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ADAPTIVE ALGORITHM FOR SOLVING SYSTEMS OF EQUATIONS WITH BLOCK CLOUD MATRICES

Volume 67, Issue 5, 2022, pages 17-31

DOI: http://doi.org/10.34229/2786-6505-2022-5-2

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Volodymyr Sydoruk, V.M. Glushkov Institute of Cybernetics of NAS of Ukraine, Kyiv, wolodymyr.sydoruk@gmail.com

Pavlo Yershov, V.M. Glushkov Institute of Cybernetics of NAS of Ukraine, Kyiv, yershov.pavel.wsk@gmail.com

ABSTRACT

Increasing requirements for the quality of design solutions, as well as theuse of new structural materials necessitates the solution of qualitatively newproblems. There is always a need to perform calculations of complex uniquestructures. Therefore, there is a growing need for new methods and approaches related to the construction and study of correct computer modelsthat adequately reflect the real operation of structures. Use of detailed mathematical models leads to a significant increase in the size of computational(discrete) problems, and hence the corresponding matrices. Usually, suchmatrices have a sparse structure and extremely large sizes. As a result, thereare problems of efficient storage, decomposition and processing of such data. Using structural regularization of matrices it is possible to solve the following problems: compact data storage; fast access and processing of largedata sets; minimization of data exchanges between computing devices. Forthe tasks with sparse symmetric matrices of block-skyscraper type, an adaptive parallel algorithm of the direct method is proposed, which provides highparallelization efficiency, takes into account the structure of sparse matricesand their data content. The developed algorithm allows to distribute betweenthe processes of calculations with blocks of non-zero elements of the triangular development of the sparse matrix so that they are carried out simultaneously by most processes. Estimates of the number of arithmetic operationsperformed by the algorithm and the speedup factor are obtained. Also obtained time characteristics and acceleration rates in solving a number ofpractical problems of modeling the strength of building structures on different numbers of processor cores using different sizes of blocks used for calculations.

Keywords: mathematical modeling, parallel algorithms, variable precision,sparse matrices.

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