Keywords: parallel preconditioning, iterative solvers, GMRES, multi-colouring, graphics processing units, multi-core CPU, convection-diffusion equation.

platform-specific implementations aggravate flexible and portable implementations and impede programmer productivity. Second, increasing core counts require fine-grained parallelism in the numerical schemes and algorithms. All software development needs to be designed towards scalability with respect to high core counts. This situation becomes particularly apparent for preconditioning techniques, where classical approaches are typically sequential or only have a limited degree of parallelism.

Our proposed concept of a multi-platform linear algebra toolbox implemented within the framework of the HiFlow₃ finite element package [1] is tackling both the aspects described. It provides unified interfaces to highly optimized implementations on diverse hardware platforms including multicore CPUs and GPUs [2]. It allows modular building of solvers for typical numerical applications. It provides cross-platform portability, flexible utilization of resources at run time, and scalable fine-grained parallelism.

In this work we detail the concept of our hybrid and portable approach for efficient parallel finite element software designed in the HiFlow₃ project. To this end, we examine a typical scenario in numerical simulation by means of a convection-diffusion equation revealing challenges for preconditioning for non-symmetric problems. Based on a two-level library with backends to multi-core CPUs and multi-GPUs, we investigate performance and efficiency of parallel preconditioning techniques for the GMRES solver based on multi-colouring and matrix reorderings for incomplete decompositions and Gauss-Seidel splitting schemes. This work extends our results for preconditioning of symmetric conjugate gradient solvers [3] on single GPUs and multicore CPUs. Our approach proves efficiency and scalability across hybrid multi-core and GPU platforms where parallelism is exploited on the level of basic linear algebra routines. Our multi-coloring schemes show considerable performance improvements, especially for the dual GPU configuration. It is a robust solution and applicable to a large class of problems, in particular to non-symmetric systems. Our methodology can be applied as an out-of-the-box approach for any matrix originating in the context of finite elements.

1

H. Anzt et al., "HiFlow₃ - A Flexible and Hardware-Aware Parallel Finite Element Package", Workshop on Parallel/High-Performance Object-Oriented Scientific Computing (POOSC10), Reno, 2010, to appear.

2

V. Heuveline, C. Subramanian, D. Lukarski, J.-P. Weiss, "A Multi-Platform Linear Algebra Toolbox for Finite Element Solvers on Heterogeneous Clusters", PPAAC'10, IEEE Cluster 2010 Workshops, Heraklion, 2010. doi:10.1109/CLUSTERWKSP.2010.5613084

3

V. Heuveline, D. Lukarski, J.-P. Weiss, "Scalable Multi-Coloring Preconditioning for Multicore CPUs and GPUs", Euro-Par 2010 Parallel Processing Workshops, UCHPC'10, 2010, to appear.

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