Статьи журнала - Вестник Южно-Уральского государственного университета. Серия: Математическое моделирование и программирование
Все статьи: 739
Статья научная
The article is devoted to the research of the class of stochastic models in mathematical physics on the basis of an abstract Sobolev type equation in Banach spaces of sequences, which are the analogues of Sobolev spaces. As operators we take polynomials with real coefficients from the analogue of the Laplace operator, and carry over the theory of linear stochastic equations of Sobolev type on the Banach spaces of sequences. The spaces of sequences of differentiable "noises" are denoted, and the existence and the uniqueness of the classical solution of Showalter - Sidorov problem for the stochastic equation of Sobolev type with a relatively bounded operator are proved. The constructed abstract scheme can be applied to the study of a wide class of stochastic models in mathematical physics, such as, for example, the Barenblatt - Zheltov - Kochina model and the Hoff model.
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Статья научная
This paper investigates the possibility of effective implementation of calculations in low-precision neural network models on the Elbrus platform with the VLIW architecture. Such models are widely used in practice to increase the computational efficiency of recognition and well suit computers with the x86 and ARM architectures. In this paper, we consider an 8-bit neural network model, in which matrix multiplication is the most resource-intensive part of the implementation. This paper presents an effective implementation of matrix multiplication that takes into account the features of the Elbrus architecture: the presence of several computational channels with various arithmetic and logic devices, an array prefetch buffer, and its own SIMD extension. We carry out theoretical and experimental comparisons of the computational efficiency of low-precision and classical neural network models, which show that Elbrus processors have much more capabilities for performing fast floating point calculations and require the development of new approaches to increase the computational efficiency of neural network models.
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Spectral problems on compact graphs
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The method of finding the eigenvalues and eigenfunctions of abstract discrete semi-bounded operators on compact graphs is developed. Linear formulas allowing to calculate the eigenvalues of these operators are obtained. The eigenvalues can be calculates starting from any of their numbers, regardless of whether the eigenvalues with previous numbers are known. Formulas allow us to solve the problem of computing all the necessary points of the spectrum of discrete semibounded operators defined on geometric graphs. The method for finding the eigenfunctions is based on the Galerkin method. The problem of choosing the basis functions underlying the construction of the solution of spectral problems generated by discrete semibounded operators is considered. An algorithm to construct the basis functions is developed. A computational experiment to find the eigenvalues and eigenfunctions of the Sturm - Liouville operator defined on a two-ribbed compact graph with standard gluing conditions is performed. The results of the computational experiment showed the high efficiency of the developed methods.
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Stability of a stationary solution to one class of non-autonomous Sobolev type equations
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The article is devoted to the study of the stability of a stationary solution to the Cauchy problem for a non-autonomous linear Sobolev type equation in a relatively bounded case. Namely, we consider the case when the relative spectrum of the equation operator can intersect with the imaginary axis. In this case, there exist no exponential dichotomies and the second Lyapunov method is used to study stability. The stability of stationary solutions makes it possible to evaluate the qualitative behavior of systems described using such equations. In addition to introduction, conclusion and list of references, the article contains two sections. Section 1 describes the construction of solutions to non-autonomous equations of the class under consideration, and Section 2 examines the stability of a stationary solution to such equations.
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Stability of solutions to the stochastic Oskolkov equation and stabilization
Статья научная
This paper studies the stability of solutions to the stochastic Oskolkov equation describing a plane-parallel flow of a viscoelastic fluid. This is the equation we consider in the form of a stochastic semilinear Sobolev type equation. First, we consider the solvability of the stochastic Oskolkov equation by the stochastic phase space method. Secondly, we consider the stability of solutions to this equation. The necessary conditions for the existence of stable and unstable invariant manifolds of the stochastic Oskolkov equation are proved. When solving the stabilization problem, this equation is considered as a reduced stochastic system of equations. The stabilization problem is solved on the basis of the feedback principle; graphs of the solution before stabilization and after stabilization are shown.
