Siberian Journal of Numerical Mathematics

Volume 7, 2004

Contents

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


Number 1, pp.1-95

Bakushinskii A.B., Kokurin M.Yu.
Continuous methods for stable approximation of solutions to nonlinear equations in the Banach space based on the regularized Newton-Kantarovich scheme
(in Russian), pp.1-12

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

Klimenko O.A.
A method for solution of elasto-electrodynamics problem
(in Russian), pp.13-24

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

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

Larin M.R. 
On a multigrid method for solving partial eigenproblems
(in English), pp.25-42

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

Marchenko M.A.
The program package MONC for distributed computations by Monte Carlo method
(in Russian), pp.43-55

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

Nechepurenko M.I.
On some characteristics of the multigraph arc coherence
(in Russian), pp.57-65

Papers [1-3] present values of the greatest arc coherence λ(p,q) and the smallest number B(p,q) of the power cuts  λ(p,q) for (p,q)-multigraphs. The present paper states the complete corrected proof of the results from [3], which brings about obtaining the asymptotic values of probabilities of coherence of one class of random multigraphs. 
Key words: multi-graph, minimum edge's connectivity, maximum edge's connectivity cuts.

Shurina E.P., Gelber M.A.
On vector finite element method for solution to electromagnetic problems
(in Russian), pp.79-95

The vector finite element method is a comparatively new approach, therefore for this method neither a general theory no computational scheme has been developed. The aim of the present paper is to analyze application of the method to solution of electromagnetic problems. Special vector variational formulations have been constructed depending on the problem. Interpolation properties of different types of elements are investigated both theoretically and experimentally. 
Key words: vector finite element method, modeling of 3D electromagnetic fields.

Smelov V.V.
On efficient approximation of piecewise smooth functions with their presentation by rapidly converging piecewise polynomial series
(in Russian), pp.67-78

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


Number 2, pp.97-185

Kamenskii G.A., Varfolomeyev E.M.
Approximate solution of variational problemsfor the mixed type nonlocal functionals (in English), pp.115-123

There are considered variational problems for the mixed type nonlocal functionals. The application of the Ritz method and the method of least square for the quadratic functionals of the above-mentioned type are investigated.
Key words: calculus of variations, nonlocal functionals, approximate solutions, methods of least squares and the Ritz method.

Kargin B.A., Sabelfeld K.K., Artemiev S.S., Voytishek A.V.
On the anniversary of Gennady Alekseevich Mikhailov (in Russian), pp.97-101

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

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

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

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

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

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

Prigarin S.M., Martin A., Winkler G.
Numerical models of binary random fields on the basis of thresholds of Gaussian functions (in English), pp.165-175

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

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

Rukavishnikov V.A., Rukavishnikova E.I.
On the error estimation of the finite element method for the third boundary value problem with singularity in the space L*{2,ν+γ} (in Russian), pp.177-185

The paper analyzes the finite element method for the third boundary value problem for non-self-conjugate second order elliptic equation with coordinated degeneration of initial data and with strong singularity of solution. The scheme of the finite element method is constructed on the basis of the definition of Rν-generalized solution to the problem, and the finite element space contains singular power functions. It is established that the rate of convergence of an approximate solution to the exact Rν-generalized solution in the norm of the Lebesgue weight space L*{2,ν+γ}(Ω) has second order.
Key words: finite element method, strong singulurity of solution, Rν-generalized solution.

Zadorin A.I., Harina O.V.
Numerical method for a system of linear equations of second order with a small parameter on a semi-infinite interval (in Russian), pp.103-114

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


Number 3, pp.186-282

On the anniversary of Sergey Konstantinovich Godunov
(in Russian), pp. 187-188

Axelsson O., Gololobov S.V. 
Monotonicity and discretization error estimates for convection-diffusion problems.
(in English), pp. 189-202

Monotone operators provide a basis for pointwise bounds of the solution and discretization errors. We apply this technique for convection-diffusion problems, including an anisotropic diffusion term and show that the discretization error has a higher order of accuracy near Dirichlet boundaries or, alternatively, the second order of the global error remains even if we use a lower order of approximation near the Dirichlet boundary.
Key words: Singularly perturbed problem, finite difference method, positive type operator, Shishkin mesh.

Gilyova L.V., Shaidurov V.V. 
Two multigrid iterative algorithms for a discrete analogue of the biharmonic equation.
(in Russian), pp. 213-228

A standard scheme of the finite element method with the use of bicubic elements on a rectangular quasi-uniform grid is considered as applied to the two-dimensional Dirichlet problem for the biharmonic equation in a rectangle. To solve this scheme, two multigrid algorithms are treated on a sequence of embedded rectangular grids: a full multigrid with V-cycle and a simpler cascadic algorithm. The presence of angular points of a rectangle results in deficiency of solution smoothness which complicates substantiation of convergence of the algorithm as compared to a smooth case. At the same time, a number of arithmetical operations remains almost optimal for the cascadic algorithm and optimal for V-cycles.
Key words: biharmonic equation, finite element method, multigrid iterative algorithm, cascadic algorithm, multigrid complexity.

