Siberian Journal of Numerical Mathematics


Volume 4, 2001

Contents

Number 1, pp.1-110
Number 2, pp.111-199
Number 3, pp.201-303
Number 4, pp.305-402


Number 1, pp.1-110

Amelkin V.A.
Algorithms for exact solving the problems of enumeration, coding, and generation of serial sequences
(in Russian), pp.1-12

Binary and any serial sequences of a specified structure are considered. For some basic types of these sequences, generalized formulas for exact solving enumerative problems without resort to generating functions are obtained. A generalized algorithm of coding and generation of the binary serial sequences with structures determined by limitations on the number of series of unities, on the weight of the sequence, on the lengths of series of unities, and on the lengths of series of zeros is proposed.

Artemiev S.S., Yakunin M.A.
Multidimensional model of the dynamics of stock prices and the problem of constructing the investment portfolio
(in Russian), pp.13-20

We consider the multidimensional model of the dynamics of stock prices in the form of the system of stochastic differential equations. We investigate the estimates of unknown parameters in model on the basis of historical prices. We introduce the characteristics of risk and profit of the investment portfolio, whose calculating by Monte Carlo method enables us to construct the set of permissible portfolios.

Bogan Yu.A.
On Fredholm integral equations in two-dimensional anisotropic theory of elasticity
(in Russian), pp.21-30

In this paper, a simple trick is worked out which immediately leads on to the derivation of the Fredholm integral equations in isotropic and anisotropic theory of elasticity for the first and the second boundary value problems. This trick is based on the circumstance that the system of equations of anisotropic theory of elasticity has simple complex characteristics. This circumstance is crucial, since it leads on to a simple system of linear algebraic equations. This approach can be extended to arbitrary second order elliptic systems with constant coefficients in the plane, when boundary operators contain only zero order or the first order derivatives. It is specific that this approach does not require a knowledge of the fundamental solution.

Znak V.I.
Sinphase-weighted median filter: evaluation of weights and processing of a harmonic signal
(in Russian), pp.31-40

We introduce the notion of the Sinphase-Weighted Median Filter which belongs to a class of weighted percentile filters, and consider specification of them under conditions of processing a harmonic signal. We offer the algorithm of computing the weights of such filter and get the estimation of its response on the sine function. Achieved inferences are confirmed by results of numerical statistical modeling.

Nikol'skii E.V.
Generalized functionally invariant solutions for equations of the elasticity theory
(in Russian), pp.41-50

For the first time, necessary and sufficient conditions of the existence of generalized functionally invariant decisions for the equation of elasticity theory are received. The investigation of these conditions is conducted, and decisions are built in a class of flat and spherical waves for longitudinal and diametrical oscillations.

Smelov V.V.
A local algorithm for smooth approximation of approximate difference and nonsmooth variational solutions
(in Russian), pp.51-60

A local algorithm for smooth approximation is proposed for approximate solutions of the one-dimensional problems which are obtained by difference methods or variational algorithms on the basis of piecewise smooth test functions. The algorithm is aimed at approximate solutions which are obtained with an error cont011.gif (1513 bytes). The algorithm has been primordially parallelized and is utmost easy both in theoretical and in practical aspects. A result of the local approximation is a twice continuous differentiable function which keeps geometric properties of the initial approximate solution. Certain advantages of the proposed algorithm as compared to the cubic splines are shown.

Sorokin S.B.
The substantiation of the method of two-sided approximations for eigenvalues of the second-order elliptic operator
(in Russian), pp.61-84

In the paper, the method of two-sided approximation for the spectral problems with the second-order elliptic operator based on the well-known fictitious method is substantiated. The expansion of eigenvalues of auxiliary problem in power series by small parameter of continuation irrespective of its sign is proved. Unimprovable error bounds of two-sided approximations are obtained. The conjugate-factorized structure of the problem operator plays the decisive role in obtaining the result. The research is realized on discrete level.

