Popov O. Computer Methods for Research of Mathematical Models with Sparse Data Structures

The thesis is devoted to the development of methods, algorithms, and software for computer research and solution of mathematical models with sparse data structures based on block and block-cyclic methods of matrix processing, structural regularization, and decomposition of sparse structure data for computers of the latest MIMD, SIMD, and hybrid MIMD + SIMD architectures. The main results of the work are as follows: highly efficient block and block-cyclic methods and algorithms of parallel calculations of research of mathematical properties and solving on computers of different architecture of linear algebra problems with sparse matrices are developed and investigated; developed methods and algorithms for computer research of the reliability of the obtained solutions; the methodology of research of mathematical models with approximate data of sparse structure on the newest high-performance parallel and distributed computer systems, including with use of multilevel parallelism is offered; algorithms of structural regularization and block-cyclic schemes of decomposition and distribution between processor devices of elements of sparse matrices are developed; developed a methodology and obtained estimates of the effectiveness of the proposed algorithms for high-performance computers of different architectures, including the use of a multilevel model of parallel computing. A methodology to solve incorrect problems with elliptic operators has been developed – the calculation of a single solution in the subspace of the first basic problem of the theory of elasticity. To solve with guaranteed accuracy systems of linear algebraic equations (which arise) with symmetric semi-defined matrices, the economic parallel algorithm of three-stage regularization for computers of different architectures is developed and investigated. The obtained fundamental results were used in the creation of intelligent software to automate the process of research and solving problems of linear algebra with approximate data and evaluation of the reliability of computer solutions (Inpartool, Inparlib). The Lira-cluster software package for numerical analysis (based on the finite element method) of the strength of building structures on high-performance computer systems has been created. New efficient methods, algorithms, and software for solving systems of linear and nonlinear equations for mathematical modeling of the life cycle of responsible welded structures on high-performance computers (MIMD and hybrid architecture) have been developed; software used in the E.O. Paton Institute of Electric Welding. of the National Academy of Sciences of Ukraine. Keywords: mathematical models, sparse data structures, linear algebra problems, block methods, block-cyclic algorithms, computer investigation of the reliability of solutions, high-performance computers, MIMD architecture, hybrid architecture.