Olga Grigorieva
Faculty: Computer Science and Technology
Speciality: Economic cibernetics
Olga GrigorievaFaculty: Computer Science and Technology Speciality: Economic cibernetics Scientific adviser: Olga Dmitrieva |
Optimization simulation of large dinamical systems(Summary of research and developments)Introduction (Motivation)Objective: To prove the effectiveness of using algorithms to work with sparse matrices in solving systems of ordinary differential equations (ODE) of large dimension.To achieve this goal in the work will address the following tasks:
Relevance of the topic chosen
Dynamic is one of the features that characterize any level of the modern economy. It can not be neglected during the planning of material, financial, human and other streams movement. Dynamic process should be balanced and optimal. In specific market relations enterprise (firm, company, industry, government) seeks to maximize profits or to meet market demand, taking into account resources, time or other constraints. At the same time enterprises seek to reducing their costs to a minimum. Researches of this problem determine the need to build an economic and mathematical models. Exactly the way of simulation helps in timely determination of the optimal solution to manage the economic system.[1] The special feature of economic models is the large number of factors, that have an influence on the system. And we need to select only the most important among them. At the same time the values of various indicators are not enough to assess the dynamics of economic system, it is necessary to have information about their changes. In research the solution of differential equations system will be used to obtain this information . Survey of research and developmentsSurvey of research and development on the world
Survey of research and development on the subject in Ukraine The problem of modeling dynamic systems in economics are discussed in the journal "Cybernetics and Systems Analysis." The magazine is unique in Ukraine periodical publishing for half a century of fundamental and applied in a wide range of problems of automatic control and informatics. The magazine is published in the creative participation: Ukrainian Association of Automatic Control, the National Space Agency of Ukraine, academic and industrial research institutions, leading universities of Ukraine and CIS countries, scientists and specialists from abroad. Sections of the magazine:
Survay of research and development on the DonNTU Masters DonNTU that research problems related to modeling danamicheskim:
Among the teachers DonNTU work with numerical methods and solution of ODE deals Associate Dmitrieva OA Developments: Parallel block algorithms for solving systems of ordinary differential equations. Publications: Tutorial Chiselnі methods іnformatitsі "textbook" Chiselnі methods. Workshop ", a monograph" Paralelnі rіznitsevі technique rozv'yazannya zadachі Cauchy, more than 30 articles in the journals Metematicheskoe modeling "," E-Mod, a collection of faculty CST and others [7] Summary of individual results available at the time of completion of the abstractsBuilding a macroeconomic modelOne of the ways of representing dynamical systems of large dimensions is a system of ordinary differential equations of the form:  In this kind of system of differential equations used to represent a dynamic model of the macroeconomy, the quality of which depends on its dimension, ie the number of equations in the system. At the same time, the selection of the factors that influence the rate of change of each individual indicator takes into account only the most important, because excessive detail in this direction may reduce the quality of the overall model. A model, based on the following premises:
The dynamic model presented in the research is as follows:  where ВВП - Gross Domestic
Product;
G-state purchases and expenses; I-total amount of investments; Xn-net exports; Dr-real income of the population; i-size inflation; r-size of the interest rate; D-amount of money in circulation; N-size of the tax population; Z-size of employment; Dn-nominal per capita income; c, n - propensity to consume and accumulate; a - coefficients; α - coefficient that reflects the change in technology; γ-coefficient, which reflects the cyclical fluctuations of the economy; β - factor reflecting the influence of time on the change of nominal income of the population. Since the resulting coefficient matrix is sparse, as is clearly shown in Table 1 ("x" - a non-zero element), then the solution of this system of differential equations will use technology to work with such matrices, as well as multi-step implicit methods for solving systems differential equations. Such an approach to decision maximizes the speed of obtaining the final result, to improve its accuracy and quality, as well as minimize the cost of machine resources for storage and handling of empty elements of the matrix. [10] Table 1 .- The coefficient
matrix system of differential equations.
 should be noted that the model most adequately reflects the macroeconomic changes that will take into account ten times more than in, respectively, and increase the degree of sparsity of coefficients. Thus, the optimization component of the work lies in finding a solution to a system of ordinary differential equations, the best method, saving resources of the machine. The very same model of macroeconomic dynamics is used to describe the behavior of the economy. Main planned resultsAlleged scientific novelty consists in increase of effectiveness of numerical methods for solving large systems of ordinary differential equations through the use of packaging formats store the nonzero elements of sparse matrices and application of algorithms to work with sparse matrices in the implementation of numerical methods for solving ODE.Expected practical results: a dynamic model of the macroeconomy, presented in the form of the ODE system, algorithm analysis of its rigidity, as well as most efficient algorithm for solving this system. Conclusions and future researchThus, modeling of dynamic processes in the economy may be effected by ODE. This raises the problem of stability of the solution, and there is a need to optimize use of existing numerical algorithms through the use of packaging formats, storage of sparse matrices and special algorithms to work with them. References
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