A generalized eigensolver based on smoothed aggregation (GES-SA) for initializing smoothed aggregation (SA) multigrid

Consider the linear system Ax = b, where A is a large, sparse, real, symmetric, and positive definite matrix and b is a known vector. Solving this system for unknown vector x using a smoothed aggregation multigrid (SA) algorithm requires a characterization of the algebraically smooth error, meaning error that is poorly attenuated by the algorithm’s relaxation process. For many common relaxation processes, algebraically smooth error corresponds to the near-nullspace of A. Therefore, having a good approximation to a minimal eigenvector is useful to characterize the algebraically smooth error when forming a linear SA solver. We discuss the details of a generalized eigensolver based on smoothed aggregation (GES-SA) that is designed to produce an approximation to a minimal eigenvector of A. GES-SA may be applied as a stand-alone eigensolver for applications that desire an approximate minimal eigenvector, but the primary purpose here is to apply an eigensolver to the specific application of forming robust, adaptive linear solvers. This paper reports the first stage in our study of incorporating eigensolvers into the existing adaptive SA framework. Copyright c © 2007 John Wiley & Sons, Ltd.