Статьи журнала - Вестник Южно-Уральского государственного университета. Серия: Математическое моделирование и программирование
Все статьи: 739
Статья научная
Of concern is an initial-boundary value problem for the modified Boussinesq equation (IMBq equation) is considered. The equation is often used to describe the propagation of waves in shallow water under the condition of mass conservation in the layer and taking into account capillary effects. In addition, it is used in the study of shock waves. The modified Boussinesq equation belongs to the Sobolev type equations. Earlier, using the theory of relatively p-bounded operators, the theorem of existence and uniqueness of the solution to the initial-boundary value problem was proved. In this paper, we will prove that the solution constructed by the Galerkin method using the system orthornormal eigenfunctions of the homogeneous Dirichlet problem for the Laplace operator converges *-weakly to an precise solution. Based on the compactness method and Gronwall's inequality, the existence and uniqueness of solutions to the Cauchy-Dirichlet and the Showalter-Sidorov-Dirichlet problems for the modified Boussinesq equation are proved.
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Anisotropic diffusion in anisotropic Stepanov spaces
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We consider a problem on the image processing and computer vision. A wide range of methods allows to solve problems of this type. The methods of partial differential equations are the most useful and interesting ones. A non-linear diffusion takes special place in these studies. In this context, fundamental theoretical foundation is a central part of this approach. Therefore, we introduce a new functional class of spaces, formulate and prove the lemma on the equivalent norms in anisotropic Stepanov spaces. Another important result of this study is the lemma that the anisotropic Stepanov spaces are Banach. In addition, we obtain the theorem on the solvability of the equation of anisotropic diffusion in anisotropic Stepanov spaces. The results can be applied to the image processing and computer vision. Also, the obtained results open the new view to this problem.
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Applicability of forecasted bankruptcy models to Russian industrial companies
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The application of effective methods for forecasting of the bankruptcy of industrial companies is always an urgent task for businesses, especially at the present stage which is characterized by an extremely high uncertainty. The paper presents the main techniques of bankruptcy modelling used in the world's practice: logit, probit and MDA-models, as well as the special private methods developed on their basis. These tools constitute the methodological foundation of our research. To assess the practical applicability of these methods to the contemporary Russian market, two sectorial companies (bankrupt and non-bankrupt) are selected as the object of study. A feature of the research is the use of financial statements of companies developed according to Russian and international standards. In the course of the calculations, we apply external and internal restrictions related to the key rate, credit history characteristics, age and regional affiliation of companies. Based on the dynamic assessment, we draw conclusions about the practical applicability and inapplicability of certain forecast models for the Russian economy. We investigate the relationship between the assessment results and the type of source data used. Research veracity is confirmed using generally recognized models and methods, as well as the practical implementation of the results obtained. We can recommend to use these results for improving the existing models for predicting bankruptcy and developing new ones, as well as for owners and investors of companies who need to make strategic decisions.
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Application of the finite volume method for calculating radiation heat transfer in applied problems
Статья научная
A number of numerical models of radiation heat transfer, based on P approximation and finite volume method, were implemented in the in-house CFD code "SigmaFlow" developed by the Krasnoyarsk group of the Institute of Thermophysics of Russian Academy of Science. The implemented finite volume method allows parallel calculations based on domain decomposition of an unstructured mesh and uses sub-mapped spatially inhomogeneous angular grids. Both conventional in CFD methods for solving linear systems such as BiCGStab, DILU, CG and a marching scheme were considered. A number of applied problems with radiation heat transfer were solved by means of CFD code "SigmaFlow". They include such problems as the numerical simulation of a gas furnace chamber, a burner and a vacuum electrical furnace.
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Статья научная
This paper is devoted to the application of the smooth approximation of the probability function in the solution of three different stochastic optimization problems: minimization of an airstrip area under the constrained probability of successful landing, minimization of the cost of water supply system with random performance and with predefined water consumption, and determination of the set of wind speed vectors which guarantees the safe landing of an aircraft in future with the given probability. The first two problems are mathematical programming problems with probability constraint, and the third one is a problem of constructing the isoquant surface of the probability function. Smooth approximation of the probability function allows to use the gradient projection method in the constrained optimization problem and to define the isoquant surface as the solution to a partial differential equation. We provide an example for each of the considered problems and compare the results with known results previously obtained using the confidence method.
