A cache‐oblivious self‐adaptive full multigrid method

Abstract

SUMMARYThis paper presents a new e cient way to implement multigrid algorithms on adaptively reÿned grids. To cope with todays demands in high-performance computing, we cannot do without such highly sophisticated numerical methods. But if we do not implement them very carefully, we lose a lot of e ciency in terms of memory usage: using trees for the storage of hierarchical multilevel data causes a large amount of non-local (in terms of the physical memory space) data accesses, and often requires the storage of pointers to neighbours to allow the evaluation of discrete operators (di erence stencils, restrictions, interpolations, etc.). The importance of this problem becomes clear if we remember that storage and not the CPUs is the bottleneck on modern computers.We established a cache-oblivious and storage-minimizing algorithm based on the concept of spacetree grids combined with a cell-oriented operator evaluation, a linear ordering of grid cells along a space-ÿlling curve, and a sophisticated construction of linearly processed data structures for vertex data. In this context, we could show that the implementation of a dynamically adaptive F-cycle is, ÿrst, very natural and, second, does not cause any overhead in terms of storage usage and access as adaptivity and multilevel data do not disturb the linear processing order of our data structures.

Complex Additive Geometric Multilevel Solvers for Helmholtz Equations on Spacetrees

“…Due to the single touch policy per unknown and the localised traversal, we can assume that each unknown is loaded into the cache exactly once. This insight mirrors previous reports on this algorithmic paradigm [Mehl et al 2006;Weinzierl 2009;Weinzierl and Mehl 2011]. We validate this at hands of the cache miss rate explaining the low bandwidth requirements.…”

“…The right-hand side is not required earlier throughout the solve process. The overall process is detailed in [Mehl et al 2006] and reiterated in Appendix B. It is completely matrix-free.…”

“…in the grid, integrated. Similar techniques have been proposed for multilevel solvers [Adams et al 2016;Mehl et al 2006;Ghysels et al 2012;Ghysels and Vanroose 2015] or Krylov solvers [Chronopoulos and Gear 1989;Hoemmen 2010;Ghysels et al 2013;Ghysels and Vanroose 2014], but, to the best of our knowledge, no other approach offers a solution representation on all levels plus single touch. Multilevel solution representations simplify the handling of hanging nodes, non-linear problems and scale-dependent discretisations [Cools et al 2014b].…”

Efficiency considerations in triangular adaptive mesh refinement

“…The method described in this paper can only be applied to meshes created by hierarchical triangular bisection. A similar method for quadrilateral meshes, refined by edge bisection of the elements, has been proposed in Mehl et al (2006).…”

Memory efficient adaptive mesh generation and implementation of multigrid algorithms using Sierpinski curves

“…Recently, our research group have presented a family of methods based on recursively substructured grids (similar to octrees), which allow the implementation of iterative multigrid solvers on fully adaptive grids, but only require a marginal amount of memory-see Günther et al (2006) and Mehl et al (2006) for rectangular grids, and Bader and Zenger (2006) for triangular grids. For efficient processing on these grids, the presented schemes combine the use of space-filling curves (Peano curves or Sierpinski curves, respectively) and a sophisticated scheme of stacks, and thus implement iterative solvers in a way such that almost no topological information needs to be stored and that it is no longer necessary to explicitly store a system of equations.…”