Parallel version of the $2^\mathrm{nd}$ order incomplete triangular factorization preconditioned conjugate gradient method using special matrix reordering

A parallel version of the stabilized $2^\mathrm{nd}$ order incomplete triangular factorization is considered for the use as a preconditioned for the conjugate gradient solution of linear algebraic systems with large sparse symmetric positive definite coefficient matrix. The reordering of the matrix is based on the use of certain domain decomposition type splitting which is further adjusted using a multi-level structure of separators. The incomplete factorization is organized using the truncation of fill-in “by position” at the separators and “by value” otherwise. Some theoretical results are given for the proposed method related, in particular, to its robustness. Numerical results are given obtained on the MVS-100K multiprocessor for the MPI implementation of the iterative linear solver.