Вестник Южно-Уральского государственного университета. Серия: Математическое моделирование и программирование @vestnik-susu-mmp
Статьи журнала - Вестник Южно-Уральского государственного университета. Серия: Математическое моделирование и программирование
Все статьи: 767
Forecasting the return of the loan portfolio on the basis of Markov model
Статья научная
We consider the problem of mathematical modelling of flows of loan portfolio payments. We assume that the change in the quality of each loan is described by a simple Markov chain with a finite number of states. In this case, the flow of loan payments is a random process, which depends on the Markov chain. On the basis of the proposed model and known relations of the stochastic systems theory, we describe the expected flows of payments of the entire loan portfolio and construct a method to forecast the expected return (net present value) of the portfolio. We analyze an accuracy of the obtained model and a sensitivity of net present value of the portfolio to a change in the transition probabilities in the Markov chain.
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Статья научная
This study aims to apply a novel technique devised by the authors to process the results of thermal physics experiments. The paper describes a two-stage technique for identifying coefficients of mathematical models from observed experimental data. The technique is based on the maximum likelihood method and is informed by the errors of all sensors used to obtain parameter measurements. Stage 1 of the technique minimizes the maximum relative error over all measured parameters, which allows gross measurement errors to be identified in qualitative terms and reduces the maximum relative error down to acceptable values. At Stage 2, we propose to use the method of weighted least absolute values to minimize the sum of absolute values of relative errors of all measured parameters. The technique was applied to process the results of thermal physics experiments aimed at generalizing the size of vapor bubbles of various types during unsteady heating of a vertical steel cylindrical heater surrounded by an upward flow of water subcooled to the saturation temperature. The numerical simulations reported in this study attest to the high quality of the proposed two-stage technique for identifying coefficients of mathematical models. The study also presents a comparative analysis of the results obtained by the classical least squares method and the novel two-stage technique.
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We study the possibility of influence on the saving of allocated funds for the elimination of consequences of natural disasters. At that, we take into account statistical data on the emergence of such phenomena and the degree of actual damage. The article describes the problem of determining the optimal share of funds that either replenish or spend the principal amount according to the distribution. We prove that, under certain conditions of the distribution and a positive mathematical expectation, it is possible to choose a share that ensures the maximum possible growth of the original deposit account. At the same time, the choice of the share allows not to lose the full provision of damage recovery. This process is presented as a serial multi-stage process based on a Markov chain that takes into account only the distribution based on the statistical data of this year to plan the size of the deposit share for the next year. For simplicity, we assume that the process is established and has a constant distribution for some time. The distribution table can be changed in the case of a major change in stochastic data. We consider a serial multi-stage process of changing the monetary amount that is purposefully deposited for the renewal, replacement and restoration of security and alarm systems at burned-out facilities. The optimal stochastic control of the change in the share of the money deposit providing this restoration is carried out based on the generalized Kelly formula. An example of model validity is shown. On the basis of statistical data, the analysis of the possibility to use this model is carried out.
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Global Schumpeterian dynamics with structural variations
Статья научная
In this paper, we present the investigations developing the schumpeterian theory of endogenous evolution of economic systems. The proposed approach allows to simulate the emergence and propagation of new technologies. We develop a mathematical model of dynamics of sector capital distribution over efficiency levels on the base of the system of nonlinear differential equations. In order to take into account the boundedness of the economic growth conditioned by the boundedness of the markets, the resource base and other factors, we introduce the notion of economical niche volume. The scenario of the emergence of the new highest efficiency level is proposed. In order to simulate the process of the emergence of the new highest efficiency level, the notion of intellectual capital is proposed. According to the proposed scenario, the new level emerges when the intellectual capital achieves the threshold value. Herewith, the dimension of the dynamic system is varied. The necessary condition for the functioning of the new level is formulated. The invariant set of the dynamic system is defined. The local stability of the equilibria is investigated. The global stability of the dynamic system is established on the base of a geometrical method. The proposed models allow to evaluate and predict the dynamics of the technological levels of the economic sector firms development.