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Статья научная
Least Absolute Deviations (LAD) method is a method alternative to the Ordinary Least Squares OLS method. It allows to obtain robust errors in case of violation of OLS assumptions. We present two types of LAD: Weighted LAD method and Generalized LAD method. The established interrelation of methods made it possible to reduce the problem of determining the GLAD estimates to an iterative procedure with WLAD estimates. The latter is calculated by solving the corresponding linear programming problem. The sufficient condition imposed on the loss function is found to ensure the stability of the GLAD estimators of the autoregressive models coefficients under emission conditions. It ensures the stability of GLAD-estimates of autoregressive models in terms of outliers. Special features of the GLAD method application for the construction of the regression equation and autoregressive equation without exogenous variables are considered early. This paper is devoted to extension of the previously discussed methods to the problem of estimating the parameters of autoregressive models with exogenous variables.
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Stationary electrochemical machining simulation applying to precision technologies
Статья научная
The modification of statement of electrochemical formation problem is offered for mathematical modelling of the precision technologies. As an example the process of cutting with a plate electrode-tool is considered. For the description of the technologies with high localization of the processes a stepwise function of current efficiency is used. It realizes for simulation of the anode dissolution process in passivating electrolytes under short impulse current. This function determines the movement rate of the anode boundary in the areas of an active electrochemical dissolution and also it defines the boundaries of the areas where dissolution is absent. The stationary and limiting-stationary machining problems are formulated and solved on the base of the offered model. The limiting model describes the maximum localization process. The stationary problem is characterized by the presence of anode surface part, on which the current density is equal to a critical value. Investigations in the whole range of ratio of the maximal and critical values of electrical field strength on the anode surface are carried out.
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Stationary solutions for the Cahn - Hilliard equation coupled with Neumann boundary conditions
Статья научная
The structure of stationary states of the one-dimensional Cahn - Hilliard equation coupled with the Neumann boundary conditions has been studied. Here the free energy is given by a fourth order polynomial. The bifurcation diagram for existence and uniqueness of monotone solutions for this problem has been constructed. Namely, we find the length of the interval on which the solution monotonically increases or decreases and has one zero for some fixed values of physical parameters. Under the non-uniqueness we understand a possibility of existence of more than one monotone solutions for the same values of physical parameters.
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Statistical analysis module for weight design of aircraft elements
Статья научная
The concept of a statistical analysis module for weight design of aircraft elements (for predicting weight characteristics of one or another aircraft elements) is proposed. Models, methods to construct single-point estimates of the predicted characteristic, quality criteria of constructed models are considered. Two approaches to the confidence estimation of the predicted characteristic are proposed. First approach is based on the assumption that errors at predicting are caused by inaccurate identification of the deterministic part of the predicted characteristic behavior. The second one is based on the assumption that the deterministic part of the predicted characteristic behavior is identified correctly and errors at predicting are caused by inaccuracy in the measurements. The structure, goals of each component of the software package that implements the statistical analysis module is considered in details. Based on the real data the problem of predicting the take-off mass of an empty equipped airliner depending on maximum pay load and the maximum flight distance at maximum pay load is given. By this problem applicability of considered models and methods is demonstrated.
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Statistical analysis of the functional status of the students
Статья научная
The South Ural is one of the most advanced industrial regions in Russia. It has huge industrial potential (metal industry, mechanical engineering, chemical process industry, oil-extracting industry, mineral resource industry) which predetermines negative ecological trends in the region. Levels of air pollution with salts of heavy metals, phenol and CO2 are 2-4 times higher than national air quality standards. Moreover, natural background radiation in the region is increased. Under the stated conditions, low levels of motor activity lead to hypoxia and cumulative disorders of the locomotor system as well as to respiratory, gastric, oncological, allergic and other diseases. Poor environmental background significantly affects demographic rates including the life expectancy which is 72 in average for the region: 60 years for men, and 75 years for women. The paper studies how the main anthropometric and physiological parameters in the local residents who are divided into three groups according to their health status and motor activity (so called health groups) influence the physical fitness. Social-biological and medical aspect of the study - to find out the correlation between parameters in the examined people from different groups - may also be of interest for specialists working on optimization of physical education practice and student's health promotion.
