Auto-tuned Krylov Methods for Sparse Matrix Computation on Cluster of Accelerated Multicore Processo

Exascale hypercomputers are expected to have highly hierarchical architectures with nodes composed by lot-of-core processors and accelerators. The different programming levels (from clusters of processors loosely connected to tightly connected lot-of-core processors and/or accelerators) will generate new difficult algorithm issues. New methods should be defined and evaluated with respect to modern state-of-the-art of scientific methods.
Hybrid Krylov Methods would allow reducing the communications along all the cores, limiting the reduction only through subsets of these cores. Added to their numerical behaviors and their fault tolerance properties, these methods are interesting candidates for Exascale matrix computing. Avoiding communication (AC) strategies may also be developed for each of the instance of these methods, generating complex methods but with high potential efficiencies. These methods have a lot of correlated parameters which may be optimized using auto-tuning strategies to accelerate convergence, minimize storage space and energy consumption.
In this talk, we first will present some basic matrix operations utilized on Krylov methods on GPU-accelerated clusters, with respect to a few choose sparse compressed formats. Then, we will discuss some results obtained on a cluster of GPU to compute eigenvalues using the MERAM method with respect to the restarting strategies. We will also analyze some experiments to build a Krylov basis using AC strategies on a cluster of GPU. As a conclusion, we will propose auto-tuning strategies for future hybrid methods on post-Petascale computers, on the road to Exascale hybrid methods.
Joint work with:
• Nahid Emad, University of Versailles, France
• Christophe Calvin, France Boillod, and Jérome Dubois, CEA Saclay France
• Tony Drummond, and Valentin Senicourt, LBNL, USA
• Maxime Hugues, INRIA Saclay, France
• Pierre-Yves Aquilenti, Agency for Science, Technology and Research, Singapore