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Gravitational model of population dynamics
Статья научная
We consider the author's mathematical model of population dynamics of territories, taking into account the migration between the territories. The model is a system of ordinary differential equations with constant coefficients. The main idea of the presented modelling is to take into account the migration interactions of territories in the form of nonlinear terms (they are the pair products of phase variables of the territories population). On the one hand, the approach allows to consider the model as a variant of expanding the gravitational approach in migration studies. On the other hand, this approach allows to apply the approaches of mathematical biology, which are successfully used in econophysics and sociodynamics. In order to verify the model, we use statistical data on population and migration between federal districts of the Russian Federation. The results of the modelling show the significance of the "repulsion" of migrants arriving in the Central and North-Western federal districts, mainly in the nearby regions (Southern, North Caucasian and Volga federal districts). Model evaluations of the migration balance are obtained. The evaluations exceed statistical ones by dozens of times and, to all appearances, describe the "latent" migration of the population, covering both long-term and short-term movements. An analysis of the change in the stationary values of the population for a linear change in the parameters is carried out. It is shown that there are such values of the parameters of migration attractiveness of the Russian Federation federal districts, under which the population increases both in the whole in the Russian Federation and in individual districts. It is established that such changes can occur due to significant differences in the opportunities, which are "provided" by different federal districts for migrants (e.g., living and working conditions, upbringing and education of children, etc.), and intra-Russian migration.
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High Accuracy Numerical Solution of Elliptic Equations with Discontinuous Coefficients
Статья научная
We develop an approach to constructing a new high-accuracy hp-version of the leastsquares collocation (LSC) method for the numerical solution of boundary value problems for elliptic equations with a coefficient discontinuity on lines of different shapes in a problem solution domain. In order to approximate the equation and the conditions on the discontinuity of its coefficient, it is proposed to use the external parts and irregular cells (i-cells) of the computational grid which are cut off by the line of discontinuity from regular rectangular cells. The proposed approach allows to obtain solutions with a high order of convergence and high accuracy by grid refining and/or increasing the degree of the approximating polynomials both in the case of the Dirichlet conditions on the boundary of the domain and in the case of the presence of Neumann conditions on a large part of the boundary. Also, we consider the case of the problem with a discontinuity of the second derivatives of the desired solution in addition to the coefficient discontinuity at the corner points of the domain. We simulate the heat transfer process in the domain where particles of the medium move in a plane-parallel manner with a phase transition and heat release at the front of the discontinuity line. An effective combination of the LSC method with various methods of accelerating the iterative process is demonstrated: the acceleration algorithm based on Krylov subspaces; the operation of prolongation along the ascending branch of the V-cycle on a multigrid complex; parallelization. The results are compared with those of other authors on solving the considered problems
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Hoff's model on a geometric graph. Simulations
Статья научная
This article studies numerically the solutions to the Showalter-Sidorov (Cauchy) initial value problem and inverse problems for the generalized Hoff model. Basing on the phase space method and a modified Galerkin method, we develop numerical algorithms to solve initial-boundary value problems and inverse problems for this model and implement them as a software bundle in the symbolic computation package Maple 15.0. Hoff's model describes the dynamics of H-beam construction. Hoff's equation, set up on each edge of a graph, describes the buckling of the H-beam. The inverse problem consists in finding the unknown coefficients using additional measurements, which account for the change of the rate in buckling dynamics at the initial and terminal points of the beam at the initial moment. This investigation rests on the results of the theory of semi-linear Sobolev-type equations, as the initial-boundary value problem for the corresponding system of partial differential equations reduces to the abstract Showalter-Sidorov (Cauchy) problem for the Sobolev-type equation. In each example we calculate the eigenvalues and eigenfunctions of the Sturm-Liouville operator on the graph and find the solution in the form of the Galerkin sum of a few first eigenfunctions. Software enables us to graph the numerical solution and visualize the phase space of the equations of the specified problems. The results may be useful for specialists in the field of mathematical physics and mathematical modelling.
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Статья научная
The motion of a hydrodynamic flow in a chemical reactor described by a one-dimensional one-parameter diffusion model is considered. Within the framework of this model, the task is set to identify the boundary condition at the reactor outlet containing an unknown concentration of the reagent under study leaving the reactor in a stream. In this case, the law of change in the concentration of the reagent over time at the reactor inlet is additionally set. After the introduction of dimensionless variables, a discrete analogue of the transformed inverse problem in the form of a system of linear algebraic equations is constructed by the method of difference approximation. The discrete analogue of the additional condition is written as a functional and the solution of a system of linear algebraic equations is presented as a variational problem with local regularization. A special representation is proposed for the numerical solution of the constructed variational problem. As a result, the system of linear equations for each discrete value of a dimensionless time splits into two independent linear subsystems, each of which is solved independently of each other. As a result of minimizing the functional, an explicit formula was obtained for determining the approximate concentration of the reagent under study in the flow leaving the reactor at each discrete value of the dimensionless time. The proposed computational algorithm has been tested on the data of a model chemical reactor.
