Geoffrey D. Sanders
Convergence Theory for Nonsymmetric Smoothed Aggregation Multigrid

Department of Applied Mathematics
526 UCB
University of Colorado
Boulder
CO 80309-0526
sandersg@colorado.edu
Thomas A. Manteuffel
Stephen F. McCormick

Applying smoothed aggregation multigrid (SA) to solve nonsymmetric linear systems, $A {\bf x} = {\bf b}$, can be problematic due to a lack of minimization principle in the coarse-grid corrections. We propose an approach that is based on approximately applying SA to the symmetric positive definite matrices $\sqrt{A^* A}$ or $\sqrt{A A^*}$. These matrices, however, are typically full and difficult to compute, and it is therefore not computationally efficient to use these matrices directly to form a coarse-grid correction. Our proposed approach approximates these coarse-grid corrections by using smoothed aggregation to accurately approximate the right and left singular vectors of $A$ that correspond to the lowest singular value. These are used to construct the interpolation and restriction operators, respectively. We present some preliminary two-level convergence theory and numerical results.