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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 A GL AGLT, where GL is a sparse approximation based on minimizing the Frobenius form \| I - GL LA \|F to the inverse of the lower triangular Cholesky factor LA 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.
Kolotilina et al. (Fri,) studied this question.