Shishkin G.I.
A decomposition method for singularly perturbed parabolic convection-diffusion equations with discontinuous initial conditions
(in Russian), pp.85-106

Grid approximations of the Cauchy one-dimensional problem for singularly perturbed parabolic equations are considered. The limit equation (with e =0, where e is the perturbation parameter multiplying the highest derivative) contains the derivative with respect to the spatial variable (convective term). The initial condition has a discontinuity of the first kind. The solution of this problem has singularities in a neighbourhood of the discontinuity for fixed values of the parameter e , and also a transient layer for small values of e. We construct special finite difference schemes which e uniformly converge on the whole domain. For this we use the domain and solution decomposition technique. The singular solution, generated by the discontinuity of the initial function, is split off and represented in the explicit form in the nearest neighbourhood of the discontinuity. In the neighbourhood of the transient layer, we employ the meshes condensing by a special way.

Yu.M.Laevsky, P.V.Banushkina
On a stability of two-level explicit schemes 
(Notes, in Russian), pp.107-109


Number 2, pp.111-199

Voytishek A.V., Uhinov S.A.
Use of the important sample in Monte Carlo method
(in Russian),
pp.111-122

The problem of the calculation of the normalize constant of density using the given sample is considered. The analog of the importance of sampling algorithm with the important (in the sense of minimization of variance of the standard Monte Carlo method) stochastic values is constructed and investigated. It is shown that in the case, when an integral is calculated by the Monte Carlo method and the important sample is not given beforehand, it is better to use not exactly important but close to important sample values. For this case, the new algorithm (called the double-sided geometric method) is proposed. This algorithm allows to reduce the cost of calculations.

Gheit V.E.
On the polynomials, the least deviating from zero in L[-1,1] metrics (Second part)
(in Russian), pp.123-136

The properties of maximum polynomials, the least deviating from zero in integral metric whose forms are calculated in the first part [5] are studied. Theorem 2.2 contains the classification of polynomials, the least deviating from zero in L[-1,1] metrics with given four leading coefficients.

Drobyshevich V.I., Katkova L.N.
Crank-Nicolson's scheme with different time-step in subdomains for the solution of parabolic problems (in Russian), pp.137-150

A method of constructing difference schemes with different time-step in subdomains based on Crank-Nicolson's scheme is suggested. It is associated with interpolation of the solution on the boundary of subdomains. It is proved that the solution of the difference problem converges in uniform norm to that of differential problem with the order O(\tau^2+h^2).

Dulov E.V., Andrianova N.A.
On a numerical solution of matrix polynomial equations
(in Russian),
pp.151-162

In this paper, we propose some direct and iterative algorithms for the solution of matrix polynomial equations of the form AX+AX  ^2+... +AX^n=C. A local convergence theorem of iterative algorithms is given, and the restrictions involved by this theorem are discussed. We give an estimation of convergence speed for these methods and make a number of useful notes for it's effective numerical implementation. Explicitly we discuss a special case, arising in problems of parameter estimation of linear dynamic stochastic systems.

Zolotukhin A.Ya., Namm R.V., Pachina A.V.
An approximate solution of the Mosolov and the Miasnikov variational problem with the Coulomb boundary friction
(in Russian), pp.163-177

The paper is devoted to the construction and the proof of stability method for the solution of the Mosolov and the Miasnikov problem with boundary friction based on the combination of iterative prox-regularization method and the modified Newton method with step regulation.

Savchenko A.O.
The optimal quadratures for numerical solving of integral Volterra equations and the Cauchy problem
(in Russian), pp.179-184

The equations for optimal subgrid points and quadrature coefficients in the problems of numerical solution of integral Volterra equations and the Cauchy problem are obtained. The optimality is regarded as the minimum of the sum of squares of approximation errors in all subgrid points under condition that the last subgrid point is equal to the right end of the subgrid. The optimal points are numerically found for various numbers of subgrid points.

Strekalovsky A.S., Kuznetsova A.A., Yakovleva T.V.
On non-convex quadratic optimization
(in Russian), pp.185-199

This paper considers the search for the global minimization of non-convex functions, in particular, quadratic functions with non-definite matrix on a parallelepiped. The global search strategy is based on the global optimality conditions connected with the classical extremum theory and is in a non-trivial combination of linearized over basic non-convexity problems, local descent problems, problems of approximation of the convex functions level surfaces, and the one-dimensional search. Various numerical calculations have been carried out, to verify the algorithm effectivity.


