

2017 year, number 1
T.A. Averina^{1,2}, K.A. Rybakov^{3}
^{1}Institute of Computational Mathematics and Mathematical Geophysics SB RAS, pr. Acad. Lavrentieva 6, Novosibirsk, 630090, Russia ^{2}Novosibirsk State University, Pirogova st., 2, Novosibirsk, 630090, Russia ^{3}Moscow Aviation Institute, Volokolamskoye sh. 4, A80, GSP3, Moscow, 125993, Russia
Keywords: апостериорная плотность вероятности, ветвящиеся процессы, метод статистических испытаний, оптимальная фильтрация, прогнозирование, стохастическая система, уравнение ДунканаМортенсенаЗакаи, уравнение КолмогороваФеллера, branching processes, conditional density, DuncanMortensenZakai equation, KolmogorovFeller equation, Monte Carlo method, optimal filtering problem, prediction problem, stochastic jumpdiffusion system
Abstract >>
In this paper we discuss the evolution of the new approach to the prediction problem for nonlinear stochastic differential systems with a Poisson component. The proposed approach is based on reducing the prediction problem to the analysis of stochastic jumpdiffusion systems with terminating and branching paths. The solution of the prediction problem can be approximately found by using numerical methods for solving stochastic differential equations and methods for modeling inhomogeneous Poisson flows.

A.E. Galashov^{1}, A.V. Kel'manov^{1,2}
^{1}Novosibirsk State University, Pirogova st., 2, Novosibirsk, 630090, Russia ^{2}Sobolev Institute of Mathematics, Acad. Koptyug avenue, 4, Novosibirsk, 630090, Russia
Keywords: поиск подмножеств, кластерный анализ, евклидово пространство, минимум суммы квадратов расстояний, NPтрудная задача, точный псевдополинимиальный алгоритм, Euclidean space, subsets search, clustering, NPhard problem, exact pseudopolynomialtime algorithm
Abstract >>
We consider a strongly NPhard problem of finding a family of disjoint subsets with given cardinalities in a finite set of points from the Euclidean space. A minimum of the sum over all required subsets of the sum of the squared distances from the elements of these subsets to their geometric centers is used as a search criterion. We have proved that if the coordinates of the input points are integer, and the space dimension and the number of required subsets are fixed (i.e. bounded by some constants), then the problem is a pseudopolynomialtime solvable one.

An.G. Marchuk
Institute of Computational Mathematics and Mathematical Geophysics SB RAS, pr. Acad. Lavrentieva 6, Novosibirsk, 630090, Russia
Keywords: распространение цунами, уравнения мелкой воды, волновой луч, кинематика волнового фронта, tsunami propagation, shallowwater equations, wave ray, wavefront kinematics
Abstract >>
In this paper, the kinematics of the tsunami wave ray and the wavefront above an uneven bottom is studied. The formula to determine the wave height along a ray tube has been obtained. The exact analytical solution for the waveray trajectory above the parabolic bottom topography has been derived. Within the waveray approach this solution gives the possibility to determine the tsunami wave heights in an area with a parabolic bottom relief. The distribution of the waveheight maxima in the area with the parabolic bottom was compared to the one obtained by the numerical computation with a shallowwater model.

H.S. Mahato
University of Georgia, 30602 Athens, USA
Keywords: периодическая среда, двухмасштабная модель, усреднение, численное моделирование, periodic medium, twoscale model, averaging, numerical simulations
Abstract >>
This paper deals with numerical simulations of a system of diffusionreaction equations in the context of a porous medium. We start by giving a microscopic model and then an upscaled version (i.e., homogenized or continuum model) of it from previous works of the author. Since with the help of homogenization we obtain a macroscopic description of a model which is microscopically heterogeneous, via these numerical simulations we show that this macroscopic description approximates the microscopic model, which contains heterogeneities and oscillating terms at the pore scale, such as diffusion coefficients.

R.V. Namm, G.I. Tsoy
Computer Centre of Far Eastern Branch RAS, Kim Yu Chena st., 65, Habarovsk, 680063, Russia
Keywords: упругая задача с трещиной, схема двойственности, модифицированный функционал Лагранжа, функционал чувствительности, соотношение двойственности, слабая полунепрерывность снизу, elastic crack problem, duality scheme, modified Lagrangian functional, sensitivity functional, duality relation, weak lower semicontinuity
Abstract >>
The dual scheme for solving a crack problem in terms of displacements is considered. The dual solution method is based on a modified Lagrangian functional. In addition, the method convergence is investigated under natural assumptions on H^{1}regularity of the crack problem solution. The duality relation for the primal and dual problems has been proposed.

