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Numerical Analysis and Applications

2015 year, number 3

Tomography of force-free fields

Alexander Leonidovich Balandin
Institute of Systems Dynamics and Control Theory, Lermontov str. 134, Irkutsk-33, Russia, 664033
Keywords: computerized tomography, spherical harmonics, inverse problems

Abstract >>
In order to investigate the force-free fields it is proposed to use the computerized tomography methods. For the inversion of the ray transformation, the method of multipole fields expansion has been developed. This method is based on the expansion of a vector field and the ray transformation over the special basis of vector-functions. Analytical expressions for the ray transform of the basis vector-functions and the results of computer simulation are given.

Stochastic cellular automata simulation of oscillations and autowaves in reaction-diffusion systems

Olga Leonidovna Bandman, Anastasiyfa Evgenevna Kireeva
Institute of Computational Mathematics and Mathematical Geophysics SB RAS, pr. Lavrentieva, 6, Novosibirsk, 630090
Keywords: stochastic cellular automata, computer simulation, asynchronous cellular automata, parallel calculations, catalytic reactions, autowaves

Abstract >>
In this paper, experience in the conducted investigation of the stochastic cellular automata models of forming stable oscillations and autowaves in active media is generalized. As a result, the concept of stochastic cellular automaton (CA), corresponding to the asynchronous CA with probabilistic transition rules, is formulated. The formal notions of a stochastic CA and a stochastic CA model are given. Properties of the CA models and methods of their synthesis, using a specified set of elementary physical and chemical transformations, are described. The possibility of the autowave and oscillatory processes simulation is shown on an example of the carbon monoxide oxidation reaction on the platinum catalyst with reconstructing its surface structure. The CA-simulation enabled to reveal the range of reaction parameters values, at which stable oscillations of the reagents concentration occur, and to observe autowaves over the platinum surface. Considerable attention has been given to a high efficiency of the stochastic CA parallel implementation, which demands preliminary transformation of the asynchronous mode to the block-synchronous one with validation of its equivalence to the asynchronous mode. The latter is done for the investigated reaction CA model by means of the comparative statistical analysis of the simulation results.

The weight coefficients in the weighted least squares method

Igor Vicheslavovich Bychkov1, Valery lvanovich Zorkaltsev2, Anna Vasilevna Kazazaeva3
1Matrosov Institute for System Dynamics and Control Theory of Siberian Branch of Russian Academy of Sciences, Lermontov St., 134, Irkutsk, 664033, Russia
2Melentiev Energy Systems Institute of Siberian Branch of the Russian Academy of Sciences, Lermontov St., 130, Irkutsk,664033, Russia
3Irkutsk State University, Karl Marx St., 1, Irkutsk, 664003, Russia
Keywords: mathematical model, agreement of parameters, the least squares method, weight coefficients

Abstract >>
We consider the problem of estimating parameters of linear mathematical models. It is proved that due to the choice of weights in the least squares method it is possible to obtain solutions by minimizing any penalty functions of a wide class, including those of the Holder norms. A limitation on a set of solutions resulting from the variation of the weights in the least squares method has been determined. The possibility of the practical use of the established theoretical facts is illustrated by the ecology-mathematical models.

The Lagrange interpolation and the Newton-Cotes formulas for functions with a boundary layer component on piecewise-uniform meshes

Alexander Ivanovich Zadorin
Omsk Branch of Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Pevtsova st., 13, Omsk, Russia, 644099
Keywords: one-variable function, boundary layer, high gradients, Shishkin mesh, Lagrange interpolation, Newton, Cotes formula, error estimate

Abstract >>
The interpolation problem of a one-variable function, which can be considered as a solution of a boundary value problem for an equation with a small parameter ε with a higher derivative is investigated. The application of the Lagrange interpolation for such a function on a uniform grid can result in serious errors. In the case of the Shishkin mesh, ε-uniform error estimates of the Lagrange interpolation are obtained. The Shishkin mesh is modified to increase the interpolation accuracy. The ε-uniform error estimates of the Newton-Cotes formulas on such meshes are obtained. Numerical experiments have been carried out. The results obtained confirm the theoretical estimates.

Stiffly stable second derivative linear multistep methods with two hybrid points

R. I. Okuonghae, M. N. O. Ikhile
Department of Mathematics, University of Benin, P.M.B. 1154, Benin City, Edo state, Nigeria
Keywords: continuous linear multistep methods, stiff problem, stiff stability, boundary locus, hybrid LMM

Abstract >>
This paper presents a family of hybrid linear multistep methods (LMM) with a second derivative term for the numerical solution of stiff initial value problems (IVPs) for ordinary differential equations (ODEs). The methods are stiffly stable for the step number k≤ 7.

An inverse eigenvalue problem for a class of matrices of second and third orders

Evgenii Aleksandrjvich Perepelkin
Polzunov Altai State Technical University, prosp. Lenina, 46, Barnaul
Keywords: eigenvalues, inverse problem, product of matrices

Abstract >>
The method for solving the inverse eigenvalue problem for the product of matrices of second and third orders is proposed. The necessary and sufficient conditions for the existence of the problem solution have been obtained.

A numerical solution of an inverse boundary value problem of heat conduction using the Volterra equations of the first kind

Svetlana V. Solodusha1, Natalia M. Yaparova2
1Melentiev Energy Systems Institute SB RAS, Russia Department of Applied Mathematics, Irkutsk, Russia
2South Ural State University, Chelyabinsk, Russia
Keywords: Volterra integral equations, numerical solution, product integration method

Abstract >>
We consider an inverse boundary value problem of heat conduction. To solve it, we propose a new approach based on the Laplace transform. This approach allows us to confine the original problem to solving the Volterra equations of the first kind. We have developed algorithms of the numerical solution to the resulting integral equations. The algorithms developed are based on the application of the product integration method and the quadrature of middle rectangles. A series of test calculations were performed to test the efficiency of the numerical methods.

Solving the traveling salesman problem using a recurrent neural network

Mikhail Sergeevich Tarkov
A.V. Rzhanov’s Institute of Semiconductor Physics SB RAS, Novosibirsk
Keywords: traveling salesman problem, recurrent neural Hopfield network, 2-opt, CUDA technology, LKH algorithm

Abstract >>
A new algorithm (NWTA-algorithm) for solving the traveling salesman problem (TSP) is proposed. The algorithm is based on the use of the Hopfield recurrent neural network, the WTA method (“Winner takes all”) for the cycle formation, and 2-opt optimization method. A special feature of the algorithm proposed is in the use of the method of partial (prefix) sums to accelerate the solution of the system of the Hopfield network equations. For attaining additional acceleration, the parallelization of the algorithm proposed is performed on GPU with the CUDA technology. Several examples from the TSPLIB library with the number of cities from 51 to 2,392 show that the algorithm proposed finds approximate solutions of the TSP (a relative increase in the length of the route with respect to the optimum is 0.03 ÷ 0.14). With a large number of cities (130 and more) the NWTA-algorithm running duration on the CPU is in 4 ÷ 24 times less than the duration of the heuristic LKH algorithm giving optimal solutions for all TSPLIB examples.