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ПРИМЕНЕНИЕ ДРОБНО-РАЦИОНАЛЬНЫХ ИНТЕРПОЛЯЦИЙ ДЛЯ РЕШЕНИЯ КРАЕВЫХ ЗАДАЧ С ОСОБЕННОСТЯМИ
стр.5-19
Статья посвящена разработке, реализации и тестированию нового метода решения сингулярно-возмущенных краевых задач для нелинейных уравнений с частными производными второго порядка в прямоугольной области. Для приближения решения в методе использованы прямые (тензорные) произведения дробно-рациональных функций, полученных из интерполяционных полиномов с узлами Чебышева, записанных в барицентрической форме, с помощью специальной замены переменной. Замена делается с целью адаптации положения узлов интерполяции к особенностям искомой функции и приводит к их сгущению в окрестности больших градиентов решения. Для аппроксимации нелинейных уравнений используется сочетание итерационного метода установления и метода коллокаций, что позволяет свести задачу на каждой итерации к решению матричного уравнения Сильвестра. Такой подход приводит к существенному снижению времени вычислений. Высокая эффективность метода продемонстрирована на примере тестовой краевой задачи в квадрате, решение которой имеет пик в центре области, обусловленный наличием у неизвестной функции полюса в комплексной плоскости.
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DIFFERENTIAL EQUATIONS OF ELLIPTIC TYPE WITH VARIABLE OPERATORS AND HOMOGENEOUS ROBIN BOUNDARY VALUE CONDITION IN UMD SPACES
стр.20-31
In this article, we give new results on the study of elliptic abstract second order differential equation with variable operators coefficients under the general Robin homogeneous boundary value conditions, in the framework of UMD spaces. Here, we do not assume the differentiability of the resolvent operators. However, we suppose that the family of variable operators verifies the Labbas-Terreni assumption inspired by the sum theory and similar to the Acquistapace-Terreni one. We use Dunford calculus, interpolation spaces and semigroup theory in order to obtain existence, uniqueness and maximal regularity results for the classical solution to the problem.
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TWO-STAGE PARAMETRIC IDENTIFICATION PROCEDURE FOR A SATELLITE MOTION MODEL BASED ON ADAPTIVE UNSCENTED KALMAN FILTERS
стр.32-43
Chernikova O.S., Grechkoseev A.K., Danchenko I.G.
The paper presents a new two-stage parametric identification procedure for constructing a navigation satellite motion model. At the first stage of the procedure, the parameters of the radiation pressure model are estimated using the maximum likelihood method and the multiple adaptive unscented Kalman filter. At the second stage, the parameters of the unaccounted perturbations model are estimated based on the results of residual differences measurements. The obtained results lead to significant improvement of prediction quality of the satellite trajectory.
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INTERNAL BONE MARROW DOSIMETRY: THE EFFECT OF THE EXPOSURE DUE TO 90SR INCORPORATED IN THE ADJACENT BONE SEGMENTS
стр.44-58
Volchkova A.Yu., Sharagin P.A., Shishkina E.A.
The paper is devoted to dosimetric modelling of the human red bone marrow (RBM) internal exposure due to beta-emitting 90Sr incorporated in spongiosa bone. The dose factor calculation (absorbed dose rate due to unit specific activity of 90Sr) is based on the modelling of radiation transport in segments of the skeleton bones with active hematopoiesis. Segmentation considerably simplifies the modelling, but can lead to an underestimation due to electron emission from the neighboring parts of the bone adjacent to the studied segment. The objective of the study is to determine this cross-fire effect on the absorbed dose in RBM. For this purpose, we analyze the results of the numerical experiment on modelling of dose absorption within the bone segments of various shape and size that were parts of the computational phantoms of skeletons of people of different sex and age. We analyze dose factor dependencies on the area of the spongiosa bone surface and the ratio of weights of bone and RBM. It is found that if the area of the spongiosa surface (SS) > 6 cm2, then the effect of neighboring bone parts exposure is negligible. For a smaller SS the extension of the linear dimensions of the spongiosa bone by 2 mean electron path lengths results in dose factor increase proportional to the ratio of the extended spongiosa bone surface area to the original one to the power of 0,28. For human computational phantoms, these values are in the range 1,03-1,21 and are used as adjustment coefficients for the dose factors. Relative standard uncertainty of the adjustment coefficient is 5%.