M. Prashanth, S. Motsa
University of KawazuluNatal, Private Bag X01, Scottsville 3209, Pietermaritzburg, South Africa
Keywords: метод Галлея, выпуклое ускорение метода Ньютона, метод продолжения, банахово пространство, условие Липшица, производная Фреше, Halley's method, convex acceleration of Newton's method, continuation method, Banach space, Lipschitz condition, FrГ©chet derivative
Abstract >>
This paper is concerned with the semilocal convergence of a continuation method between two thirdorder iterative methods, namely, Halley's method and the convex acceleration of Newton's method, also known as superHalley's method. This convergence analysis is discussed using a recurrence relations approach. This approach simplifies the analysis and leads to improved results. The convergence is established under the assumption that the second Fréchet derivative satisfies the Lipschitz continuity condition. An existenceuniqueness theorem is given. Also, a closed form of error bounds is derived in terms of a real parameter α ∈ [0,1]. Two numerical examples are worked out to demonstrate the efficiency of our approach. On comparing the existence and uniqueness region and error bounds for the solution obtained by our analysis with those obtained by using majorizing sequences [15], we observed that our analysis gives better results. Further, we observed that for particular values of α our analysis reduces to Halley's method (α = 0) and convex acceleration of Newton's method (α = 1), respectively, with improved results.

E.M. Rudoy, N.A. Kazarinov, V.Yu. Slesarenko
Lavrentyev Institute of Hydrodynamics of SB RAS, Lavrentyev str., 15, Novosibirsk, 630090, Russia
Keywords: двухслойная конструкция, трещина, условие непроникания, вариационное неравенство, метод декомпозиции области, алгоритм Удзавы, twolayer structure, crack, nonpenetration condition, variational inequality, domain decomposition method, Uzawa algorithm
Abstract >>
The equilibrium problem for two elastic bodies pasted together along some curve is considered. There exists a crack on a part of the curve. Nonlinear boundary conditions providing a mutual nonpenetration between crack faces are set. The main objective of the paper is to construct and to approve an algorithm for the numerical solution of the equilibrium problem. The algorithm is based on the two approaches: the domain decomposition method and the Uzawa method. The numerical experiment illustrates the efficiency of the algorithm.

N. Choubey^{1}, J.P. Jaiswal^{2}
^{1}Oriental Institute of Science and Technology, Bhopal, M.P. India462021 ^{2}Maulana Azad National Institute of Technology, Bhopal, M.P. India462051
Keywords: итерационный метод, схема без памяти, схема с памятью, вычислительная эффективность, численный результат, iterative method, without memory scheme, with memory scheme, computational efficiency, numerical result
Abstract >>
The main objective and inspiration in the construction of two and threepoint with memory methods is to attain the ut computational efficiency without any additional function evaluations. At this juncture, we have modified the existing fourth and eighth order without memory methods with optimal order of convergence by means of different approximations of selfaccelerating parameters. The parameters are calculated by a Hermite interpolating polynomial, which accelerates the order of convergence of the without memory methods. In particular, the Rorder convergence of the proposed two and threestep with memory methods is increased from four to five and eight to ten. One more advantage of these methods is that the condition f'(x) ≠ 0 in the neighborhood of the required root, imposed on Newton's method, can be removed. Numerical comparison is also stated to confirm the theoretical results.

B.M. Shumilov
Tomsk State University of Architecture and Building, pl. Solyanaya, 2, Tomsk, 634003, Russia
Keywords: Bсплайны, вейвлеты, неявные соотношения разложения, Bsplines, wavelets, implicit decomposition relations
Abstract >>
This paper deals with the use of a scalar product with derivatives for constructing semiorthogonal splinewavelets. The reduction of supports of such wavelets in comparison with classical semiorthogonal wavelets is shown. For the splines of the 3rd degree, the algorithm of wavelettransformation in the form of the solution to a threediagonal system of the linear equations with strict diagonal prevalence has been obtained. The results of numerical experiments on the calculation of derivatives of a discretely set function are presented.