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Approximation of solutions to the boundary value problems for the generalized Boussinesq equation
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The paper is devoted to one of the Sobolev type mathematical models of fluid filtration in a porous layer. Results that allow to obtain numerical solutions are significant for applied problems. We propose the following algorithm to solve the initial-boundary value problems describing the motion of a free surface filtered in a fluid layer having finite depth. First, the boundary value problems are reduced to the Cauchy problems for integro-differential equations, and then the problems are numerically integrated. However, numerous computational experiments show that the algorithm can be simplified by replacing the integro-differential equations with the corresponding approximating Riccati differential equations, whose solutions can also be found explicitly. In this case, the numerical values of the solution to the integro-differential equation are concluded between successive values of approximating solutions. Therefore, we can pointwise estimate the approximation errors. Examples of results of numerical integration and corresponding approximations are given.
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The paper is focused on the modelling of a one-dimensional chaotic process which dynamics is described by a one-parameter nonlinear map. The problem is to estimate the initial condition and model parameter from measurements corrupted by additive errors. The considered guaranteed (set-membership) approach assumes that the prior information about the unknown variables (initial condition, model parameter and measurement errors) is presented as interval estimates. In this context, the estimation problem can be stated as a problem of solving a system of nonlinear inequalities. Due to the nonlinearity, it is not possible to obtain an exact characterization of the solution set. The developed algorithm computes an outer approximation as a union of non-overlapping boxes.
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Assessment of economic sustainability of an enterprise
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In a turbulent economy and increasing economic crises, the question of assessing the economic sustainability of a corporation is quite relevant and in demand in scientific research. At the same time, with a certain degree of conditionality, all works in this direction can be divided into two clusters: works based on traditional indicators of financial stability and ones using original scientific approaches to determining the economic condition of an enterprise. This article belongs to the works of the second cluster. The basis for assessing the economic sustainability of an enterprise is the economic and mathematical modelling of the operating activities of the enterprise based on the well-known Cobb - Douglas production function. A methodology is developed for constructing a mathematical model of the enterprise's activity, for determining elasticity indicators of output according to the resources used in the production process and elasticity of production, on the basis of the dynamics of which an approach to assessing economic stability is proposed. The methodology was tested on the data of PJSC Chelyabinsk Tube Rolling Plant during the implementation of the innovation project "Height 239".
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Asymptotic estimate of a Petrov - Galerkin method for nonlinear operator-differential equation
Статья научная
In the current paper, we study a Petrov - Galerkin method for a Cauchy problem for an operator-differential equation with a monotone operator in a separable Hilbert space. The existence and the uniqueness of a strong solution of the Cauchy problem are proved. New asymptotic estimates for the convergence rate of approximate solutions are obtained in uniform topology. The minimal requirements to the operators of the equation were demanded, which guaranteed the convergence of the approximate solutions. There were no assumptions of the structure of the operators. Therefore, the method, specified in this paper, can be applied to a wide class of the parabolic equations as well as to the integral-differential equations. The initial boundary value problem for nonlinear parabolic equations of the fourth order on space variables was considered as the application.
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Balance on Nash in "soft" control problems for systems of the hierarchal type
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This paper discusses the hierarchical system, which includes the Center, symbolizing the global interests of the system and control objects (CO), perform guidance Center, but at the same time pursuing its own local interests. In the first phase, the Centre produces a decision, taking into account global interests, and in the second stage CO chooses a specific version of the decision, taking into account their own administration and their local interests. It is anticipated that decision is consistent if the interests of each of the parties are counted so that a departure from it cannot improve the performance of any of the parties, i.e. it is balance on Nash. «Soft» control suggests that for each pair of «Center-facility control» global and local interests are taken into account in varying degrees. The method of finding equilibrium solutions depending on the extent of the accounting of interests and levels of reflection is found.
