Combining incomplete factorization strategies “by value” and “by position” for the $2^{\mathrm{nd}}$ order incomplete triangular factorization in parallel algorithms for the preconditioned conjugate gradient method

Some earlier and newly developed parallel versions of the stabilized $2^{\mathrm{nd}}$ order incomplete triangular factorization are considered as preconditioners 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 with separators. The incomplete factorization is organized using the truncation of fill-in “by value” within the subdomains and “by position” and “by value” at the separators. Some theoretical results are given for the proposed preconditionings related to its quality and robustness. For an MPI implementation of the iterative linear solver, numerical results are given obtained for matrices from the University of Florida collection.