Relaxation methods such as Jacobi or Gauss-Seidel are often applied as smoothers in algebraic multigrid. Incomplete factorizations can also be employed, however, direct triangular solves are comparatively slow on GPUs. Previous work by Antz et al. [1] proposed an iterative approach for solving such sparse triangular systems. However, when using the stationary Jacobi iteration, if the upper or lower triangular factor is highly non-normal, the iterations will diverge. An ILUT smoother is introduced for classical Ruge-Stüben C-AMG that applies Ruiz scaling to mitigate the non-normality of the upper triangular factor. Our approach facilitates the use of Jacobi iteration in place of the inherently sequential triangular solve. Because the scaling is applied to the upper triangular factor as opposed to the global matrix, it can be done locally on an MPI-rank for a diagonal block of the global matrix. A performance model is provided along with numerical results for matrices extracted from the PeleLM [15] pressure continuity solver.
In many applications, linear systems arise where the coefficient matrix takes the special form I + K + E, where I is the identity matrix of dimension n, rank(K) = p ≪ n, and ∥E∥ ≤ ϵ < 1. GMRES convergence rates for linear systems with coefficient matrices of the forms I+K and I+E are guaranteed by well‐known theory, but only relatively weak convergence bounds specific to matrices of the form I + K + E currently exist. In this paper, we explore the convergence properties of linear systems with such coefficient matrices by considering the pseudospectrum of I + K. We derive a bound for the GMRES residual in terms of ϵ when approximately solving the linear system (I + K + E)x = b and identify the eigenvalues of I + K that are sensitive to perturbation. In particular, while a clustered spectrum away from the origin is often a good indicator of fast GMRES convergence, that convergence may be slow when some of those eigenvalues are ill‐conditioned. We show there can be at most 2p eigenvalues of I + K that are sensitive to small perturbations. We present numerical results when using GMRES to solve a sequence of linear systems of the form (I + Kj + Ej)xj = bj that arise from the application of Broyden's method to solve a nonlinear partial differential equation.
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