  Dropping tolerance criteria play a central role in Sparse Approximate Inverse
preconditioning. Such criteria have received, however, little attention and
have been treated heuristically in the following manner: If the size of an
entry is below some empirically small positive quantity, then it is set to
zero. The meaning of "small" is vague and has not been considered rigorously.
It has not been clear how dropping tolerances affect the quality and
effectiveness of a preconditioner $M$. In this paper, we focus on the adaptive
Power Sparse Approximate Inverse algorithm and establish a mathematical theory
on robust selection criteria for dropping tolerances. Using the theory, we
derive an adaptive dropping criterion that is used to drop entries of small
magnitude dynamically during the setup process of $M$. The proposed criterion
enables us to make $M$ both as sparse as possible as well as to be of
comparable quality to the potentially denser matrix which is obtained without
dropping. As a byproduct, the theory applies to static F-norm minimization
based preconditioning procedures, and a similar dropping criterion is given
that can be used to sparsify a matrix after it has been computed by a static
sparse approximate inverse procedure. In contrast to the adaptive procedure,
dropping in the static procedure does not reduce the setup time of the matrix
but makes the application of the sparser $M$ for Krylov iterations cheaper.
Numerical experiments reported confirm the theory and illustrate the robustness
and effectiveness of the dropping criteria.