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DECOMPOSITION OF THE PROBLEM IN THE NUMERICAL SOLUTION OF DIFFERENTIAL-ALGEBRAIC SYSTEMS FOR CHEMICAL REACTIONS WITH PARTIAL EQUILIBRIA
стр.59-70
The paper considers two simple systems of differential-algebraic equations that appear in the study of chemical kinetics problems with partial equilibria: some of the variables are determined from the condition argmin for some system function state, which depends on all variables of the problem. For such a statement, we can write a differential-algebraic system of equations where the algebraic subproblem expresses the conditions for the minimality of the state function at each moment. It is convenient to use splitting methods in numerical solving, i.e. to solve dynamic and optimization subproblems separately. In this work, we investigate the applicability of differential-algebraic splitting using two simplified systems. The convergence and order of accuracy of the numerical method are determined. Different decomposition options are considered. Calculations show that the numerical solution of the split system of equations has the same order of accuracy as the numerical solution of the joint problem. The constraints are fulfilled with sufficient accuracy if the procedure of the numerical method ends with the solution of the optimization subproblem. The results obtained can be used in the numerical solving of more complex chemical kinetics problems.
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ALGORITHM FOR PROCESSING THE RESULTS OF CALCULATIONS FOR DETERMINING THE BODY OF OPTIMAL PARAMETERS IN THE WEIGHTED FINITE ELEMENT METHOD
стр.71-79
Rukavishnikov V.A., Seleznev D.S., Guseinov A.A.
The weighted finite element method allows to find an approximate solution to a boundary value problem with a singularity faster in 106 times than the classical finite element method for a given error equal to 10-3. In this case, it is required to apply the necessary control parameters in the weighted finite element method. The body of optimal parameters is determined on the basis of carrying out and analysing a series of numerical experiments. In this paper we propose an algorithm for processing the results of calculations and determining the body of optimal parameters for the Dirichlet problem and the Lamé system in a domain with one reentrant corner on the boundary taking values from π to 2π.
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METHOD FOR ANALYZING THE STRUCTURE OF NOISY IMAGES OF ADMINISTRATIVE DOCUMENTS
стр.80-89
Slavin O.A., Pliskin E.L.
The problem of extracting content elements (fields) from the images of administrative documents via descriptions of anchoring elements is considered. Administrative documents contain static elements and content elements (filled information). The static objects of the document model are the lines of the document structure and the words. Sets of objects united by properties and relationships are described. The text descriptor can contain attributes that distinguish it from similar descriptors. We suggest using combined descriptors of line segments and words. We showed experimentally that the extraction of object sets improves the recognition accuracy of the document fields by 17% and the accuracy of information extraction by 16%. For optical character recognition, we employed SDK Smart Document Engine in the experiment.
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NUMERICAL METHOD FOR SOLVING THE INVERSE PROBLEM OF NON-STATIONARY FLOW OF VISCOELASTIC FLUID IN THE PIPE
стр.90-98
Aliev A.R., Gamzaev Kh.M., Darwish A.A., Nofal T.A.
The process of unsteady flow of incompressible viscoelastic fluid in a cylindrical tube of constant cross-section is considered. To describe the rheological properties of a viscoelastic fluid, the Kelvin–Voigt model is used and the mathematical model of this process is presented as an integro-differential partial differential equation. Within the framework of this model, the problem is to determine the pressure drop along the length of the pipe, which ensures the passage of a given flow rate of viscoelastic fluid through the pipe. This problem belongs to the class of inverse problems related to the recovery of the right parts
of integro-differential equations. By replacing variables, the integro-differential equation is transformed into a third-order partial differential equation. First, a discrete analog of the problem is constructed using finite-difference approximations. To solve the resulting difference problem, we propose a special representation that allows splitting the problems into two mutually independent second-order difference problems. As a result, an explicit formula is obtained for determining the approximate value of the pressure drop along the length of the pipeline for each discrete value of the time variable. Based on the proposed
computational algorithm, numerical experiments were performed for model problems.
