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Numerical Analysis and Applications

2013 year, number 2

A modified algorithm for statistical modeling of systems with random structure with distributed transitions

T.A. Averina
Keywords: numerical methods, stochastic differential equations, systems with random structure

Abstract >>
An algorithm for statistical simulation of random-structure 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 cross-section method when the transition intensity depends on the vector of state.

Algorithms for solving inverse geophysical problems on parallel computing systems

E.N. Akimova, D.V. Belousov, V.E. Misilov
Keywords: inverse gravimetry problems, parallel algorithms, direct and iterative methods, parallel computing systems

Abstract >>
For solving inverse gravimetry problems, efficient stable parallel algorithms based on iterative gradient methods are proposed. For solving systems of linear algebraic equations with block-tridiagonal 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 MVS-IMM parallel computing system, NVIDIA graphics processors, and the Intel multi-core CPU with the use of new computing technologies. The parallel algorithms are incorporated into a developed system of remote computations «Specialized Web-Portal for Solving Geophysical Problems on Multiprocessor Computers». Problems with «quasi-model» and real data are solved.

Convergence of splitting method for the nonlinear Boltzmann equation

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

Abstract >>
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.

Transferring a system with unknown disturbance under optimal control to a state of dynamic balance and to Оµ-vicinity of a final state

V.M. Aleksandrov
Keywords: optimal control, speed, computing time, disturbance, phase trajectory, dynamic balance, limit cycle, transferring accuracy, linear system

Abstract >>
The problem of transferring a linear system to a state of dynamic balance under simultaneous action of an unknown disturbance and time-optimal 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.

Using Zernike moments for analysis of images

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.

Preconditioner for a Laplace grid operator on a condensed grid

A.M. Matsokin
Keywords: Dirichlet boundary value problem for the Poisson equation, finite element method with piecewise-linear functions, condensed grid (topologically equivalent to a rectangular grid), preconditioner

Abstract >>
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 piecewise-linear 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 5-point difference scheme on a uniform grid.

Mathematical modeling of matter distribution in cells assembling into a ring

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, auto-oscillations, model for continuation with respect to parameters

Abstract >>
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.

Superconvergence and a posteriori error estimates of RT1 mixed methods for elliptic control problems with an integral constraint

T. Hou
Keywords: elliptic equations, optimal control problems, superconvergence, a posteriori error estimates, mixed finite element methods, postprocessing

Abstract >>
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 co-state are approximated by order k=1 Raviart-Thomas 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 h2. 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.