Nonsymmetric Reduction-Based Algebraic Multigrid

Abstract

Algebraic multigrid (AMG) is often an effective solver for symmetric positive definite (SPD) linear systems resulting from the discretization of general elliptic PDEs, or the spatial discretization of parabolic PDEs. However, convergence theory and most variations of AMG rely on A being SPD. Hyperbolic PDEs, which arise often in large-scale scientific simulations, remain a challenge for AMG, as well as other fast linear solvers, in part because the resulting linear systems are often highly nonsymmetric. Here, a novel convergence framework is developed for nonsymmetric, reduction-based AMG, and sufficient conditions derived for 2 -convergence of error and residual. In particular, classical multigrid approximation properties are connected with reduction-based measures to develop a robust framework for nonsymmetric, reduction-based AMG.Matrices with block-triangular structure are then recognized as being amenable to reductiontype algorithms, and a reduction-based AMG method is developed for upwind discretizations of hyperbolic PDEs, based on the concept of a Neumann approximation to ideal restriction (nAIR). nAIR can be seen as a variation of local AIR ( AIR) introduced in previous work, specifically targeting matrices with triangular structure. Although less versatile than AIR, setup times for nAIR can be substantially faster for problems with high connectivity. nAIR is shown to be an effective and scalable solver of steady state transport for discontinuous, upwind discretizations, with unstructured meshes, and up to 6th-order finite elements, offering a significant improvement over existing AMG methods. nAIR is also shown to be effective on several classes of "nearly triangular" matrices, resulting from curvilinear finite elements and artificial diffusion.

Necessary Conditions and Tight Two-level Convergence Bounds for Parareal and Multigrid Reduction in Time

“…The restriction operator defined through Z = −A cf A −1 f f is referred to as "ideal restriction," denoted R ideal , where it is ideal in being the unique restriction operator that yields an exact coarse-grid correction at C-points. Following this with an exact solve on F-points as a relaxation scheme then yields an exact solution at F-points, without modifying the solution at C-points [25,26]. Thus, the solution is exact and we have a two-grid reduction, where solving Ax = b is reduced to solving one system based on A f f and one system based on RAP .…”

“…Thus, coarse-grid correction with P ideal and restriction by injection, preceded by an exact solve on F-points, also yields an exact two-level reduction [26]. 5 In the algebraic setting, A −1 f f is often not easily computed, so approximations are made, such as in AMG methods based on an approximate ideal restriction (AIR) [25,26]. MGRiT and the system in (1) are unique in that the action of A −1 f f can be computed, so ideal interpolation and exact F-relaxation are feasible choices.…”

“…The following section derives error and residual-propagation operators for MGRiT. Further details on reduction-based multigrid methods can be found in [24,25,26,31], and further details on the MGRiT algorithm can be found in, for example, [5,8,9,12]. The reduction properties rely on the ideal interpolation and restriction operators,…”

An Efficient Sweep-Based Solver for the S N Equations on High-Order Meshes

“…In (17) and 16, n is used in the reference element integration to emphasize that the reference element indexing in mesh element κ e is not the same as in κ e . The face Jacobian J Γ is derived through the dim − 1 versions of (3) and (15).…”

“…In particular, Gauss-Legendre quadrature is often used with great success to evaluate the polynomial (assuming constant material properties) integrands of (12)- (14). However, for linear mesh elements in 3D, and more generally HO mesh elements in multi-dimensional geometry, (16) and (17) e,e integrands are non-smooth on re-entrant faces where Ω d ·n e changes sign. In such cases standard quadrature schemes designed to integrate smoothly varying functions can converge slowly.…”