ILU Smoothers for Low Mach Navier-Stokes Pressure Solvers

Abstract

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.

Neumann Series in GMRES and Algebraic Multigrid Smoothers

“…Once the transfer matrices are determined, the coarse-matrix representations are computed recursively from A through sparse triple-matrix multiplication. This describes a V -cycle, the simplest complete AMG cycle; refer to Algorithm 1 in the recent paper by Thomas et al [12] for a complete description. AMG methods achieve optimality (constant work per degree of freedom in A 0 ) through error reductions by the smoother and corrections propagated from coarser levels.…”

“…In [12], an extensive empirical analysis shows that for certain drop tolerances, and amounts of fill, U s 2 < 1. Thus, the convergence of the Neumann series is guaranteed.…”

“…Of course, for strictly upper triangular matrices, U p s is guaranteed to be the zero matrix when p = n, however, numerical nilpotence occurs much sooner (i.e., p n). Building a theoretical framework to describe the relationship between these metrics is part of our ongoing work, and we refer to [12] for more empirical analyses of U p s 2 . In particular, a relationship between larger drop tolerances and smaller fill-in when computing the ILU factorizations is observed when using row scaling.…”

“…In [12], a method for scaling the U factor to form U = I + U s in order to mitigate a high degree of non-normality and condition number is proposed. In general, fast convergence of the Jacobi iteration can then be expected [13] when iteratively computing the triangular solution, thus avoiding the comparatively high cost on the GPU of a direct triangular solve.…”

“…In general, fast convergence of the Jacobi iteration can then be expected [13] when iteratively computing the triangular solution, thus avoiding the comparatively high cost on the GPU of a direct triangular solve. Here, the analysis in [12] is extended and theoretical upper bounds on the departure from normality [14] of a (row and/or column) scaled triangular matrix are provided. In the case of nonsymmetric coefficient matrices, reordering prior to scaling may be employed to enforce a symmetric sparsity pattern.…”