Neumann Series in GMRES and Algebraic Multigrid Smoothers

Abstract

Neumann series underlie both Krylov methods and algebraic multigrid smoothers. A low-synch modified Gram-Schmidt (MGS)-GMRES algorithm is described that employs a Neumann series to accelerate the projection step. A corollary to the backward stability result of Paige et al. (2006) demonstrates that the truncated Neumann series approximation is sufficient for convergence of GMRES. The lower triangular solver associated with the correction matrix $T_m = (\: I + L_m \:)^{-1}$ may then be replaced by a matrix-vector product with $T_m = I - L_m$. Next, Neumann series are applied to accelerate the classical R\"uge-Stuben algebraic multigrid preconditioner using both a polynomial Gauss-Seidel or incomplete ILU smoother. The sparse triangular solver employed in these smoothers is replaced by an inner iteration based upon matrix-vector products. Henrici's departure from normality of the associated iteration matrices leads to a better understanding of these series. Connections are made between the (non)normality of the $L$ and $U$ factors and nonlinear stability analysis, as well as the pseudospectra of the coefficient matrix. Furthermore, re-orderings that preserve structural symmetry also reduce the departure from normality of the upper triangular factor and improve the relative residual of the triangular solves. To demonstrate the effectiveness of this approach on many-core architectures, the proposed solver and preconditioner are applied to the pressure continuity equation for the incompressible Navier-Stokes equations of fluid motion. The pressure solve time is reduced considerably with no change in the convergence rate and the polynomial Gauss-Seidel smoother is compared with a Jacobi smoother. Numerical and timing results are presented for Nalu-Wind and the PeleLM combustion codes, where ILU with iterative triangular solvers is shown to be much more effective than polynomial Gauss-Seidel.