Approximate inverse preconditioners with adaptive dropping

It is well-known that analysis of incomplete Cholesky and LU decompositions with a general dropping is very difficult and of limited applicability, see, for example, the results on modified decompositions [1], [2], [3] and later results based on similar concepts. This is true not only for the dropping based on magnitude of entries but it also applies to algorithms that use a prescribed sparsity pattern. This paper deals with dropping strategies for a class of AINV-type incomplete decompositions [4] that are based on the generalized Gram–Schmidt process. Its behavior in finite precision arithmetic has been discussed in [5]. This analysis enables better understanding of the incomplete process, and the main goal of the paper is to propose a new adaptive dropping strategy and to illustrate its efficiency for problems in structural mechanics.