Number 3, pp. 201-303

Bespalov A.Yu., Rukavishnikov V.A.
The use of singular functions in the h-p version of the finite element method for a Dirichlet problem with degeneration of the input data

(in Russian), pp.201-228

The paper is devoted to a Dirichlet problem for a second-order non-self-adjoint elliptic equation with a strong singularity of the solution caused by a coordinated degeneration of input data at boundary points of a two-dimensional domain. The h-p version of the finite element method is used to approximate this problem. We introduce a finite element space with a singular basis that depends on the space to which the solution to the problem belongs. An exponential convergence rate in the norm of a weighted Sobolev space is proved.

Blatov I.A., Kitaeva E.V.
An incomplete factorization method with the fast Fourier transform for discrete Poisson equations with different boundary conditions
(in Russian), pp.229-242

For the discrete Laplacian on a rectangular grid with Dirichlet and Dirichlet-Neumann boundary conditions, a spectral equivalent preconditioner of incomplete block-factorization type is constructed. The inversion of this preconditioner is realized with the help of the fast Fourier transform with O(N ln N) arithmetical operations.

Iskakov K.T.
An optimization approach to solving a discrete inverse problem for a one-dimensional hyperbolic equation (in Russian), pp.243-258

In this paper, an optimization approach to solving a discrete problem of determining the coefficient of a one-dimensional hyperbolic equation in integral formulation is studied. The properties of the solutions to the inverse and direct discrete problems are investigated. Estimates both for the cost function and its gradient are obtained. Also, the convergence of the steepest descent method minimizing the misfit functional is proved.

Karchevsky A.L., Fatianov A.G.
Numerical solution of the inverse problem for a system of elasticity with the aftereffect for a vertically inhomogeneous medium (in Russian),
pp.259-268

In this paper, the results of numerical solution of a 1-D inverse problem for a system of elasticity with the aftereffect for a vertically inhomogeneous medium are presented. The Boltzmann model is chosen to take into account the aftereffect. The purpose of the inverse problem is to reconstruct the longitudinal and transverse velocities by using the known longitudinal and transverse displacements at the surface. A series of model calculations is carried out, and the results of reconstructions of a velocity model of the medium typical for Western Siberia are presented.

Leonov A.S.
Numerical implementation of special regularizing algorithms for solving a class of ill-posed problems with sourcewise represented solutions
(in Russian), pp.269-177

An important class of ill-posed problems, namely, inverse problems of continuation for the values of an abstract function, is under consideration. The class is associated with certain semigroups of operators. A priori, the problems of this class have sourcewise represented solutions with known powers. For solving these problems we apply special Tikhonov's regularizing algorithms of the generalized residual principle. The approximate solutions obtained by the use of the algorithms and modifications proposed are of optimal order of accuracy without regard to the positive sourcewise representation power. The algorithm is numerically implemented in an efficient way. An example of application of the algorithm is given.

Popov A.S.
New cubature formulae invariant under the octahedral group of rotations for a sphere (in Russian), pp.281-284

In this paper, new cubature formulae of Gaussian type invariant under the octahedral group of rotations for a sphere are constructed. The parameters of the cubature formulae of the 12th, 13th, 14th, 16th, 18th, 20th, 21st, 22nd, 24th, and 25th orders of accuracy are given to 16 significant digits.

Rozhenko A.I.
On the construction of a normal pseudo-solution for a system of linear equations with a rectangular matrix (in Russian), pp.285-294

A modified QR factorization algorithm is proposed. It allows us to construct a normal pseudo-solution of a System of Linear Algebraic Equations (SLAE) for a rectangular or square degenerate matrix with the same efficiency as for SLAE with a square nonsingular matrix. As an application, the construction of an analytic spline on a degenerate mesh is studied, and a modified algorithm is proposed to provide the "best" spline solution.

Rysbaiuly B.R.
A finite difference method for a viscous one-dimensional conducting compressible gas with a contact discontinuity (in Russian), pp.295-303

In this paper, a priori estimates for the solution of one class of the difference scheme for a viscous one-dimensional conducting gas with a contact discontinuity are obtained. The stability and convergence of the difference problem are demonstrated by using the a priori estimates obtained. The Newton method for the system of nonlinear equations under study is justified.


