On the peculiarities of using sparse matrices in problems of mathematical modeling

Abstract

Mathematical modeling and the related computer experiment are now one of the main means of studying objects, processes and phenomena of various nature: in science, engineering, economics, society, etc. A significant improvement in the quality of mathematical modeling in many areas of science and engineering is possible only with the use of fundamentally new three-dimensional models, the transition from computer simulation of individual components and assemblies to the calculation and optimization of the product as a whole. It is obvious that the consideration of problems in such a formulation leads to discrete mathematical models of super-large sizes. Existing supercomputers of different parallel architectures make it possible to efficiently solve such problems. However, the time for solving problems on parallel computers consists of the time of the actual solution and the time of performing additional operations, that are necessary for the exchange of information between computing devices, that is overhead costs. This is especially true for problems of linear algebra with different structures of sparse matrices of large volumes, that arise in the mathematical modeling of processes. Sparse matrix compaction schemes, decomposition of data arrays between processors are one of the main factors for the effective solution of these problems on parallel computers. The paper considers efficient methods for processing sparse matrices of arbitrary structure for the purpose of effective mathematical modeling of structural strength problems on parallel computers. Various methods of regularization and decomposition of sparse matrices of arbitrary structure, efficient data storage schemes, technology for studying the conditionality of a matrix with approximate data on a computer are proposed. This way of using sparse matrices in mathematical modeling ensures more efficient use of computing resources and reliability of computer results. Problems of mathematical modeling are presented, where the considered methods of processing sparse matrices were effectively applied.