Supercomputer Implementations of Preconditioned Krylov Subspace Methods

Preconditioned Krylov subspace methods are among the preferred iterative techniques for solving large sparse linear systems of equations. As computer architectures are evolving and problems are becoming more complex, iterative techniques are undergoing several mutations. In particular, there has been much recent work on new preconditioned that yield higher parallelism or on new implementations of the standard ones. In the past, a conservative and well understood approach has consisted of porting standard preconditioners to the new computers. However, in addition to their limited parallelism, such preconditioners have other drawbacks. The simple ILU(O) pre-conditioner may fail to converge for realistic problems arising from applications such as Computational Fluid Dynamics. The more robust analogues ILU(k) [30] or ILUT(k) [40] that allow more fill-in are not always a good alternative since they tend to be sequential in nature. In this paper we discuss these issues and give an overview of the standard approaches. Then we will propose a number of alternatives. It will be argued that some approaches based on multi-coloring can offer good compromises between generality and efficiency.