Number 4, pp.305-402

OBITUARY Vladimir Vasilenko (in Russian), pp.305-306

OBITUARY Vladimir Vasilenko, 1947-2001 (in English), pp.307-308

Antyufeev V.S.
On identification of optical parameters of a plane slab (in Russian),
pp.309-316

A new method for the identification of the phase function and optical thickness of a homogeneous plane slab by using the data of measurement of angular distribution of the intensity of transmitted radiation is proposed. The equation of radiation transport is regarded as an evolution equation that specifies a semigroup of the operator. The generating operator of this semigroup is restored by using the transmission operator obtained from measurements. This operator contains the entire information about the optical parameters of the slab. The desired optical parameters can be found by calculating the generating operator.

Bakushinskii A.B., Kokurin M.Yu., Yusupova N.A.
On Iterative Methods of Gradient Type for Solving Nonlinear Ill-Posed Equations (in Russian), pp.317-329

Iterative methods of gradient type for the approximate solution of noisy nonlinear equations without the property of regularity are proposed and investigated. We prove the convergence of the approximations generated by the methods to a neighborhood of the solution with a diameter proportional to the magnitude of errors in the input data and in a sourcewise representation of the initial residual. This is ensured by a suitable combination of the method of gradient descent for a residual functional with approximate projecting onto special finite-dimensional subspaces.

Globisch G.
Multigrid methods for interface problems (in English), pp.331-352

We analyze multigrid convergence when 2D-elliptic boundary value problems with interfaces are discretized using finite element methods where coarse meshes do not approximate the interface geometry. Starting with the initial mesh, the used interface adapted mesh generator constructs a finite element mesh up to a certain refinement level where the interface lines are approximated with a sufficient precision. It is shown that multigrid cycles based on SOR-smoothing and specific interpolation and restriction operators converge independently of the meshsize parameter. Moreover, in practice the convergence is also independent of the ratio of the jumping coefficients. We demonstrate the efficiency of our method by means of numerical examples.

Godunov S.K.
Partition of the spectrum by Hermite forms and one-dimensional spectral matrix portraits (in English), pp.353-360

There exist classes of (in general) nonselfadjoint matrix operators whose eigenvalues of a spectral cluster are ill-conditioned. In applications, it is convenient to describe properties of such operators in terms of some criteria for spectral dichotomy. It is convenient to divide the spectrum by a series of plane curves depending on a single parameter. The graphical dependence of a criterion for dichotomy on this parameter is naturally regarded as spectral portrait. Criteria for dichotomy are connected with Hermite forms. (Recall that Hermite forms appeared in 1856 in solving a similar problem studied by Hermite).

Kuznetsov Yu.I.
Cyclical matrices and the Chebyshev polynomials (in Russian), pp.361-371

A sequence of polynomials generated by a cyclical Jacobi matrix is treated as a function of an integer argument, which is the order of the polynomials. Formulas for the sum and difference of the functions and their arguments that generalize similar formulas for trigonometrical functions are constructed. An expression for such sequences of polynomials with the use of Chebyshev polynomials of the second kind is obtained. A divisibility formula for Chebyshev polynomials of the second kind is obtained. A solution of the inverse problem for Chebyshev polynomials, i.e. a description of the corresponding functions of integer arguments by using their properties is presented.

Makhotkin O.A.
An effective algorithm for the simulation of 1D finite probability densities by using the two-side rejection method (in Russian), pp.373-388

An effective general algorithm for the simulation of one-dimensional probability densities defined on a finite interval is presented. The two-side rejection algorithm has been constructed. It uses piecewise linear interval approximations of structurally simple functions. The algorithm has been investigated in detail for the beta-distribution with non-negative parameters. A modification of the algorithm for infinite densities and infinite intervals is proposed.

Simonov N.A.
Monte Carlo solution of a parabolic equation with a random coefficient
(in English), pp.389-402

A parabolic equation is considered. Its coefficient in the linear term, the right-hand side, and the initial value of the equation are random functions. Some Monte Carlo estimators for sample values of its solution and for some functionals are constructed and verified.