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Sibirskii Zhurnal Industrial'noi Matematiki
2014, vol. 17, No 2 (58)Contents
UDC 517.958
D.S. Anikonov, V.G. Nazarov, I.V. Prokhorov
The integro-differential indicator for a problem of single-beam tomographyThe article continues the authors' investigations of the problem of X-ray tomography with few radiographic projections (beams). In their previous works, the authors suggested a special integro-differential operators for processing the available information. A new formula for such an operator is proposed. The article contains a theoretical justification of the algorithm and the results of the corresponding numerical experiments.
Keywords: inverse problem, tomography, location, radiation, transport equation
Pp. 3–10.Anikonov Dmitrii Sergeevich
Sobolev Institute of Mathematics SB RAS, 4 Koptyug av., 630090 Novosibirsk;
Novosibirsk State University, 2 Pirogova st., 630090 Novosibirsk.
Nazarov Vasilii Gennadievich
Prokhorov Igor' Vasilievich
Institute of Applied Mathematics FEB RAS, 7 Radio st., 690041, Vladivostok; Far Eastern Federal University, 8 Sukhanova st., 690091, Vladivostok.
E-mail: anik@math.nsc.ru; naz@iam.dvo.ru; prh@iam.dvo.ru
UDC 517.9
Yu.E. Anikonov
Differential identities for partial differential equationsWe give some new representations for the solutions and coefficients of second order partial differential equations. We also find formulas for the solution and the source function for a wide class of linear systems of partial differential equations.
Keywords: differential identity, orthogonal polynomial, particular solution.
Pp. 11–17.Anikonov Yurii Evgen'evich
Sobolev Institute of Mathematics SB RAS, 4 Koptyug av., 630090 Novosibirsk. E-mail: anikon@math.nsc.ru
UDC 517.958
T.V. Bugueva
A multidimensional inverse problem of determining two coefficients in the acoustic equationWe consider a linearized inverse problem of finding the coefficients of the acoustic equation; namely, the problem of determining two functions depending on three space variables, the propagation speed of acoustic waves, and the density of the acoustic medium. Some algorithm is obtained for solving the problem as well as a conditional stability estimate for a solution to the inverse problem.
Keywords: acoustic equation, inverse problem, determination of the density of the acoustic medium and the propagation speed of acoustic waves
Pp. 18–31.Bugueva Tat'yana Vladimirovna
Sobolev Institute of Mathematics SB RAS, 4 Koptyug av., 630090 Novosibirsk;
Novosibirsk State University, 2 Pirogova st., 630090 Novosibirsk. E-mail: bugueva@math.nsc.ru
UDC 517.968.22
A.F. Voronin
Reconstruction of the convolution operator from the right-hand side on the real half-axisWe study a Volterra integral equation of the first kind in convolutions on a semi-infinite interval. Under rather natural constraints on the kernel and the right-hand side of a Volterra integral equation (the kernel has bounded support and the support of the right-hand side may be unbounded), it is possible to reconstruct the integral operator of the equation (the solution and the kernel of the integral operator) from the right-hand side of the equation. Some uniqueness theorem is proved, as well as necessary and sufficient conditions for solvability and the explicit formulas for the solution and the kernel are obtained.
Keywords: Volterra integral equation of the first kind, convolution, uniqueness, reconstruction formula for the convolution operator.
Pp. 32–40.Voronin Anatolii Fedorovich
Sobolev Institute of Mathematics SB RAS, 4 Koptyug av., 630090 Novosibirsk E-mail: voronin@math.nsc.ru
UDC 519.688
K.V. Voronin
Numerical study of MPI/OpenMP implementation with postman threads for a three-dimensional splitting scheme in heat transfer problemsWe present the results of a study of the parallel algorithms based on MPI and OpenMP for vector splitting schemes in heat transfer problems. The following parallel implementations are compared: MPI, ``simple'' MPI/OpenMP (#pragma omp directives applied to MPI-based code), and MPI/OpenMP with ``postman'' threads. The main idea of the last algorithm is to split off one thread within each computatinal node to perform data transfer. This approach allows one to implement overlapping of useful computations and data transfer. The presented results show that introducing postman threads can significantly improve performance of MPI/OpenMP implementation; nevertheless, for the considered class of numerical algorithms, it is more reasonable to use MPI implementation.