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Implementation of surface-related multiple prediction problem on reconfigurable computer systems
Статья научная
The traditional methodology of computer-aided synthesis of parallel-pipeline programs for reconfigurable computer systems based on field programmable gate arrays (FPGAs) is aimed at the highest possible computer system performance, achieved on available hardware resource. Application of such an approach to real-time problems can lead to inefficient use of system hardware resource. Frequently, this fact leads to idle stand of occupied equipment and to higher requirements to power consumption, size and cost of the end product. We suggest a new methodology to synthesize of parallel-pipeline programs for solution of real-time computationally intensive problems. The methodology provides data processing having a specified rate which depends on a specified time interval. With the help of the developed methodology, it is possible to synthesize a problem computing structure, which requires the minimum hardware resource for the specified system performance. In order to illustrate the suggested methodology, we give the solution of the real-time surface-related multiple prediction problem. We evaluate various configurations of reconfigurable computer systems based on Xilinx Kintex UltraScale FPGAs.
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Income modelling of enterprises on the basic of vector prediction
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This paper proposes a new approach to income modelling of enterprises, based on the vector prediction methods. Existing approaches to the income modelling are based on the use of traditional methods of economic dynamics prediction such as the average absolute increase and average growth rate. Quantitative inaccuracy and highly approximate nature of predictions are inherent for traditional methods. The authors propose the economic-mathematical model for the enterprise income planning for a year ahead on a quarterly basis. For revenue prediction two methods of vector prediction are used (the method of orthogonal differences and multiplicative Holt - Winters' method). This model provides prediction for a few steps forward at the same time. Individually, each of these methods doesn't take into account the diversity of the process. Only in conjunction the methods allow to take into account the demolition of trends and seasonal nature of income, thus providing the necessary stability of the prediction. To summarize two predictions their linear combination is calculated, the choice of weighting coefficients being based on the accuracy of private predictions. The accuracy of the private prediction is defined as the average relative error of the forecast.
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Статья научная
Preparation of the target-protein, particularly the protein protonation method can affect considerably the spatial arrangement of the attached hydrogen atoms and the charge state of individual molecular groups in amino acid residues. This means that the calculated protein-ligand binding energies can vary significantly depending on the method of the protein preparation, and it also can lead to the different docked positions of the ligand in the case of docking (positioning of the ligand in the protein active site). This work investigates the effect of the hydrogen atoms arrangement method in the target-protein on the protein-ligand binding energy. All hydrogen atoms of target-protein are fixed or movable. The comparison of the protein-ligand binding energies obtained for the test set of target-proteins prepared using six different programs is performed and it is shown that the protein-ligand binding energy depends significantly on the method of hydrogen atoms incorporation, and differences can reach 100 kcal/mol. It is also shown that taking into account solvent in the frame of one of the two continuum implicit models smooths out these differences, but they are still about 10-20 kcal/mol. Moreover, we carried out the docking of the crystallized (native) ligands from the protein-ligand complexes using the SOL program and showed that the different methods of the hydrogen atoms addition to the protein can give significantly different results both for the positioning of the native ligand and for its protein-ligand binding energy.
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Influence to new formulas gradient for removing impulse noise images
Статья научная
In conjugate gradient techniques, the conjugate formula is often the primary point of concentration. The conjugate gradient technique is used to solve problems that arise during the process of picture restoration. By using the quadratic model, a brand-new coefficient conjugate will be produced for the operation. The algorithms demonstrate both local and global convergence and descent. The numerical testing revealed that the newly developed method is much superior to the one that came before it. The recently created conjugate gradient strategy has better performance than the FR conjugate gradient technique, which is the industry standard.
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Intellectual mathematical support software and inner architecture of LMS MAI Class.Net
Статья научная
Distance education prove to be effective in improving the learning and teaching environment. One of the main advantages of distance learning is that web-based courses can be taken anytime and anywhere. The implementation of an e-learning management system (LMS) requires not only good and fast hardware, but also the use of modern software technologies and architectural solutions. This article outlines the main ways of forming the LMS architecture based on a microservice approach, which allows the achievement of high performance and fault tolerance. A distinctive feature of the CLASS.NET system is the presence of a special mathematical software package that allows the optimization of educational processes and tasks (such as students tests generation, students progress analysis, knowledge level assessment, task difficulty analysis, personal learning curve planning). The process of interaction between the LMS system and mathematical software package, as well as the main ways of forming such software as completely independent applications for their further integration into other learning management systems, are thoroughly described. The efficiency of the microservice architecture in terms of scaling, performance and general behavior in case of critical errors in comparison with other systems based on classical architectural approaches is shown. The algorithm of predicting the time a student spends to answer the tasks, which is included in the mathematical software package, is considered.
