MPI+OpenMP implementation of the conjugate gradient method with a factorized preconditioner based on the use of grid node reordering

The paper proposes ways to use MPI and MPI+OpenMP technologies for constructing and inverting a factorized preconditioner of the type incomplete triangular decomposition for solving systems of linear algebraic equations with an arbitrary symmetric positive definite matrix. Methods of using MPI and MPI+OpenMP technologies are based on the use of grid node orderings consistent with the division of the calculation area. The use of OpenMP technology in the construction and inversion of the preconditioner is carried out for most rows of the matrix. Comparative timing results for the MPI+OpenMP and MPI implementations of the proposed preconditioning used with the conjugate gradient method for a model problems and the sparse matrix collections SuiteSparse are presented.