Title: Adaptive smoothed aggregation for nonsymmetric problems
Speaker:
Geoffrey Sanders, University of Colorado, Boulder
Date/Time: Monday, September 14, 2009 at 10:30am
Location: CSRI Building, Room 90 (Sandia NM)
Brief Abstract: Multigrid methods are known to provide optimal solvers for a large class of these problems. The favorable convergence properties of these methods stem from combining two complementary error-reduction processes: a local relaxation process and a coarse-grid correction. It is expected that the relaxation process efficiently attenuates most of the error components. The remaining error is referred to as algebraically smooth with respect to the given relaxation. The aim is then to construct coarser spaces with much smaller dimension and sparse bases to accurately represent the algebraically smooth error. Choosing a relaxation with good smoothing properties and constructing coarse subspaces with adequate approximation properties (and the associated intergrid transfer operators) are the general goals when designing a multigrid method.
Instead of making assumptions regarding the geometric smoothness of algebraically smooth error, the coarsening techniques considered here focus on constructing coarse spaces within the smoothed aggregation (SA) framework, where interpolation operators are chosen to accurately approximate a certain set of algebraically smooth prototypes. Previously, adaptive smoothed aggregation (aSA) was successfully applied to symmetric applications where a representative set algebraically smooth error vectors is neither obvious nor supplied.
This talk discusses the design of an aSA framework for nonsymmetric problems, based on the singular value decomposition of A, that reduces to the original aSA framework for SPD problems. Within this framework, we provide a nonsymmetric version of the strong approximation property, which is commonly employed for convergence analysis for SPD matrices. Preliminary two-level convergence analysis is presented, suggesting more relaxation should be applied at each level within a multigrid solver for highly nonsymmetric problems. Several features new to the SA framework are explored and developed in the design of a nonsymmetric version of aSA. We report numerical results that show our framework successfully creates optimal aSA solvers for certain nonsymmetric problems.
CSRI POC: Jonathan Hu, (925) 294-2931 |