fractures; discrete fracture models with matrix-fracture exchange (in Russian), pp. 197-204

 

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.

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Kabanikhin S.I., Krivorotko O.I. 

Singular value decomposition in the source problem (in Russian), pp. 205-211

 

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.

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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 Russian), pp. 213-221

 

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.

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Kalinkin A.A., Laevsky Yu.M. 

Iterative solver for systems of linear equations with a sparse stiffness matrix for clusters (in Russian), pp. 223-228

  

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.

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Fatyanov A.G.

A wave method for multiple waves suppression for any complex subsurface geometries (in Russian), pp. 229-233

  

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.


Number 3, pp. 235-344

 

Andreev A.B., Racheva M.R.

Lower bounds for eigenvalues and postprocessing by an integral type nonconforming FEM (in Russian), pp. 235-249

 

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.
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Voevodin A.F.
The method of conjugate operators for solving boundary value problems for ordinary second order differential equations
(in Russian), pp. 251-260 

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.

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

Modeling of wave processes in a vapor-liquid medium (in Russian), pp. 261-270 

 

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.

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Laevsky Yu.M., Kandryukova T.A.

On approximation of discontinuous solutions to the Buckley-Leverett equation (in Russian), pp. 271-280 

 

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.

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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 Russian), pp. 281-292

 

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.

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Ostapenko V.V. 

On compact approximations of divergent differential equations (in Russian), pp. 293-306 

 

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.

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Perevarukha A.U.

The cyclic and unstable chaotic dynamics in models of two populations of sturgeon fish (in Russian), pp. 307-320 

 

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.

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Savchenko A.O., Savchenko O.Ya.

Calculation of charges screening an external coaxial electric field on the surface of a conducting axial symmetric body (in Russian), pp. 321-327 

 

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.

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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 (in Russian), pp. 329-344 

 

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.


Number 4, pp. 345-447

 

Antonova T.V.

Localization method for lines of discontinuity of approximately defined in the small function of two variables (in Russian), pp. 345-357

 

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 L2 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 L2. 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

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

Explicit-implicit schemes for convection-diffusion-reaction problems (in Russian), pp. 359-369

 

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 splitting schemes in terms of physical processes, when separated by convection and diffusion transfers, the reaction process. In this paper, unconditionally stable schemes for unsteady convection-diffusion-reaction equations are used, when explicit-implicit approximations are applied in splitting the operator reaction. An example of a model 2D problem in the rectangle is presented.

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

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Egorshin A.O.

On counter orthogonalization processes (in Russian), pp. 371-385

 

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.

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Moskalensky E.D.

On the evolution of wavefront of a plane wave passing through an area with heterogeneities (in Russian), pp. 387-392

 

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.

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Penenko A.V.

Discrete-analytic schemes for solving an inverse coefficient heatconduction problem in a layered medium with gradient methods (in Russian), pp. 393-408

 

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.

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Potapov D.K.

On solutions of the Gol'dshtik problem (in Russian), pp. 409-415

 

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.

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Savelev L.Ya., Balakin S.V.

A stochastic model of a digit transfer by computing (in Russian), pp. 417-423

 

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.

Error estimates and superconvergence of semidiscrete mixed methods for optimal control problems governed by hyperbolic equations (in Russian), pp. 425-440

 

In this paper, we investigate L(L2) -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.

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