Towards Adaptive Smoothed Aggregation (AlphaSA) for Nonsymmetric Problems

Applying smoothed aggregation (SA) multigrid to solve a nonsymmetric linear system, $A\mathbf{x} =\mathbf{b}$, is often impeded by the lack of a minimization principle that can be used as a basis for the coarse-grid correction process. This paper proposes a Petrov-Galerkin (PG) approach based on applying SA to either of two symmetric positive definite (SPD) matrices, $\sqrt{A^{t}A}$ or $\sqrt{AA^{t}}$. These matrices, however, are typically full and difficult to compute, so it is not computationally efficient to use them directly to form a coarse-grid correction. The proposed approach approximates these coarse-grid corrections by using SA to accurately approximate the right and left singular vectors of $A$ that correspond to the lowest singular value. These left and right singular vectors are used to construct the restriction and interpolation operators, respectively. A preliminary two-level convergence theory is presented, suggesting that more relaxation should be applied than for an SPD problem. Additionally, a nonsymmetric version of adaptive SA ($\alpha$SA) is given that automatically constructs SA multigrid hierarchies using a stationary relaxation process on all levels. Numerical results are reported for convection-diffusion problems in two dimensions with varying amounts of convection for constant, variable, and recirculating convection fields. The results suggest that the proposed approach is algorithmically scalable for problems coming from these nonsymmetric scalar PDEs (with the exception of recirculating flow). This paper serves as a first step for nonsymmetric $\alpha$SA. The long-term goal of this effort is to develop nonsymmetric $\alpha$SA for systems of PDEs, where the SA framework has proven to be well suited for adaptivity in SPD problems.