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Статья научная
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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Статья научная
This work considers hypersonic aircraft open-loop control problem in a presence of terminal and phase constraints. By the discretization process this problem is transformed into a nonlinear programming problem which is solved numerically by the interval explosion search algorithm. This algorithm belongs to metaheuristic algorithms of interval global optimization. Desired control is constructed in a class of interval piecewise-constant and piecewise-linear functions. Also this work demonstrates the comparison of results obtained by the proposed method and by Galerkin projection technique. This comparison confirms the efficiency of the interval based control algorithm.
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Introducing a power of the operator in direct spectral problems
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The resolvent method, proposed by Sadovnichiy and Dubrovsky in the 1990s, is successfully applied in the direct spectral problem to calculate the asymptotics of eigenvalues of the perturbed operator, find formulas for the regularized trace, and recover perturbation. But the application of this method faces difficulties when the resolvent of the unperturbed operator is non-nuclear. Therefore, a number of physical problems could only be considered on the interval. This article describes a justification of the transition to the power of an operator in order to expand the area of possible applications of the resolvent method. Considering the problem of calculating the regularized trace of the Laplace operator on a parallelepiped of arbitrary dimension, we show that for every fixed dimension it is possible to choose the required power of the operator and to calculate the regularized traces. These studies are relevant due to the need to study important applied problems, particularly in hydrodynamics, electronics, elasticity theory, quantum mechanics, and other fields.
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Invariant Manifolds of the Hoff Model in "Noise"
Статья научная
The work is devoted to the study the stochastic analogue of the Hoff equation, which is a model of the deviation of an I-beam from the equilibrium position. The stability of the model is shown for some values of the parameters of this model. In the study, the model is considered as a stochastic semilinear Sobolev type equation. The obtained results are transferred to the Hoff equation, considered in specially constructed “noise” spaces. It is proved that, in the vicinity of the zero point, there exist finite-dimensional unstable and infinite-dimensional stable invariant manifolds of the Hoff equation with positive values of parameters characterizing the properties of the beam material and the load on the beam
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Invariant description of control in a Gaussian one-armed bandit problem
Статья научная
We consider the one-armed bandit problem in application to batch data processing if there are two alternative processing methods with different efficiencies and the efficiency of the second method is a priori unknown. During the processing, it is necessary to determine the most effective method and ensure its preferential use. Processing is performed in batches, so the distributions of incomes are Gaussian. We consider the case of a priori unknown mathematical expectation and the variance of income corresponding to the second action. This case describes a situation when the batches themselves and their number have moderate or small volumes. We obtain recursive equations for computing the Bayesian risk and regret, which we then present in an invariant form with a control horizon equal to one. This makes it possible to obtain the estimates of Bayesian and minimax risk that are valid for all control horizons multiples to the number of processed batches.
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Invariant manifolds of semilinear Sobolev type equations
Статья научная
The article is devoted to a review of the author's results in studying the stability of semilinear Sobolev type equations with a relatively bounded operator. We consider the initial-boundary value problems for the Hoff equation, for the Oskolkov equation of nonlinear fluid filtration, for the Oskolkov equation of plane-parallel fluid flow, for the Benjamin-Bon-Mahoney equation. Under an appropriate choice of function spaces, these problems can be considered as special cases of the Cauchy problem for a semilinear Sobolev type equation. When studying stability, we use phase space methods based on the theory of degenerate (semi)groups of operators and apply a generalization of the classical Hadamard-Perron theorem. We show the existence of stable and unstable invariant manifolds modeled by stable and unstable invariant spaces of the linear part of the Sobolev type equations in the case when the phase space is simple and the relative spectrum and the imaginary axis do not have common points.
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Inverse problem for Sobolev type mathematical models
Статья научная
The work is devoted to the study of an inverse problem for the linear Sobolev type equation of higher order with an unknown coefficient depending on time. Since the equation might be degenerate the phase space method is used. It consists in construction of projectors splitting initial spaces into a direct sum of subspaces. Actions of operators also split. Therefore, the initial problem is reduced to two problems: regular and singular. The regular one is reduced to the first order nondegenerate problem which is solved via approximations. The needed smoothness of the solution is obtained. Then it is substituted into the singular problem which is solved using the methods of relatively polynomially bounded operator pencils theory. The main result of the work contains sufficient conditions for the existence and uniqueness of the solution to the inverse problem for a complete Sobolev type model of the second order. This technique can be used to investigate inverse problems of the considered type for Boussinesq-Love mathematical model.
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