Incomplete inverse triangular factorization in parallel algorithms of preconditioned conjugate gradient methods

A preconditioner for large sparse symmetric positive definite coefficient matrix is considered based on its approximate inverse in the form of product of a lower triangular sparse matrix by its transpose. A parallel algorithm for the construction and application of the preconditioner is proposed. A new approximate block Jacobi preconditioning method is proposed based on the use of the incomplete inverse triangular factorization of diagonal blocks. Timing results are presented for a model problem and test problems with matrices from the collection of the university of Florida for the proposed preconditioning in comparison with the 2$^{\mathrm{nd}}$ order Block Incomplene Inverse Cholesky and the standard point Jacobi preconditionings.