The problems of high performance technologies for solving large sparse linear systems

The technological problems of implementation of the parallel algorithms for solving the systems of linear algebraic equations with sparse matrices of high order that arise in grid approximations in the large problems of mathematical modeling are considered. Classification of algebraic systems and comparative analysis of complexity of the direct, iterative and, combined methods of their solution are given with consideration of different matrix data structures and access formats. The bottle necks and basic approaches to develop the high-performance mathematical software for multi-processor computing systems with shared and distributed memory are described on the basis of using MPI, OpenMP, and hybrid programming. The paper was prepared on the basis of the author's report at the International Conference on Parallel Computing Technologies (PaVT-2009; http://agora.guru.ru/pavt2009).