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scientific edition of Bauman MSTU


Bauman Moscow State Technical University.   El № FS 77 - 48211.   ISSN 1994-0408

Mathematics and Mathematical Modeling

Hooke–Jeeves Method-used Local Search in a Hybrid Global Optimization Algorithm
# 06, June 2014
DOI: 10.7463/0614.0716155
V.D. Sulimov, P.M. Shkapov, S.K. Nosachev
Modern methods for optimization investigation of complex systems are based on development and updating the systems’ mathematical models in connection with solving the corresponding inverse problems. The optimization approach is one of the main approaches to solving the inverse problems. In the main case it is necessary to find a global extremum of not everywhere differentiable criterion function. When the number of variables is large they use the stochastic global optimization algorithms. As stochastic algorithms yield too expensive solutions, so this drawback restricts their applications. Developing hybrid algorithms that combine a stochastic algorithm for scanning the variable space with deterministic local search method is a promising way. A new hybrid algorithm that integrates a multiple Metropolis algorithm and the Hooke–Jeeves method for the local search is proposed. Some results on solving the global optimization benchmark are presented.
Software Package Nesvetay-3D for Modeling Three-Dimensional Flows of Monatomic Rarefied Gas
# 06, June 2014
DOI: 10.7463/0614.0712314
V.A. Titarev
The paper contains a review of the recent developments of a three-dimensional computational code "Nesvetay-3D" for modelling rarefied gas flows. The code solves the Boltzmann kinetic equation with various model collision integrals using an implicit finite-volume scheme of Godunov type.  Arbitraryunstructured spatial meshes can be used so that flows over arbitrary geometrical shapes can be computed, including those of. Large-scale computations can be run on hundreds of CPU cores using MPI. Performance and robustness of the numerical scheme and computer code are demonstrated on a number of examples, including gas flows into vacuum through micropipes and external flows over re-entry craftsat high altitude.
Solution of Direct Kinematic Problem for Stewart-Gough Platform with the Use of Analytical Equation of Plane
# 04, April 2014
DOI: 10.7463/0414.0706936
A.L. Lapikov, V.N. Paschenko
The article examines existing approaches to solution of direct kinematic problem for the Stewart-Gough platform. It suggests the method to solve this problem for the one type of simplified model of Stewart-Gough platform (type 6-3), which is based on determination of analytical equation of moving platform plane. The task is formalized as a system of nonlinear algebraic equations a characteristic property of which is equations of the same structure, with the same type of nonlinearity. The solution of the direct kinematic problem (pose of the moving platform) is given as homogenous transformation matrix.
Model-Based Approach to EEG Classification
# 04, April 2014
DOI: 10.7463/0414.0705745
A.G.Trofimov, S.L. Shishkin, A.E. Ossadtchi
The paper offers a developed method of constructing a feature space for electroencephalogram classification. It is based on the localization of brain’s electrical activity sources. The simplest statistical characteristics of dipole moments for equivalent current dipoles are chosen as features for classification, and the nearest neighbour algorithm is used for classification. The research on real electroencephalograms reveals that the accuracy of the proposed method is comparable to the accuracy of the existing classical approaches in brain-computer interfaces at the same time giving a number of opportunities to further increase it and having clear neurophysiological interpretation.
On Some Anisotropy-Based Analysis Problems for Linear Discrete-Time Descriptor Systems with Nonzero-Mean Input Signals
# 04, April 2014
DOI: 10.7463/0414.0704850
O.G. Andrianova
The paper presents a novel concept of anisotropy-based analysis for linear discrete-time descriptor systems with nonzero-mean input signals. Descriptor systems are a general case of normal systems. They contain both differential (difference) equations and algebraic ones. The paper offers an algorithm for mean anisotropic computation of the Gaussian stationary random sequence with nonzero mean (on the assumption that, the signal was generated by the shaping filter in descriptor form). The equations for anisotropy norm computation (in the frequency domain) for descriptor systems are developed. Numerical examples are given.
Refinement of Some Estimates for Values of the Hypergeometric Functions
# 04, April 2014
DOI: 10.7463/0414.0704694
P.L. Ivankov
To study the arithmetic properties of values of the generalized hypergeometric functions with irrational parameters it is impossible to use directly Siegel's method known in the theory of tran-scendental numbers. The reason is a too fast growing minimal common denominator of the ex-pansion coefficients of such functions in formal power series. In some cases, however, this diffi-culty can be overcome by means of special reasoning, and a linear approximating form (or sim-ultaneous approximations) can be constructed using a Dirichlet principle. Here, some techniques related to the effective construction methods of the abovementioned approximations are applied too. In case of inhomogeneous forms, using these considerations leads to insufficiently accurate estimates. In this paper for refining such estimates we construct simultaneous approximations with the optimal choice of zero polynomial degree.
Feynman formulae for evolution semigroups
# 03, March 2014
DOI: 10.7463/0314.0701581
Ya.A. Butko
