

2013 year, number 2
T.A. Averina^{}
Keywords: numerical methods, stochastic differential equations, systems with random structure
Abstract >>
An algorithm for statistical simulation of randomstructure systems with distributed transitions has been constructed. The proposed algorithm is based on numerical methods for solving stochastic differential equations, and uses a modified maximum crosssection method when the transition intensity depends on the vector of state.

E.N. Akimova^{}, D.V. Belousov^{}, V.E. Misilov^{}
Keywords: inverse gravimetry problems, parallel algorithms, direct and iterative methods, parallel computing systems
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For solving inverse gravimetry problems, efficient stable parallel algorithms based on iterative gradient methods are proposed. For solving systems of linear algebraic equations with blocktridiagonal matrices arising in geoelectrics problems, a parallel matrix sweep algorithm, a square root method, and a conjugate gradient method with preconditioner are proposed. The algorithms are implemented numerically on the MVSIMM parallel computing system, NVIDIA graphics processors, and the Intel multicore CPU with the use of new computing technologies. The parallel algorithms are incorporated into a developed system of remote computations «Specialized WebPortal for Solving Geophysical Problems on Multiprocessor Computers». Problems with «quasimodel» and real data are solved.

A.Sh. Akysh
Keywords: splitting method, convergence of the splitting method scheme, nonlinear Boltzmann equation, global solvability of the nonlinear Boltzmann equation in time, existence and uniqueness of a solution to the Boltzmann equation, a priori estimates
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The question of convergence of the splitting method scheme for the nonlinear Boltzmann equation is considered. On the basis of the splitting method scheme, boundedness of positive solutions in the space of continuous functions is obtained. By means of the solution boundedness and found a priori estimates, convergence of the splitting method scheme and uniqueness of the limiting element are proved. The found limiting element satisfies the equivalent integral Boltzmann equation. Thereby global solvability of the nonlinear Boltzmann equation in time is shown.

V.M. Aleksandrov
Keywords: optimal control, speed, computing time, disturbance, phase trajectory, dynamic balance, limit cycle, transferring accuracy, linear system
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The problem of transferring a linear system to a state of dynamic balance under simultaneous action of an unknown disturbance and timeoptimal control is considered. Optimal control is calculated along the phase trajectory, and it is periodically updated for discrete phase coordinate values. It is proved that the phase trajectory comes to the dynamic equilibrium point and makes undamped periodic motions (a stable limit cycle). The location of the dynamic equilibrium point and the limit cycle form are considered as functions of different parameters. With the disturbance calculated in the process of control, the accuracy of transferring to the required final state increases. A method for estimating attainable accuracy is presented. Results of simulation and numerical calculations are given.

L.A Babkina^{}, Yu.P. Garmai^{}, D.V. Lebedev^{}, Pantina^{}, R.A^{}, M.V. Filatov^{}
Keywords: image analysis, Zernike moments, atomic force microscopy, cell nuclei of higher organisms, PCA
Abstract >>
A method for analyzing AFM images of the cell nuclei of higher organisms by expanding these images by Zernike moments is proposed. This method allows for expanding the pilot image by Zernike moments whose spatial harmonics are Zernike polynomials. It is shown that the reverse procedure of image reconstruction using Zernike polynomials converges to the experimental image and the expansion amplitude is a quantitative spectral characteristic when comparing the morphological features of different images. It is shown that expansion amplitudes can be used as input vectors for cluster analysis of images by PCA.

A.M. Matsokin
Keywords: Dirichlet boundary value problem for the Poisson equation, finite element method with piecewiselinear functions, condensed grid (topologically equivalent to a rectangular grid), preconditioner
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In this paper, it is proved that a Laplace grid operator approximating a Dirichlet boundary value problem for the Poisson equation by the finite element method with piecewiselinear functions on an evenly condensed grid that is topologically equivalent to a rectangular grid (i.e. obtained by shifting the rectangular grid nodes) is equivalent, in the range, to the operator of a 5point difference scheme on a uniform grid.

S.I. Fadeev^{}, V.V. Kogai^{}, V.V. Mironova^{}, N.A. Omelyanchuk^{}, V.A. Likhoshvai^{}
Keywords: cell ensemble, gene networks, autonomous system, circular model, stationary solution, autooscillations, model for continuation with respect to parameters
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In this paper, a mathematical model describing substance transport in a circular cell ensemble is considered. The model is represented by an autonomous system of equations. With a model of continuation with respect to a parameter, it is shown that stationary solutions may have different symmetry representing closed curves. Periodic solutions have the same property, whereas the component plots repeat each other by a simple shift.

T. Hou
Keywords: elliptic equations, optimal control problems, superconvergence, a posteriori error estimates, mixed finite element methods, postprocessing
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In this paper, we investigate the superconvergence property and a posteriori error estimates of mixed finite element methods for a linear elliptic control problem with an integral constraint. The state and costate are approximated by order k=1 RaviartThomas mixed finite element spaces, and the control variable is approximated by piecewise constant functions. Approximations of the optimal control of the continuous optimal control problem will be constructed by a projection of the discrete adjoint state. It is proved that these approximations have convergence order h^{2}. Moreover, we derive a posteriori error estimates both for the control variable and the state variables. Finally, a numerical example is given to demonstrate the theoretical results.

