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Siberian Journal of Numerical Mathematics

2013 year, number 1

Theorem of training for a competition algorithm

V.S. Antyufeev
Keywords: theorem of training, probabilistic convergence, artificial neural network

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

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

A.I. Zadorin, S.V. Tikhovskaya
Keywords: second order nonlinear ordinary differential equation, singular perturbation, Newton method, Picard method, Samarskii scheme, Shishkin mesh, uniform convergence, two-grid algorithm

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

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

V.I. Zorkaltsev, S.M. Perzhabinsky
Keywords: interior point method, weighted Euclidean rate, linearization

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

A numerical method for solving inverse thermoacoustic problem

S.I. Kabanikhin, O.I. Krivorotko, M.A. Shishlenin
Keywords: thermoacoustic problem, inverse and ill-posed problems, wave equation, method of simple iteration

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

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

V.I. Kostin, V.V. Lisitsa, G.V. Reshetova, V.A. Tcheverda
Keywords: seismic waves, finite difference techniques, domain decomposition, interpolation, groups of processor elements

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

The cubature formulas on a sphere invariant with respect to a dihedral group of rotations with inversion D6h

A.S. Popov
Keywords: numerical integration, invariant cubature formulas, invariant polynomials, dihedral group of rotations

Abstract >>
An algorithm of searching for the best (in a sense) cubature formulas on a sphere that are invariant with respect to a dihedral group of rotations with inversion D6h has been veloped. This algorithm was applied to find parameters of all the best cubature formulas of this group of symmetry up to the 23rd order of accuracy n . In the course of the study carried out, the exact values of parameters of the corresponding cubature formulas were found for n ≤ 11, and the approximate ones were obtained by the numerical solution of systems of nonlinear algebraic equations by a Newton-type method for the other n. For the first time, the ways of obtaining the best cubature formulas for the sphere were systematically investigated for the case of the group which is not a subgroup of the groups of symmetry of the regular polyhedrons.

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

A.N. Pchelintsev
Keywords: generally-periodical solution, system of ordinary differential equations, Fourier series, almost periodical solution, irrational winding of torus

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

Comparison analysis for improving preconditioned SOR-type iterative method

Najafi H. Saberi, S.A. Edalatpanah
Keywords: preconditioning, comparison theorems, spectral radius, SOR, L-, M-matrix

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

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

V.F. Chistyakov, E.V. Chistyakova
Keywords: differential-algebraic equations, index, least square method, gradient methods

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