Keywords: high-performance computing, parallel algorithms, MPI, OpenMP, mixed finite element method, heat transfer, splitting scheme.
Pp. 41–49.Voronin Kirill Vladislavovich
Institute of Computational Mathematics and Mathematical Geophysics SB RAS, 6 Lavrent'ev av.,
Novosibirsk State University, 2 Pirogova st., 630090 Novosibirsk. E-mail: kvoronin@labchem.sscc.ru
UDC 519.95
N.G. Zagoruyko, V.V. Tatarnikov
Detecting mistakes and filling gaps in data cubesWe consider the family of ZET algorithms, designed to detect gross errors and gaps in ''object-parameter'' data tables and ''object-parameter-time'' data cubes. To work with each element of the table, we use information not from the entire table but only out of its ''competent'' subtable. We also consider methods of selecting a competent subtable having tools for avoiding falling into local extrema. The example is exhibited of a ZET algorithm for solving some applied problem.
Keyword: gap filling, rival similarity function.
Pp. 50–58.Zagoruyko Nikolay Grigor'evich
Sobolev Institute of Mathematics SB RAS, 4 Koptyug av., 630090 Novosibirsk.
Tatarnikov Vadim Vladimirovich
Novosibirsk State University, 2 Pirogova st., 630090 Novosibirsk. zag@math.nsc.ru ; vadim.tatarnikov@gmail.com
UDC 517.929
A.A. Korobov
Justification of an effective algorithm for determining the pointwise completeness and pointwise degeneracy of linear delay systems of order 12We justify an effective algorithm for the reduction of a linear system or order 12 with delay h to the normal form. In case the active matrix of the system is degenerate, we also justify a modified algorithm for checking the poitwise completeness of the system. To this end, it suffices to find all real roots of one polynomial of degree 4 whose all coefficients depend on the geometric invariants of the normal form and the exponent of the product of one-half of the active matrix by h.
Keywords: pointwise completeness of a delay system of ordinary differential equations.
Pp. 59–73.Korobov Aleksey Alexandrovich
Sobolev Institute of Mathematics SB RAS, 4 Koptyug av., 630090 Novosibirsk;
Novosibirsk State University, 2 Pirogova st., 630090 Novosibirsk. E-mail: korobov@math.nsc.ru
UDC 544.43:519.6
A. M. Lipanov, A.A. Bolkisev
On solving systems of chemical kinetics equations by explicit methodsWe present some versions of the methods for solving systems of differential equations of chemical kinetics that base on approximations of a quasisteady state and a partial equilibrium. These methods are explicit and guarantee the positivity and boundness of solutions for each time step. In contrast to the existing methods, the methods we propose ensure mass conservation and the fulfillment of the stoichiometric relations, which makes it possible to significantly increase the time meshsize while preserving accuracy.
Keywords: chemical kinetics, ordinary differential equation, explicit method, quasisteady state, partial equlibrium.
Pp. 74–86Lipanov A.M. , Bolkisev A.A.
Institute of Mechanics UB RAS, 34 Baramzinoy st., 426067 Izhevsk
E-mail: ipm@udman.ru; celsior.izh@gmail.com
UDC 539.3
Ragozina V.E., Ivanova Yu.E.
On the impact deformation of an incompressible half-space under the action of a shear load of variable directionA loading process in which the shear action on the boundary plane changes both in intensity and direction is considered on the example of a one-dimensional plane problem for a nonlinear elastic incompressible half-space. We show that, in the domains of the space where the nonlinearity of the system becomes a substantial factor, in the front region of the shock wave, the solution is determined by a system of nonlinear evolution equations. We obtain the general solution to the evolution system. As an example, we consider a particular solution to the evolution system for one of the simplest boundary conditions. We also expose a parametric method for finding the displacements on the basis of a solution to the evolution system.