Khudаyarov B.A. 
Numerical solution of nonlinear problems in the filter of viscoelastic shells.
(in Russian), pp. 277-282

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

Moghrabi I.A. 
Symmetric-rank-one multi-step quasi-Newton implicit update algorithms.
(in English), pp. 241-248 

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

Nechepurenko M.I. 
Refinement of convergence conditions of the Chebyshev method.

(in Russian), pp. 249-260 

The iterative Chebyshev method of an approximate solution of equations of the form F(x)=0 in Banach spaces is studied, assuming that F'' satisfies the Lipschitz condition. Accurate (attainable) estimates of the domains of existence and uniqueness of solution, non-refinable conditions of existence and convergence of the Chebyshev method as well as asymptotic estimates of the rate of convergence have been obtained.
Key words: equations in Banach spaces, iterative Chebyshev method, accurate estimates, domains of existence and uniqueness.

Pevnyi A.B. 
Multiresolution analysis in the space \ell2(\Bbb Z) using discrete splines.
(in Russian), pp. 261-275

A non-stationary multiresolution analysis Vk_(k≥0) in the space \ell2(\Bbb z) is performed, the subspaces Vk consisting of discrete splines. In each Vk, there is a function υk such that the system {υk(-l2k): \ l\in\Bbb Z} forms the Riesz base of Vk. A system of wavelets  ψkl (j)= ψk (j-l2k),  l \in \Bbb Z, k=1,2,..., is not generated by shifts and dilations of the unique function. The subspaces Wk=span {ψkl:\ l\in\Bbb Z} form an orthogonal expansion of the space: \ell2(\Bbb Z)=\oplus_{k=1} Wk.

The space Vk is the same as the space of discrete splines {\bmm S}_{p,2k} of order p with a distance between the knots 2k. For every p, a multiresolution analysis is obtained (for p=1 - the Haar multiresolution analysis).
Key words: discrete splines, discrete wavelets, multiresolution analysis.

Voronina T.A. 
Determination of spatial distribution of oscillation sources by remote measurements on a finite set of points.
(in Russian), pp. 203-211 

The paper deals with determination of spatial distribution of oscillation sources by remote measurements on a finite set of points. This problem is assumed to be a problem of reconstruction of the original tsunami waveform from the measurement of the arrived wave on a finite set of coastal receivers. The propagation of the wave is described by linearized shallow-water equations when the depth depends on two variables. The direct problem is approximated by the explicit-implicit finite difference scheme. The ill-posed inverse problem of reconstruction is regularized by means of singular value decomposition, so r-solution is a result of the numerical process. Numerical experiments for the model bottom relief, having some basic morphological features typical of the island arc regions are presented. The quality of the solution obtained is assessed as relative errors (in L2-norm) in restoration of the source function.
Key words: wave propagation, ill-posed inverse problem, regularization, singular value decomposition, r-solution, finite diference scheme, numerical modelling.

Zhukovskii E.L. 
Solution to integral equations with \delta-like kernel.
(in Russian), pp. 229-240 

The subject of consideration is the integral equations with \delta-like kernel which result from the processing of  physical process spectra, in the impulse technique, as well as in the time series analysis. Solution and estimates of the integral equations by the Gaussian or the Legendre least squares methods and by their regularized forms, like the orthogonal projections method, are well known. Here, contrary to the mentioned methods, the analysis focuses on the form of the east-squares method which uses the integral function theory, when the spectrum and length of the function are in the uncertainty principle relation.
Key words: integral equation, uncertainty principle, least squares method, regularization, spectrum.


Number 4, pp.283-376

Bogulskii I.O., Cheverda V.A. 
Time iterative procedure of modeling time-dependent processes in essentially inhomogeneous media.
(in Russian), pp.283-286 

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

Bubyakin A.A., Laevsky Yu.M. 
On one approach to constructing schemes of increased order of accuracy in the finite element method.
(in Russian), pp.287-300 

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

Gorunescu F., Gorunescu M., Gorunescu R.  
A metaheuristic GAs method as a decision support for the choice of cancer treatment.
(in English), pp.301-307 

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

Kotel'nikov E.A. 
Searching for the global maximum of a quadratic function with linear constraints.
(in Russian), pp.327-334 

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

Larin M., Padiy A. 
On the theory of the generalized augmented matrix preconditioning method.
(in English), pp.335-343 

This paper is devoted to an improvement of the theory of the recently proposed generalized augmented matrix preconditioning method*. Namely, we compute a sharp lower bound on the eigenvalues of a preconditioned matrix based on the properties of a projector involved in its definition.
(*
Padiy A., Axelsson O., Polman B. Generalized augmented matrix preconditioning approach and its application to iterative solution of ill-conditioned algebraic systems// SIAM J. Matrix Anal. Appl.- 2000.- N 22.- P.793-818.)
 Key words: linear system, augmented matrix, preconditioning method.

Prigarin S.M., Fedchenko N.V. 
Solution of boundary value problems for linear systems of stochastic differential equations.
(in Russian), pp.345-361 

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

Shary S.P. 
Solving tied interval linear systems.
(in Russian), pp.363-376 

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

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

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