The paper describes an approach to solve initial and initial-boundary value problems for evolution equations. This approach is based on representation of solutions of such equations by limits of n-fold iterated integrals when n tends to infinity (such representations are called Feynman formulae). These formulae allow to calculate solutions of evolution equations directly, are suitable for approximation of transition probabilities of stochastic processes, can be used for computer modeling of classical, quantum and stochastic dynamics. In the present note Feynman formulae are constructed both for evolution semigroups, obtained by additive and multiplicative perturbations of generators of some original semigroups, and for the Cauchy-Dirichlet initial-boundary value problem for a differential equation of parabolic type. In particular, the paper presents Feynman formulae for the Cauchy and the Cauchy-Dirichlet problems for a second order parabolic equation with variable coefficients and for the Cauchy problem for Schrödinger equation.
The covering method for the solution of terminal control problem
# 02, February 2014
DOI: 10.7463/0214.0699730
V.N. Chetverikov
The paper proposes a new method to solve the terminal control problem for dynamical systems. The method is based on the complement of initial system by equations for the derivatives of the control and on the reformulation of the terminal control problem into two associated Cauchy problems. It is shown that this method is applied to the flat systems and generalizes the previously used approach. Here, the solution of the terminal control problem lies among the solutions of an arbitrary determined system of ordinary differential equations of appropriate order. This feature of the new method can be used for control design in the case of flat systems taking into consideration the constraints. In addition, an example is given to demonstrate the possibility to apply this method to the non-flat systems.
A local search with smoothing approximation in hybrid algorithms of diagnostics of hydromechanical systems
# 02, February 2014
DOI: 10.7463/0214.0699149
V.D. Sulimov, P.M. Shkapov
This article deals with the global optimization issues on the computational diagnostics of hydromechanical systems. The criterion functions are assumed to be continuous, Lipschitzian, multiextremal, and not always differentiable. The article proposes two novel hybrid algorithms with scanning a search space by the stochastic Multi-Particle Collision Algorithm that uses the analogy to the absorption and scattering processes for nuclear particles. The local search is implemented using the hyperbolic smoothing function method for the first algorithm, and the linearization method with two-parametric smoothing approximations of criteria for the second one. There are some results on the model solution of computational diagnostics of the coolant phase composition in the reactor primary circuit.
About one method of extension operator construction
# 02, February 2014
DOI: 10.7463/0214.0697588
E.M. Popova
This article examines S.L. Sobolev anisotropic spaces. It proposes a method to construct the operator extension T from the banach space of functions to Sobolev anisotropic space. The operator T is the best one in terms of growth rate of high order derivatives from extension function, and its construction is based on any extension operator Ext. The construction method is to apply the approximation operator saving boundary values to the operator Ext. V.I. Burenkov offers this method in the isotropic case while E.M. Popova’s proposal is to use it in the anisotropic one.
A dynamic model of a cell population system
# 12, December 2013
DOI: 10.7463/1213.0646463
M.S. Vinogradova
This paper deals with development of a mathematical model of population dynamics of human stem cells being cultured in vitro. This investigation was inspired by intensive development of stem cell transplantation. The authors investigated formation processes of a cell population with chromosome abnormalities in a population of normal cells along with co-development of these populations in vitro. In this paper a refined continuous dynamic model of an isolated population system consisting of a population of normal stem cells and a population of cells with chromosomal abnormalities was proposed. An essential feature of this model is a necessity for biological characteristics of processes that occur in a cell population system for its creation. These characteristics are:  a portion of cells divided within a specified time, a portion of dead cells and a portion of cells passed into a population of abnormal cells from normal cell populations. This approach allows to provide a more detailed analysis of the impact of different "primary" parameters on dynamics of a population system. Parametric analysis of the model was carried out; the main scenarios of its development were described. The proposed model allows to simulate correctly the main evolution scenarios of a population system which are known from experimental investigations.
Thermo-physical model of bio-tissue and its numerical implementation
# 10, October 2013
DOI: 10.7463/1013.0645537
S.Yu. Makarov
An adaptive mathematical model of bio tissue for heat calculations was introduced; this model takes into account hemoperfusion, heat exchange with surrounding air and other specific for living tissues internal heat sources which are calculated from boundary conditions of a steady state. Discretization procedure for this model was described; results of numerical simulation by the example of single and double layer bio-tissue were presented. In order to validate the discrete model corresponding analytical solution for a particular case was obtained. It was shown that the existent Pennes model is a special case of the proposed model. Possible application domains were also discussed in this paper.
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