Using a generalized eigensolver based on smoothed aggregation (GES-SA) to initialize smoothed aggregation multigrid

Geoffrey Sanders

Campus Box 526
University of Colorado at Boulder
Boulder, CO 80309-0526


Panayot Vassilevski (LLNL)
Thomas Manteuffel (CU)
Steve McCormick (CU)
Marian Brezina (CU)
John Ruge (CU)

Abstract Consider the linear system Ax=b, where A is a large, sparse and symmetric positive definite matrix. Solving this system for uknown vector x using a smoothed aggregation multigrid (SA) algorithm requires a characterization of the algebraically smooth error that is poorly attenuated by the algorithmÕs relaxation process. For many relaxation processes of interest, algebraically smooth error corresponds to the near null-space of A. An eigenvector corresponding to the smallest eigenvalue of A, or minimal eigenvector, is an essential part of the near-nullspace of A. Therefore, having an approximate minimal eigenvector can be useful to characterize the algebraically smooth error when forming the SA method. This talk discusses 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 might be very useful as a stand-alone eigensolver for applications that desire an approximate minimal eigenvector, but the intention here is to use GES-SA to produce an initial algebraically smooth eigenvector that can be used to initialize SA or an adaptive version of SA