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2015 year, number 1

1.
Computing of optimal inertial control with a linear system

Vladimir Mikhailovich Aleksandrov
Sobolev Institute Mathematics of SB RAS, 4 Acad. Koptyug avenue, 630090, Novosibirsk, Russia
Keywords: optimal control, speed, switching moment, inertial switching, non-inertial switching, phase trajectory

Abstract >>
Computing of time-optimal inertial control amounts to solving the three problems: 1) computing of optimal control on the assumption that the control is without inertia; 2) finding the optimal switching time of the control; 3) calculating of the error induced by the time lag of the control followed by correcting the control time and switching moments. Characteristics of the problems are considered and methods of their solution are given. A way of assignment of initial approximation is presented. A computational algorithm, results of modeling and numerical computations are performed.



2.
New frequency characteristics of the numerical solution to stochastic differential equations

Sergey Semenovich Artemiev1,2, Aleksandr Aleksandrovich Ivanov1, Dmitriy Dmitrievich Smirnov2
1Institute of Computational Mathematics and Mathematical Geophysics SB RAS, pr. Lavrentieva, 6, Novosibirsk, Russia, 630090
2Novosibirsk State University, Pirogova 2, Novosibirsk, Russia, 630090
Keywords: stochastic differential equations, cumulative frequency curve, frequency phase portrait, generalized Euler's method

Abstract >>
The problems of the numerical analysis of stochastic differential equations with oscillatory solutions trajectories are studied. For the analysis of the numerical solution it is proposed to use the frequency response of generalizing the integral curve and the phase portrait. The results of numerical experiments carried out on a cluster of NCC-30T Siberian Supercomputer Center at the ICM\&MG SB RAS using a set of programs PARMONC are presented.



3.
Simulation of the response of the Hadley cell and extratropical troposphere stratification on the climate changes using a general atmosphere circulation of medium complexity

IrinaVladimirovna Borovko1, Vladimir Nikolaevich Krupchatnikov2,3
1Institute of Computational Mathematics and Mathematical Geophysics SB RAS, pr. Lavrentieva, 6, Novosibirsk, Russia, 630090
2Siberian Regional Hydrometeorological Research Institute, str. Sovetskaya, 30, Novosibirsk, Russia, 630099
3Institute of Monitoring of Climatic and Ecological Systems of the Siberian Branch of the Russian Academy of Sciences, Academichesky ave., 10/3, Tomsk, 634055
Keywords: Hadley cell, atmosphere stratification, climate changes

Abstract >>
In this paper, the response of the general atmosphere circulation to the climate changes is simulated with use of a medium complexity model. It is shown that with a temperature gradient decrease, the Hadley circulation weakens and its boundary moves to the poles. The troposphere height dynamics depending on atmosphere temperature is investigated. A relation between characteristics determining the atmosphere baroclinic instability is obtained.



4.
Automatic program parallelization with block data distribution

Lev Romanovich Gervich, Evgeniy Nikolaevich Kravchenko, Boris Yakovlevich Steinberg, Mikhail Viktorovich Yurushkin
Southern Federal University, 105/42 Bolshaya Sadovaya Str., Rostov-on-Don, 344006
Keywords: automatic parallelization, tiling, memory, distributed memory, block distribution of arrays, optimization of memory, distribution with overlapping

Abstract >>
This paper discusses several automated methods of acceleration programs. The acceleration is achieved by parallelization and optimization of memory access. Optimization of accesses to RAM is achieved by switching to a block code and block placements arrays. When using a distributed memory, the automated distribution of arrays and array distribution with overlapping are employed. Automation is implemented using the C language with pragmas in Open Parallelizing System. This paper presents the numerical results for linear algebra and mathematical physics. Some features of this demonstration converter have a remote access to the Internet.



5.
Approximate solution of large systems of equations with multi–dimensional Toeplitz matrices

Anatoliy Vsevolodovich Kozak, Dmitriy Igorevich Khanin
Southern Federal University, 105/42 Bolshaya Sadovaya Str., Rostov-on-Don, 344006
Keywords: approximate solution, Toeplitz matrices, multi-dimensional cyclic matrices, multi-dimensional convolution operators on polyhedral

Abstract >>
The conditions using the inverse operator and its form in the truncated two-dimensional convolution operators on sets with smooth boundaries are known. The presence of the corner points adds complexity to the task. The equations with multi-dimensional convolution operators on polyhedra is considered. The approximate method for them is proposed, and estimates for the errors are obtained. The possibility of approximation solutions of these equations with multi-dimensional cyclic matrices is also investigated.



6.
A new coupled reduced alternating group explicit method for non–linear singular two point boundary value problems on a variable mesh

R. K. Mohanty1, Jyoti Talwar2
1Department of Applied Mathematics Faculty of Mathematics and Computer Science South Asian University Akbar Bhawan, Chanakyapuri New Delhi, 110021, India
2Department of Mathematics Faculty of Mathematical Sciences University of Delhi, Delhi, 110 007, India
Keywords: two point singular boundary value problems, geometric mesh, third order method, singular equation, CRAGE method, Newton-CRAGE method, Burgers equation, RMS errors

Abstract >>
In this paper, we discuss a new coupled reduced alternating group explicit (CRAGE) and Newton-CRAGE iteration methods to solve non-linear singular two point boundary value problems u″ = ƒ(r,u,u′), 0< r<1, subject to given natural boundary conditions u(0) = A1, u(1) = A2, where A1 and A2 are finite constants, along with a third order numerical method on a geometric mesh. The proposed method is applicable to singular and non-singular problems. We discuss the convergence of the CRAGE iteration method in detail. The results obtained from the proposed CRAGE iteration method are compared with the results of the corresponding two parameter alternating group explicit (TAGE) iteration methods to demonstrate computationally the efficiency of the proposed method.



7.
Iterative algorithm for calculation of spectral parameters of a quadratic bunch of operators in the Hilbert space

Victor Ivanovich Tarakanov, Alexey Olegovich Dubovik
Surgut State University, pr. Lenin, 1, Surgut, Tyumen region, 628400
Keywords: operator, spectrum, iterative algorithm

Abstract >>
A new iterative algorithm is suggested for calculating spectral parameters of a quadratic bunch of partially symmetrical compact operators in the Hilbert space.



8.
Method for description of heat transfer processes in fractal systems using scale variable

Olga Nikolaevna Khatuntseva1,2
1S.P. Korolev rocket and space corporation «Energia», 4A Lenin Street, Korolev, Moscow area, 141070
2Moscow Institute of Physics and Technology, 9 Institutskiy per., Dolgoprudny, Moscow Region, 141700
Keywords: fractal, fractional dimension, scaling, heat conduction, diffusion

Abstract >>
A method is offered for the description of heat transfer (diffusion) processes in fractal systems based on the heat conductivity equation enhanced by an additional variable specifying a scale of the consideration of the fractal.