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МЕТОДЫ ИССЛЕДОВАНИЯ УСТОЙЧИВОСТИ И СТАБИЛИЗАЦИИ НЕКОТОРЫХ СИСТЕМ С БОЛЬШИМ ЗАПАЗДЫВАНИЕМ
стр.99-108
Гребенщиков Б.Г., Ложников А.Б.
Статья посвящена исследованию свойств систем дифференциальных уравнений, содержащих большое (в частности, линейное) запаздывание. Системы с линейным запаздыванием имеют достаточно широкое применение в биологии, в частности, при моделировании распределения клеток в ткани организма; а также в теории нейронных сетей. Уравнения подобного типа встречаются также в задачах физики и механики, где важным моментом является асимптотическое поведение решения (в частности, асимптотическая устойчивость). При неустойчивости таких систем возникает задача стабилизации. Оптимальный алгоритм стабилизации основан на совокупности стабилизации систем обыкновенных дифференциальных уравнений и в дальнейшем разностных систем. Данный алгоритм достаточно просто реализуется с использованием численных методов решения систем дифференциальных уравнений с запаздыванием и решения матричных уравнений. Авторами составлена программа, позволяющая достаточно эффективно находить управляющее воздействие, осуществляющее стабилизацию некоторых систем.
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PARALLEL DELIVERY OPERATIONS MODELLING
стр.109-114
Zavalishchin D.S., Vakolyuk K.K.
Some delivery organization scheme is considered. The key point is the principle of routes parallelization using several carriers at the same time and these auxiliary carriers can be based on the main carrier. An example of such a delivery system is a van carrying several autonomous carriers, which in turn can carry out simultaneous so–called parallel deliveries. Delivery routes are determined based on the coordinates of customers, the determination of acceptable starting points for auxiliary carriers, the technical and energy limitations of the main and auxiliary carriers, and the minimization of the amount of time spent on delivery
operations. The developed algorithm for solving the problem on routing of delivery using primary and secondary carriers allows to reduce delivery time and resources. The algorithm is implemented in Python using the libraries for processing and visualization of trajectories and other space–time data, packages for extracting, modelling, analyzing and visualizing street networks on the example of the Yekaterinburg city.
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SMOOTH APPROXIMATION OF THE QUANTILE FUNCTION DERIVATIVES
стр.115-122
Sobol V.R., Torishnyy R.O.
In this paper, a smooth approximation of the second-order derivatives of quantile
function is provided. The convergence of approximations of the first and second order derivatives of quantile function is studied in cases when there exists a deterministic equivalent for the corresponding stochastic programming problem. The quantile function is one of common criteria in stochastic programming problems. The first-order derivative of quantile function can be represented as a ratio of partial derivatives of probability function. Using smooth approximation of probability function and its derivatives we obtain approximations of these derivatives in the form of volume integrals. Approximation of the second-order derivative is obtained directly as derivative of the first-order derivative. A
numerical example is provided to evaluate the accuracy of the presented
approximations.
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A NEW FORMULA ON THE CONJUGATE GRADIENT METHOD FOR REMOVING IMPULSE NOISE IMAGES
стр.123-130
A variety of conjugate gradient algorithms are constructed on the coefficient conjugate. In this paper, a new coefficient conjugate based on the quadratic model for impulse noise removal is proposed. Its global convergence results might be achieved under Wolfe line search circumstances. To demonstrate the performance of the conjugate gradient approach for impulse noise reduction, numerical experiments are provided.
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