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2016 year, number 1

1.
A singular solution to the problem of minimizing resource consumption

Vladimir Mikhailovich Aleksandrov
Sobolev Institute Mathematics of SB RAS, pr. Acad. Koptyug 4, 630090, Novosibirsk, Russia
Keywords: оптимальное управление, финитное управление, быстродействие, время перевода, расход ресурсов, момент включения управления, момент выключения, итерационный процесс, сопряженная система, фазовая траектория, optimal control, finite control, speed, transfer time, resource consumption, control start moment, control stop moment, iterative process, adjoint system, phase trajectory

Abstract >>
An iterative method of finding a singular solution to the problem of minimizing resource consumption has been developed. This method is based on the information about the finite control structure. A condition for existence of a singular solution is obtained. The limit value for transferring the time between the normal and the singular solutions is found. A relation between the variations of the control switching moments and the variations of the initial conditions of the adjoint system is determined. A system of linear algebraic equations relating the variations of the initial conditions of the adjoint system to the variations of the phase coordinates from a given final state of the system is obtained. The computational algorithm, the modeling results and the numerical calculations are presented.



2.
Numerical models of mosaic homogeneous isotropic random fields and problems of radiative transfer

Andrey Yurevich Ambos
Institute of Computational Mathematics and Mathematical Geophysics SB RAS, pr. Acad. Lavrentieva 6, Novosibirsk, Russia, 630090
Keywords: пуассоновский ансамбль, случайное поле, корреляционная длина, перенос излучения, метод максимального сечения, Poisson ensemble, random field, correlation function, radiative transfer, maximum cross-section method

Abstract >>
The new algorithms of statistical modeling of radiative transfer through different types of stochastic homogeneous isotropic media have been created. To this end a special geometric implementation of «the maximum cross-section method» has been developed. This implementation allows one to take into account the radiation absorption by the exponential multiplier factor. The dependence of a certain class of solution functionals of the radiative transfer equation on the correlation length and the field type is studied theoretically and by means of numerical experiments. The theorem about the convergence of these functionals to the corresponding functionals for an average field with decreasing the correlation length up to zero has been proved.



3.
Analysis of the accuracy of estimates of the first moments of solving SDE with Wiener and Poisson components by Monte Carlo method

Sergey Semenovich Artemiev, Mikhail Aleksandrovich Yakunin
Institute of Computational Mathematics and Mathematical Geophysics SB RAS, pr. Acad. Lavrentieva 6, Novosibirsk, Russia, 630090
Keywords: стохастические дифференциальные уравнения, винеровская и пуассоновская составляющие, метод Монте-Карло, обобщенный метод Эйлера, ансамбль траекторий, шаг интегрирования, оценки моментов, stochastic differential equations, Wiener and Poisson components, Monte Carlo method, generalized Euler method, ensemble of trajectories, integration step, estimates of moments

Abstract >>
In this paper, we investigate the accuracy of estimates of the first moments of a numerical solution to SDE with the Wiener and the Poisson components by the generalized Euler explicit method. The exact expressions for the mathematical expectation and variance of the test SDE solution are obtained. These expressions allow us to investigate the dependence of the accuracy of estimates obtained by Monte Carlo method on the values of SDE parameters, the size of an integration step, and the size of an ensemble of simulated trajectories of the solution. The results of the numerical experiments are presented.



4.
Convergence of the adapting grid method of Bakhvalov's type for singularly perturbed boundary value problems

Igor Anatolevich Blatov, Elena Victorovna Kitaeva
Keywords: сингулярно возмущенная краевая задача, галеркинский проектор, сетка Бахвалова, алгоритмы адаптации, singularly perturbed boundary value problem, Galerkin projection, Bakhvalov's grid, adaptation algorithms

Abstract >>
We consider the Galerkin finite element method for non-self-adjoint boundary value problems on Bakhvalov's grids. Using the Galerkin projections method the convergence of a sequence of computational grids with an unknown boundary of the boundary layer has been proved. Numerical examples are presented.



5.
Numerical modeling of a fluid flow in anisotropic fractured porous media

Petr Nikolaevich Vabishchevich, Alexander Vissarionovich Grigoriev
North-Eastern Federal University, 58 Belinsky str., Yakutsk, 677000
Keywords: модель двойной пористости, анизотропная фильтрация, трещиновато-пористые среды, градиентная функция перетока, double porosity model, anisotropic filtration, fractured porous media, gradient flow function

Abstract >>
A model of double porosity in the case of an anisotropic fractured porous medium is considered (Dmitriev, Maksimov; 2007). The function of the exchange flow between fractures and porous blocks, which depends on the direction of a flow, is investigated. The flow function is based on the difference between pressure gradients. This feature enables one to take into account anisotropic filtering properties in a more general form. The results of the numerical solution of the model two-dimensional problem are presented. The computational algorithm is based on the finite element spatial approximation and the explicit-implicit temporal approximation.



