SkvG: Cache-Optimal Parallel Solution of PDEs on High Performance Computers Using Space-Trees and Space-Filling Curves

Abstract

Facing the problem of implementing an efficient solver for partial differential equations, we are, in general, confronted with a certain quandary between numerical efficiency and efficiency in the usage of hardware resources: Modern numerical methods require the handling of hierarchical multilevel data on adaptively refined data structures, which are mostly represented by trees. On the other hand, as data access is one of the most important bottlenecks in high performance computing, we would wish to process data linearly with a high locality in time and space to be able to exploit the capability of cache hierarchies. In this paper, we show an approach based on space-filling curves as an odering mechanism for the cells of space-tree grids, with the help of which we can transform our (inherently highly non-local) data respresentation by trees to a few linearly processed data sets. As a consequence, we reach extremely high cache hit-rates above 99, 9%. In addition, the used methods make both parallelization and multigrid algorithms on adaptive grids with hierarchical data very straightforward and efficient.