Petascale elliptic solvers for anisotropic PDEs on GPU clusters

Eike Hermann Muller, Robert Scheichl, Benson Muite, Eero Vainikko
Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, United Kingdom
arXiv:1402.3545 [cs.DC], (14 Feb 2014)


   author={M{"u}ller}, E.~H. and {Scheichl}, R. and {Muite}, B. and {Vainikko}, E.},

   title={"{Petascale elliptic solvers for anisotropic PDEs on GPU clusters}"},

   journal={ArXiv e-prints},




   keywords={Computer Science – Distributed, Parallel, and Cluster Computing, Computer Science – Numerical Analysis, Mathematics – Numerical Analysis, 65Y05 (Primary), 65N55 (Secondary), G.1.0, G.1.8, C.1},




   adsnote={Provided by the SAO/NASA Astrophysics Data System}


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Memory bound applications such as solvers for large sparse systems of equations remain a challenge for GPUs. Fast solvers should be based on numerically efficient algorithms and implemented such that global memory access is minimised. To solve systems with up to one trillion (10^12) unknowns the code has to make efficient use of several million individual processor cores on large GPU clusters. We describe the multi-GPU implementation of two algorithmically optimal iterative solvers for anisotropic elliptic PDEs. For this we parallelise the solvers and adapt them to the specific features of GPU architectures, paying particular attention to efficient global memory access. We achieve a performance of up to 0.78 PFLOPs when solving an equation with 0.55×10^12 unknowns on 16384 GPUs; this corresponds to about 3% of the theoretical peak performance of the machine and we use more than 40% of the peak memory bandwidth with a Conjugate Gradient (CG) solver. Although the other solver, a geometric multigrid algorithm, has a slightly worse performance in terms of FLOPs per second, overall it is faster as it needs less iterations to converge; the multigrid algorithm can solve a linear PDE with half a trillion unknowns in about one second.
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