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2014 year, number 4

1.
Cellular automata with a dynamic structure for simulating the biological tissues growth

A. A. Vitvitsky
Institute of Computational Mathematics and Mathematical Geophysics SB RAS, pr. Lavrentieva, 6, Novosibirsk, Russia, 630090
Keywords: computer simulation, cellular automata, cellular automata with dynamic structure, orphogenesis, apical meristem escape, Arabidopsis Thaliana

Abstract >>
The concept of cellular automata with the dynamic structure of a cellular space (DCA) is proposed. The DCA extends the capabilities of classical cellular automata (CA), and allows using the cellular automata approach to the problems of simulating the biological tissues growth. A DCA differs from a classical CA in that its cells are not located on a regular lattice, and intercellular connections are explicitly described by the neighborhood matrix. In addition, insertion and partition operators are introduced for the DCA. These operators allow one to dynamically change the cell space-structure. Based on this extension, the DCA-model of the apical meristem escape of Arabidopsis Thaliana growth is constructed, being a parallel composition of the two DCA: the asynchronous two-dimensional DCA simulating self-regulation in biological cells, and the synchronous one-dimensional DCA simulating growth and division of biological cells. The results of computer simulations have shown that the behavior of the proposed DCA-model matches the behavior of the existing model based on the composition of differential equations and the method of L-system (Lindenmayer system). Furthermore, the proposed DCA-model allows one to simulate growth of individual biological cells and to visualize the substances dynamics in these cells (decay, synthesis and diffusion).



2.
Minimization of a quadratic function on the sphere

E. A. Kotel'nikov
Institute of Computational Mathematics and Mathematical Geophysics SB RAS, pr. Lavrentieva, 6, Novosibirsk, Russia, 630090
Keywords: quadratic optimization on sphere, Cholesky decomposition, trust region, step trajectory, quadratic model

Abstract >>
In this paper, a sequential algorithm for solving the problem of minimization of a quadratic function on a sphere is proposed. At each iteration of the scheme, a two-dimensional problem of minimization is solved. Numerical comparisons with other methods are presented.



3.
Which of inverse problems can have a priori approximate solution accuracy estimates comparable in order with the data accuracy

A. S. Leonov
National Research Nuclear University MEPhI (Moscow Engineering Physics Institute), 115409, Moscow, Kashirskoe shosse, 31
Keywords: linear inverse problems, correctness in the sense of Tikhonov, a priori and a posteriori accuracy estimate

Abstract >>
It is proved that a priori global accuracy estimate for approximate solutions to linear inverse problems with perturbed data can be of the same order as approximate data errors for well-posed in the sense of Tikhonov problems only. A method for assessing the quality of selected sets of correctness is proposed. The use of the generalized residual method on a set of correctness allows us to solve the inverse problem and to obtain a posteriori accuracy estimate for approximate solutions, which is comparable with the accuracy of the problem data. The approach proposed is illustrated by a numerical example.



4.
Numerical stochastic models of the sea surface undulation and extreme ocean waves

K. V. Litvenko1,2, S. M. Prigarin1,2
1Institute of Computational Mathematics and Mathematical Geophysics SB RAS, pr. Lavrentieva, 6, Novosibirsk, Russia, 630090
2Novosibirsk State University, Pirogova 2, Novosibirsk, Russia, 630090
Keywords: simulation of random fields, conditional spectral models, time series, sea surface undulation, extreme ocean waves (freak-waves, rogue waves)

Abstract >>
The paper deals with simulation of the time-space stochastic structure of the sea surface undulation and extreme ocean waves. Numerical algorithms are constructed on the basis of conditional spectral models and models of time series adapting data of observations. Estimates for frequencies of extreme waves appearance are studied on the basis of the theory of random processes and fields.



5.
On finding exact solutions of the two-dimensional eikonal equation when the front of the wave propagating in a medium is a circle

E. D. Moskalensky
Institute of Computational Mathematics and Mathematical Geophysics SB RAS, pr. Lavrentieva, 6, Novosibirsk, Russia, 630090
Keywords: wave propagation, front of wave, eikonal equation

Abstract >>
Wave propagation in a two-dimensional medium is considered in the case when the front of the wave is a circle with the center (a (t),0) and radius r(t). A question is posed: what is the distribution of velocity in the medium? Common characteristics and examples of such media are given.



6.
L(α)–stable variable order implicit second derivative Runge Kutta methods

R. I. Okuonghae, M. N. O. Ikhile
University of Benin, P.M.B 1154, Benin City, Edo state, Nigeria
Keywords: second derivative, Runge Kutta method, collocation, interpolation

Abstract >>
This paper considers the extension of the popular Runge Kutta methods (RKMs) to second derivative Runge Kutta methods (SDRKMs) for the direct solution of stiff initial value problems (IVPs) of ordinary differential equations (ODEs). The methods are based on using collocation and interpolation techniques. The last stage of the input approximation is identical to the output method. The SDRKMs are L(α)-stable for the methods examined. Numerical experiments are given comparing one of these methods with a two derivative Runge Kutta method (TDRKM) and a second derivative linear multistep method (SDLMM).



7.
A new three-sample non-parametric statistical test with its special case equivalent to the Whitney test

G. I. Salov
Institute of Computational Mathematics and Mathematical Geophysics SB RAS, pr. Lavrentieva, 6, Novosibirsk, Russia, 630090
Keywords: three samples, homogeneity test, nonparametric statistical test

Abstract >>
In this paper, we propose a new non-parametric statistical test for the problem of homogeneity of three samples. We consider an alternative for which one sample values tend to be stochastically larger than every one from the two other samples values. The Whitney test is equivalent to special (linear) case of this test. Some comparisons are made for the cases with samples from exponential and uniform distributions.



8.
A network version of the non-standard trigonometric basis and its advantages with respect to a similar polynomial basis

V. V. Smelov
Institute of Computational Mathematics and Mathematical Geophysics SB RAS, pr. Lavrentieva, 6, Novosibirsk, Russia, 630090
Keywords: functional basis, elliptic operator, energy scalar product, functional, generalized solution, conjugation condition

Abstract >>
A trigonometry-based functional basis as a network version is proposed. It is aimed at the approximation with high orders of accuracy of smooth and piecewise-smooth functions. A comparative analysis of the features of the basis proposed and a polynomial one is made. The trigonometric version offers considerable advantages over the polynomial bases.



9.
Iterative scheme of finding a spectrum of the product of two non-commutative operators

V. I. Tarakanov, S. A. Lysenkova, M. V. Nesterenko
Surgut State University of KHMAO, Lenien Ave. 1, Surgut, 628400
Keywords: operator, spectrum, iterative algorithm

Abstract >>
We consider spectral features and an iterative scheme of finding a spectrum of the product of two non-commutative partially symmetric operators in the Hilbert space H. In this case it is assumed that one of operators is compact, the second not necessarily being compact and even restricted in H. Numerical implementation of the iterative scheme for finding the operator spectrum of the problem of eigen-oscillations of the Rayleigh beam is presented.