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Stochastic Leontief type equations with impulse actions
Статья научная
By a stochastic Leontief type equation we mean a special class of stochastic differential equations in the Ito form, in which there is a degenerate constant linear operator in the left-hand side and a non-degenerate constant linear operator in the right-hand side. In addition, in the right-hand side there is a deterministic term that depends only on time, as well as impulse effects. It is assumed that the diffusion coefficient of this system is given by a square matrix, which depends only on time. To study the equations under consideration, it is required to consider derivatives of sufficiently high orders from the free terms, including the Wiener process. In connection with this, to differentiate the Wiener process, we apply the machinery of Nelson mean derivatives of random processes, which makes it possible to avoid using the theory of generalized functions to the study of equations. As a result, analytical formulas are obtained for solving the equation in terms of mean derivatives of random processes.
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Stochastic Leontieff type equations and mean derivatives of stochastic processes
Статья научная
We understand the Leontieff type stochastic differential equations as a special sort of Ito stochastic differential equations, in which the left-hand side contains a degenerate constant linear operator and the right-hand side has a non-degenerate constant linear operator. In the right-hand side there is also a summand with a term depending only on time. Its physical meaning is the incoming signal into the device described by the operators mentioned above. In the papers by A.L. Shestakov and G.A. Sviridyuk the dynamical distortion of signals is described by such equations. Transition to stochastic differential equations arise where it is necessary to take into account the interference (noise). Note that the investigation of solutions of such equations requires the use of derivatives of the incoming signal and the noise of any order. In this paper for differentiation of noise we apply the machinery of the so-called Nelson's mean derivatives of stochastic processes. This allows us to avoid using the machinery of the theory of generalized functions. We present a brief introduction to the theory of mean derivatives, investigate the transformation of the equations to canonical form and find formulae for solutions in terms of Nelson's mean derivatives of Wiener process.
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Stochastic Leontieff type equations in terms of current velocities of the solution II
Статья научная
In papers by A.L. Shestakov and G.A. Sviridyuk a new model of the description of dynamically distorted signals in some radio devices is suggested in terms of so-called Leontieff type equations (a particular case of algebraic-differential equations). In that model the influence of noise is taken into account in terms of the so-called symmetric mean derivatives of the Wiener process instead of using white noise. This allows the authors to avoid using the generalized function. It should be pointed out that by physical meaning, the current velocity is a direct analog of physical velocity for the deterministic processes. Note that the use of current velocity of the Wiener process means that in the construction of mean derivatives the σ-algebra "present" for the Wiener process is under consideration while there is also another possibility: to deal with the σ-algebra "present" of the solution as it is usually done in the theory of stochastic differential equation with mean derivatives. This approach was previously suggested by the authors under the assumption that the matrix pencil, that determines the equation, satisfies the so-called "rank-degree" condition. In this paper we consider stochastic Leontieff type equation given in terms of current velocities of the solution without this assumption.
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Stochastic Leontieff-type equations with multiplicative effect in spaces of complex-valued «noises»
Статья научная
We consider a Leontieff-type stochastic equation, that is, a system of differential equations implicit with respect to the time derivative in the spaces of random processes. The concepts previously introduced for the spaces of differentiable «noise» using the Nelson-Gliklikh derivative carry over to the case of complex-valued «noise»; in addition, the right-hand side of the equation is subject to multiplicative effect of a special form. We construct a solution to the Showalter-Sidorov problem for Leontieff-type equations with multiplicative effect of a complex-valued process of special form. Aside from the introduction and references, the article consists of two parts. In the first part we carry over various concepts of the space of real-valued differentiable «noise» to the complex-valued case. In the second part we construct a Showalter-Sidorov solution to a Leontieff-type equation with multiplicative effect of a complex-valued process of special form. The list of references is not intended to be complete and reflects only the authors'' personal preferences.
