Mathematics – Numerical Analysis
Scientific paper
2012-03-11
Mathematics
Numerical Analysis
19 pages
Scientific paper
Dropping tolerance criteria play a key role in the power sparse approximate inverse (PSAI) preconditioning and many other sparse approximate inverse (SAI) preconditioning techniques, such as all the static F-norm minimization based SAI preconditioning procedures. However, such criteria have received little attention and have often been treated heuristically in such a way: If an entry is below some empirically small quantity in magnitude, then it is considered to be small and set as zero in these techniques. The meaning of "small" is vague and loose and has not been considered rigorously and systematically. It has not been clear how dropping tolerances affect the quality and effectiveness of a preconditioner $M$. In this paper, a solid theory on robust selection criteria is established for PSAI in this paper. It serves two important purposes, one of which is to guarantee the non-singularity of $M$ and make it as sparse as possible and the other is to simultaneously make $M$ have comparable preconditioning quality to the possibly much denser one obtained by PSAI without dropping. The theory directly applies to all the static F-norm minimization based SAI preconditioning techniques. Numerical experiments are reported to confirm the theory and illustrate the effectiveness of selection criteria for dropping tolerances. Particularly, it is addressed that (i) smaller tolerances make $M$'s possibly much denser, their construction more expensive but do not improve the preconditioning quality and (ii) larger tolerances can produce very poor and even (numerically) {\em singular} $M$'s, so that preconditioning is ineffective and even fails completely.
Jia Zhongxiao
Zhang Qian
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