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Baltic Sea water dynamics model acceleration
Статья научная
Industrial Baltic sea water dynamics modelling program optimization and parallelization is described. Program is based on solving the system of partial differential equations of shallow water with numerical methods. Mechanical approach to program modernization is demonstrated involving building module dependency graph and rewriting every module in specific order. To achieve desired speed-up the program is translated into another language and several key optimization methods are used, including parallelization of most time-consuming loop nests. The theory of optimizing and parallelizing program transformations is used to achieve best performance boost with given amount of work. The list of applied program transformations is presented along with achieved speed-up for most time-consuming subroutines. Entire program speed-up results on shared memory computer system are presented.
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Статья научная
Camera-traps is a relatively new but already popular instrument in the estimation of abundance of non-identifiable animals. Although camera-traps are convenient in application, there remain both theoretical complications (such as spatial autocorrelation or false negative problem) and practical difficulties, for example, laborious random sampling. In the article I propose an alternative way to bypass the mentioned problems. In the proposed approach, the raw video information collected from the camera-traps situated at the spots of natural attraction is turned into the frequency of visits, and the latter is transformed into the desired abundance estimate. The key for such a transformation is the application of the correction coefficients, computed for each particular observation environment using the Bayesian approach and the massive database (DB) of observations under various conditions. The proposed method is based on automated video-capturing at a moderate number of easy to reach spots, so in the long term many laborious census works may be conducted easier, cheaper and cause less disturbance for the wild life. Information post-processing is strictly formalized, which leaves little chance for subjective alterations. However, the method heavily relies on the volume and quality of the DB, which in its turn heavily relies on the efforts of the community. Although the construction of such DB could be rather difficult and controversial, it is much easier than the solution of the initial abundance estimation problem. Moreover, such a rich DB of visits might benefit not only censuses, but also many behavioral studies.
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Boundary inverse problem for star-shaped graph with different densities strings-edges
Статья научная
The paper is devoted to the mathematical modelling of star-shaped geometric graphs with n rib-strings of different density and the solution of the boundary inverse spectral problem for Sturm-Liouville differential operators on these graphs. Earlier it was shown that if strings have the same length and densities, then the stiffness coefficients of springs at the ends of graph strings are not uniquely recovered from natural frequencies. They are found up to permutations of their places. We show, that if the strings have different densities, then the stiffness coefficients of springs on the ends of graph strings are uniquely recovered from all natural frequencies. Counterexamples are shown that for the unique recovery of the stiffness coefficients of springs on n dead ends of the graph, it is not enough to use n natural frequencies. Examples are also given showing that it is sufficient to use n+1 natural frequencies for the uniqueness of the recovery of the stiffness coefficients of springs at the n ends of the strings. Those, the uniqueness or non-uniqueness of the restoration of the stiffness coefficients of springs at the ends of the strings depends on whether the string densities are identical or different.
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Calculation of aircraft area on satellite images by genetic algorithm
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Classical genetic algorithm is used in a computational experiment, which is described in this article, to measure an area of the aircraft Boeing 737-300. Satellite images containing objects of interest are selected as initial data. The computational experiment consists of two steps. The first step presents the generalized formulas of values. These formulas are needed to select initial images, using the theory of modified descriptive algebra of images. After that, the initial data is formed according to the calculated values, i.e. the satellite image is chosen in the needed scale and the shooting angle. At the second step, a model of technical vision system (computer vision system) in the modified descriptive images algebra and a fitness function for the genetic algorithm are developed. Then varying parameters of the model are chosen and their optimization is carried out in MATLAB. The article demonstrates development of the model due to its complexity by additional imaging techniques. Experimentally it was found that the evolution of the model improves optimization results.
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Causal relations in support of implicit evolution equations
Статья научная
This is a brief exposition of dynamic systems approaches that form the basis for linear implicit evolution equations with some indication of interesting applications. Examples in infinite-dimensional dissipative systems and stochastic processes illustrate the fundamental notions underlying the use of double families of evolution equations intertwined by the empathy relation. Kisynski's equivalent formulation of the Hille-Yosida theorem highlights the essential differences between semigroup theory and the theory of empathy. The notion of K-bounded semigroups, a more direct approach to implicit equations, and related to empathy in a different way, is included in the survey.
