A. Yu. Yeremin

Factorized sparse approximate inverse preconditionings I: theory

This paper considers construction and properties of factorized sparse approximate inverse preconditionings well suited for implementation on modern parallel computers. In the symmetric case such preconditionings have the form

Eigenvalue translation based preconditioners for the GMRES(k) method

A \to G_L AG_L^T

Variable Block CG Algorithms for Solving Large Sparse Symmetric Positive Definite Linear Systems on Parallel Computers, I: General Iterative Scheme

, where

FACTORIZED SPARSE APPROXIMATE INVERSE PRECONDITIONING II: SOLUTION OF 3D FE SYSTEMS ON MASSIVELY PARALLEL COMPUTERS

G_L

Matrix-free iterative solution strategies for large dense linear systems

is a sparse approximation based on minimizing the Frobenius form

On a family of two-level preconditionings of the incomplete block factorization type

\| I - G_L L_A \|_F

Clusters, preconditioners, convergence

to the inverse of the lower triangular Cholesky factor

Block SSOR preconditionings for high order 3D FE systems

L_A

Block SSOR preconditionings for high-order 3D FE systems. II. Incomplete BSSOR preconditionings

of A, which is not assumed to be known explicitly. These preconditionings preserve symmetry and/or positive definiteness of the original matrix and, in the case of M-, H-, or block H-matrices, lead to convergent splittings.

Incomplete Block Factorizations as Preconditioners for Sparse SPD Matrices

The paper considers a possible approach to the construction of high-quality preconditionings for solving large sparse unsymmetric offdiagonally dominant, possibly indefinite linear systems. We are interested in the construction of an efficient iterative method which does not require from the user a prescription of several problem-dependent parameters to ensure the convergence, which can be used in the case when only a procedure for multiplying the coefficient matrix by a vector is available and which allows for an efficient parallel/vector implementation with only one additional assumption that the most of eigenvalues of the coefficient matrix are condensed in a vicinity of the point 1 of the complex plane. The suggested preconditioning strategy is based on consecutive translations of groups of spread eigenvalues into a vicinity of the point 1. Approximations to eigenvalues to be translated are computed by the Arnoldi procedure at several GMRES(k) iterations. We formulate the optimization problem to find optimal translations, present its suboptimal solution and prove the numerical stability of consecutive translations. The results of numerical experiments with the model CFD problem show the efficiency of the suggested preconditioning strategy.