New methods for adapting and for approximating inverses as preconditioners Original Research Article

We consider projection methods—a class of inherently parallel methods for calculating approximate inverses and we consider the properties of these approximate inverses applied as explicit preconditioners of Krylov subspace methods. In the first part, we discuss a theoretical framework for general projection methods including an explicit representation of an approximate inverse given by a projection method and, importantly, a statement on the quality of the approximation of such an approximate inverse to the exact inverse in form of a minimisation property. Further, we consider strategies for the adaptive generation of sparsity patterns for general projection methods. Since these strategies depend on parameters, they are tunable with regards to the available computer architecture and to the characteristics of the considered linear system. Then, we focus on two particular projection methods, namely FSAI-projection and Plain projection proposed by C. Koschinski [Ph.D. Thesis, University of Karlsruhe, 1999] in greater detail. We compare the performance of these preconditioning techniques to both implicit and explicit state-of-the-art preconditioning methods. The results of these numerical experiments indicate that the new projection methods are competitive to the state-of-the-art preconditioning techniques.