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Parallel implementation of the conjugate gradient method with the IC1 preconditioner based on the use of grid node reordering

O. Yu. Milyukova


Abstract: The paper proposes a method for using MPI+OpenMP technology for constructing and inverting a first-order incomplete triangular Cholesky expansion preconditioner IC1($\tau$) for solving system of linear algebraic equations with an arbitrary symmetric positive definite matrix on a small number of processors. 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. When constructing the preconditioning matrix IC1($\tau$) using MPI, cutting is performed at positions in some of its rows. The use of OpenMP technology in the construction and inversion of the preconditioner is carried out for most rows of the matrix. The time taken to solve problems using the conjugate gradient method with the IC1($\tau$) preconditioner using MPI and hybrid MPI+OpenMP technology is compared using the example of a model problem and a number of problems from the SuiteSparse collection of sparse matrices.

Keywords: incomplete Cholesky factorization, Domain Decomposition ordering, parallel preconditioning, conjugate gradient method.

DOI: 10.20948/prepr-2023-61