Factorized Approximate Inverses with Adaptive Dropping
Jiří Kopal, Miroslav Rozložńık, Miroslav Tůma · SIAM Journal on Scientific Computing · 2016
This paper presents a new approach to constructing factorized approximate inverses for a symmetric and positive definite matrix $A$. The proposed strategy is based on adaptive dropping that reflects the quality of preserving the relation $UZ = I$ between the direct factor $U$ and the inverse factor $Z$ satisfying $A = U^TU$ and $A^{-1}=ZZ^T$. An important part of the approach is column pivoting, used to minimize the growth of the condition number of leading principal submatrices of $U$ that occurs explicitly in the dropping criterion. Numerical experiments demonstrate that the resulting approximate inverse factorization is robust as a preconditioner for solving large and sparse systems of linear equations.