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Choosing average values when determining characteristics of the unsteady boiling of liquid
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This paper presents an analysis of the issues associated with constructing mathematical models for processes of intense phase transformations and, in particular, focuses on the aspect of using closing relations of empirical origin. The main trend in the implementation of modern numerical algorithms for practical problems is aimed at improving the accuracy of calculation results. The latter is usually achieved by refining a certain set of coefficients in mathematical models. These refinements are carried out both on the basis of the modernization of existing approaches, and with the involvement of new empirical information obtained for a limited number of regime conditions. Predictive models for describing the dynamics of phase transformations, as one of the most difficult in the mathematical formulations, refer to a particularly striking manifestation of the problem under study. In this research, we discuss the existing and widely used experimental work devoted to the extraction of primary information about the dynamics of vapor bubbles on the surface of metal heaters. Their example reveals the presence of a simplified approach in the existing development methodology, and shows a way to determine the correct generalization of empirical information that has a pseudo-stochastic nature.
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Circular shift of loop body - programme transformation, promoting parallelism
Статья научная
The article deals with the programme transformation executing the circular shift of loop body statements. It can be used for vectorizing or parallelizing. This becomes possible due to the fact that when the order of loop body statements is changed, some of the bottom-up arcs become top-down arcs. Besides, sometimes loop carried dependence arcs are substituted by loop independent ones. It should be pointed out that in executing the circular shift the number of loop iterations is reduced by one. The transformation can be used both independently and in conjunction with other transformations promoting parallelism. These could be "forward substitution", "scalar expansion", "privatization", "array expansion", etc. The transformation under consideration in this article can be used both in hand parallelization and added to a paralleling (optimizing) compiler. Moreover, the application of the transformation results in the equivalent code only for the loops where loop unrolling is the equivalent transformation. Thus, they can contain nested loops, if statements and other programming language statements.
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Статья научная
Inverse problems of identification of the fractional diffusivity and the order of fractional differentiation are considered for linear fractional anomalous diffusion equations with the Riemann - Liouville and Caputo fractional derivatives. As an additional information about the anomalous diffusion process, the concentration functions are assumed to be known at several arbitrary inner points of calculation domain. Numerically-analytical algorithms are constructed for identification of two required parameters of the fractional diffusion equations by approximately known initial data. These algorithms are based on the method of time integral characteristics and use the Laplace transform in time. The Laplace variable can be considered as a regularization parameter in these algorithms. It is shown that the inverse problems under consideration are reduced to the identification problem for a new single parameter which is formed by the fractional diffusivity, the order of fractional differentiation and the Laplace variable. Estimations of the upper error bound for this parameter are derived. A technique of optimal Laplace variable determination based on minimization of these estimations is described. The proposed algorithms are implemented in the AD-TIC package for the Maple software. A brief discussion of this package is also presented.
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Commutative encryption method based on hidden logarithm problem
Статья научная
A candidate for post-quantum commutative encryption algorithm is proposed, which is based on the hidden discrete logarithm problem defined in a new 6-dimensional finite non-commutative associative algebra. The properties of the algebra are investigated in detail and used in the design of the proposed commutative cipher. The formulas describing the set of p2 different global right-sided units contained in the algebra and local left-sided units are derived. Homomorphisms of two different types are considered and used in the commutative cipher. The encrypted message is represented in the form of a locally invertible element T of the algebra and encryption procedure includes performing the exponentiation operation and homomorphism map followed by the left-sided multiplication by a randomly selected local right-sided unit. The introduced commutative cipher is secure to the known-plaintext attacks and has been used to develop the post-quantum no-key encryption protocol providing possibility to send securely a secret message via a public channel without using any pre-agreed key. The proposed commutative encryption algorithm is characterized in using the single-use keys that are selected at random directly during the encryption process.
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