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Stochastic inclusions with forward mean derivatives having decomposable right-hand sides
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In this paper, we prove a theorem on the existence of solutions for stochastic differential inclusions given in terms of the forward mean derivatives and the quadratic mean derivatives. These derivatives present information on the drift and the diffusion coefficient, respectively. The forward mean derivatives were introduced by E. Nelson for the needs of the so-called stochastic mechanics (a version of quantum mechanics), while the quadratic mean derivatives were introduced by Yu.E. Gliklich and S.V. Azarina. In the case of both the forward mean derivatives and the quadratic mean derivatives, we assume that the right-hand side is set-valued and lower semi-continuous, but not necessarily convex. Instead of this, we assume that the right-hand side is decomposable. Such inclusions naturally arise in many models of physical processes.
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Stochastic mathematical model of internal waves
Статья научная
The paper studies a mathematical model of internal gravitational waves with additive "white noise'', which models the fluctuations and random heterogeneity of the medium. The mathematical model is based on the Sobolev stochastic equation, Dirichlet boundary conditions and the initial Cauchy condition. The Sobolev equation is obtained from the assumption of the propagation of waves in a uniform incompressible rotation with a constant angular velocity of the fluid. The solution to this problem is called the inertial (gyroscopic) wave, since it arises due to the Archimedes's law and under the influence of inertia forces. By "white noise'' we mean the Nelson-Gliklikh derivative of the Wiener process. The study was conducted in the framework of the theory of relatively bounded operators, the theory of stochastic equations of Sobolev type and the theory of (semi) groups of operators. It is shown that the relative spectrum of the operator is bounded, and the solution of the Cauchy-Dirichlet problem for the Sobolev stochastic equation is constructed in the operator form.
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Stochastic model of optimal dynamic measurements
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Under consideration is the stochastic model of optimal dynamic measurements. To solve this problem, the theory of optimal dynamic measurements which has actively been developing for the deterministic problems is extended to the stochastic case. The main purpose of the model is to restore a dynamically distorted input signal from a given observation using methods of the theory of dynamic measurements and the optimal control theory for Leontief type systems. Based on the results obtained by the authors earlier it is shown that optimal dynamic measurement as a minimum point of the cost functional doesn't depend on stochastic interference such as resonances in chains and random interference at the output of measuring transducer.
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Strongly continuous operator semigroups. Alternative approach
Статья научная
Inheriting and continuing the tradition, dating back to the Hill-Iosida-Feller-Phillips-Miyadera theorem, the new way of construction of the approximations for strongly continuous operator semigroups with kernels is suggested in this paper in the framework of the Sobolev type equations theory, which experiences an epoch of blossoming. We introduce the concept of relatively radial operator, containing condition in the form of estimates for the derivatives of the relative resolvent, the existence of C 0-semigroup on some subspace of the original space is shown, the sufficient conditions of its coincidence with the whole space are given. The results are very useful in numerical study of different nonclassical mathematical models considered in the framework of the theory of the first order Sobolev type equations, and also to spread the ideas and methods to the higher order Sobolev type equations.
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Structural parametric modelling of an information-analytical system
Статья научная
We develop a method for studying relations between interior and exterior characteristics of an information-analytical system. As interior characteristics, we use the balance estimates accounting for conflict interactions among the elements. We solve the problem on assuming that the interior and exterior characteristics of the system are related. We justify the expediency of using the least squares method to find an explicit functional dependence among the characteristics. We develop a method for calculating the functional dependence in the case of a matrix representation of the values of characteristics basing on pseudoinverse matrices. A numerical example illustrates how the proposed method works in the structure parametric modeling of an information-analytical system in an internal affairs department.
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Studying the model of air and water filtration in a melting or freezing snowpack
Статья научная
The article is devoted to a theoretical study of a non-stationary problem on thermomechanical processes in snow taking into account the effects of melting and freezing. Snow is modeled as a continuous medium consisting of water, air and porous ice skeleton. The governing equations of snow are based on the fundamental conservation laws of continuum mechanics. For the one-dimensional setting, the Rothe scheme is constructed as an approximation of the considered problem and the Rothe method is formally justified, i.e., convergence of approximate solutions to the solution of the considered problem is established under some additional regularity requirements.
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