6.
The skew-symmetric iterative method for solving the convection-diffusion-reaction equation with the alternating-sign reaction coefficient

Lev Abramovich Krukier1, Boris L’vovich Krukier1, Yu-Mei Huang2
1Instityte Mathematic, Mechanic and computer science of I.I. Vorovich, Stachki 200/1, k. 2, Rostov on Done, 344090
2Scool mathematic and statistics, Landjou, China, 730000
Keywords: уравнение конвекции-диффузии-реакции, знакопеременный коэффициент реакции, центрально-разностная схема, итерационные методы, convection-diffusion-reaction equation, alternating sign coefficient of reaction, central difference scheme, iterative method

Abstract >>
The iterative product, that is, the triangular skew-symmetric method (PTSM) is used to solve linear algebraic equation systems obtained by approximation of a central-difference scheme of the first boundary value problem of convection-diffusion-reaction and standard grid ordering. Sufficient conditions of a non-negative definiteness of the matrix resulting from this approximation have been obtained for a non-stationary sign of the reaction coefficient. This feature ensures the convergence of a sufficiently wide class of iterative methods, in particular, the PTSM. In the test problems, the compliance of the theory with computational experiments is verified, and comparison of the PTSM and the SSOR is made.



7.
Probability density function of leaky integrate-and-fire model with Lиvy noise and its numerical approximation

P. Singh1, M.K. Kadalbajoo2, K. Sharma3
1Thapar University, Patiala, India
2Indian Institute of Technology, Kanpur, India
3South Asian University, New Delhi, India
Keywords: модель с утечками «интегрировать-и-сработать», уравнение переноса, конечно-объемная аппроксимация, шум Леви, Leaky integrate-and-fire model, transport equation, finite volume approximation, Lиvy noise

Abstract >>
We investigate a numerical analysis of a leaky integrate-and-fire model with Lèvy noise. We consider a neuron model in which the probability density function of a neuron in some potential at any time is modeled by a transport equation. Lèvy noise is included due to jumps by excitatory and inhibitory impulses. Due to these jumps the resulting equation is a transport equation containing two integrals in the right-hand side (jumps). We design, implement, and analyze numerical methods of finite volume type. Some numerical examples are also included.



8.
About an approximate solution to the Fredholm integral equation of the first kind by the residual method

Vitaliy Pavlovich Tanana, Evgeniy Yurevich Vishnyakov, Anna Ivanovna Sidikova
Keywords: регуляризация, метод невязки, модуль непрерывности, оценка погрешности, некорректная задача, regularization, method of residuals, module of continuity, evaluation of inaccuracy, ill-posed problem

Abstract >>
The Tikhonov finite-dimensional approximation was applied to an integral equation of the first kind. This allowed us to use the variation regularization method of choosing the regularization parameter residuals from the principle of reducing the problem to a system of linear algebraic equations. The estimate of accuracy of the approximate solution with allowance for the error of the finite-dimensional problem approximation has been obtained. The use of this approach is illustrated on an example of solving an inverse boundary value problem for the heat conductivity equation.



9.
Theoretical basis of the iterative process of the joint assessment of difficulties in tasks and levels of training students

Igor Semyonovich Shreifel, Ivan Nikolaevich Eliseev
Don State Technical University, Rostov region, the Russian Federation 346500, Shevchenko st., 147, Shakhty, Rostov region, Russia
Keywords: итерационный процесс, итерационная последовательность, трудность задания, уровень подготовки студента, дихотомическая матрица ответов, iterative process, an iterative sequence, difficulty of test questions, level of training students, dichotomous response matrix

Abstract >>
In this paper, we study the iterative process of the joint numerical assessment of levels of training students and difficulties in tasks of diagnostic tools using the dichotomous response matrix A of N x M size, with allowance for the contribution of tasks of different difficulty to the assessments obtained. It is shown that not for any matrix A there exist infinite iterative sequences, and in the case of their existence, they do not always converge. A wide range of sufficient conditions for their convergence have been obtained, which are based on the following: 1) matrix A contains at least three different columns; 2) if one places the columns of A in non-decreasing order of column sums, then for any position of the vertical dividing line between the columns there exists a row, which has, at least, one unity to the left of the line, and at least one zero to the right of the line. It is established that the response matrix A obtained as a result of testing reliability satisfies these two conditions. The properties of such matrices have been studied. In particular, the equivalence of the above conditions of primitiveness of the square matrix B of order M with the entries bij N ell 1 (1- ali ) alj has been proved. Using the matrix analysis we have proved that the primitiveness of the matrix B ensures the convergence of iterative sequences, as well as independence of their limits of the choice of the initial approximation. We have estimated the rate of convergence of these sequences and found their limits.