Keywords: nonlinear elastic incompressible medium, transverse shock wave, shear load of variable direction, evolution equations for the change of shear intensity and shear direction
Pp. 87–96.Ragozina Victoria Evgen'evna
Ivanova Yulia Evgen'evna
Institute of Applied Mathematics FEB RAS, 7 Radio st., 690041, Vladivostok. E-mail: ragozina@vlc.ru ; ivanova@iacp.dvo.ru
UDC 519.21:51.74
L.Ya. Savel'ev
Simple stochastic models of fracturesStochastic Markov models describe various natural and technical processes. They are often used in various fields. In particular, there are considered Markov models with discrete time and few states. In particular cases, such models make it possible to implement efficient computations. The article deals with special models with two, four, and eight states as well as the processes with hydraulic fracturing and the development of a wavefront.
Keywords: stochastic modeling, Markov models, finite Markov chain, crack, probability, mean.
Pp. 97–106.Savel'ev Lev Yakovlevich
Sobolev Institute of Mathematics SB RAS, 4 Koptyug av., 630090 Novosibirsk;
Novosibirsk State University, 2 Pirogova st., 630090 Novosibirsk. E-mail: savelev@math.nsc.ru
UDC 519.95
M.G. Sadovskii, M.Yu. Senashova
To the problem of modeling of reflexive behaviour in a conflict on the example of a biological communityWe consider a two species model of a spatially distributed community. The species in the community are prey-predator related, and each species is distributed over two stations. Transfers between stations (migrations) are targeted and maximize the reproduction coefficient of each species. Both species are supposed to implement a reflexive behavior strategy for determining an optimal migration flux.
Keywords: two-species community, targeted migration, maximazation, evolution optimality, parity.
Pp. 107–118.Sadovskii Mikhail Georgievich
Senashova Maria Yur'evna
Institute of Computational Modeling SB RAS, Akademgorodok, 50/44, 660036 Krasnoyarsk. E-mail: msad@icm.krasn.ru ; msen@icm.krasn.ru
UDC 532.516
V.L. Sennitskii
On a relaxing motion of a hydromechanical systemWe obtain the exact solution to the problem on a relaxing rotation of a solid (a cylinder) and the viscous fluid surrounding it that is at rest at infinity. A series of nontrivial peculiarities of the dynamics of this hydromechanical system is revealed. In particular, we prove that the fluid can move at a velocity exceeding that of the boundary of the solid.
Key words: solid, viscous fluid, free relaxing motion, wave of maximum fluid velocity.
Pp. 119–124.Sennitskii Vladimir Leonidovich
Lavrent'ev Institute of Hydrodynamics SB RAS, 15 Lavrent'ev av., 630090 Novosibirsk. E-mail: sennitskii@yandex.ru
UDC 517.948
V.P. Tanana, A.A. Erygina
Estimation of the precision for the Tikhonov regularization method in solving an inverse problem of solid-state physicsWe obtain an accuracy estimate for the Tikhonov regularization method with the regularization parameter chosen by the error principle for solving the problem of finding of the phonon spectrum of a crystal from its heat capacity. A numerical solution of this problem is exposed that reconstructs the ``fine structure'' of the solution.
Keywords: regularization, continuity modulus, error estimate, ill-posed problem.
Pp. 125–136.Tanana Vitalii Pavlovich
South Ural State University, 76 Lenin av., 454080 Chelyabinsk
Erygina Anna Aleksandrovna
Chelyabinsk State University, 129 Bratiev Kashirinykh, 454001 Chelyabinsk E-mail: tvpa@susu.ac.ru; anya.erygina174@gmail.com
UDC 519.632
M.V. Urev
Convergence of the finite element method for elliptic equations with strong degenerationWe consider the questions of numerical solution by the finite element method (FEM) for the first boundary value problem for an elliptic equation with degeneration on a part of the boundary. We pose the weak and strong variational statements in the function spaces with compatible weights that correspond to the problem. Using the method of the multiplicative separation of singularity for the finite element method that utilizes piecewise linear elements, we prove that the convergence of the approximate solutions to the exact solution in the weighted norm is no worse than in the case of an elliptic equation without degeneracy.
Keywords: elliptic equation with degeneracy, weighted Sobolev space, multiplicative separation of singularity, finite element method, convergence.
Pp. 137–148.Urev Mikhail Vadimovich
Institute of Computational Mathematics and Mathematical Geophysics SB RAS, 6 Lavrentiev av., 630090 Novosibirsk E-mail: mih.urev2010@yandex.ru