AIR multigrid with GMRES polynomials (AIRG) and additive preconditioners for Boltzmann transport

We develop a reduction multigrid based on approximate ideal restriction (AIR) for use with asymmetric linear systems. We use fixed-order GMRES polynomials to approximate Aff−1 and we use these polynomials to build grid transfer operators and perform F-point smoothing. We can also apply a fixed sparsity to these polynomials to prevent fill-in. When applied in the streaming limit of the Boltzmann Transport Equation (BTE), with a P0 DG FEM angular discretisation and a low-memory spatial discretisation on unstructured grids, preconditioning an outer GMRES iteration with a single V-cycle of this “AIRG” multigrid outperforms the ℓAIR implementation in hypre, with two to three times less work. We measure one “work unit” as the the FLOPs required to compute a single matrix-vector product. We also measure the growth in work required to perform the setup/solve with space/angle refinement by computing f in O(nf), where n is the number of space/angle unknowns; We find that AIRG is very close to scalable. We find f=1.092 for the setup and f=1.003 for the solve, with growth from 83 work units to 84 in the solve across six levels of spatial refinement; in comparison, distance 2 ℓAIR with FCF-Jacobi smoothing gives f=1.1 in the solve, with growth from 112 work units to 219. Using AIRG and fixed-sparsity GMRES polynomials gives f=1.076 for the setup and f=1.018 for the solve. Three levels of angular refinement with AIRG and fixed-sparsity GMRES polynomials gives f=1.055 in the solve. We then present an iterative method designed for use with scattering which uses the additive combination of two preconditioners applied to the angular flux; a single AIRG V-cycle on the streaming/removal operator and a single V-cycle of BoomerAMG from hypre on a DSA-type operator. We find our iterative method is very close to scalable, with f=1.022 and f=1.02 in the solve with spatial and angular refinement, respectively. Similarly the memory use with space/angle refinement grows with a maximum of f=1.006.