Exponential Integrators on Graphics Processing Units

Lukas Einkemmer, Alexander Ostermann
Department of Mathematics, University of Innsbruck, Innsbruck, Austria
arXiv:1309.4616 [cs.NA], (18 Sep 2013)
@article{2013arXiv1309.4616E,

   author={Einkemmer}, L. and {Ostermann}, A.},

   title={"{Exponential Integrators on Graphic Processing Units}"},

   journal={ArXiv e-prints},

   archivePrefix={"arXiv"},

   eprint={1309.4616},

   primaryClass={"cs.NA"},

   keywords={Computer Science – Numerical Analysis},

   year={2013},

   month={sep},

   adsurl={http://adsabs.harvard.edu/abs/2013arXiv1309.4616E},

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

}

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In this paper we revisit stencil methods on GPUs in the context of exponential integrators. We further discuss boundary conditions, in the same context, and show that simple boundary conditions (for example, homogeneous Dirichlet or homogeneous Neumann boundary conditions) do not affect the performance if implemented directly into the CUDA kernel. In addition, we show that stencil methods with position-dependent coefficients can be implemented efficiently as well. As an application, we discuss the implementation of exponential integrators for different classes of problems in a single and multi GPU setup (up to 4 GPUs). We further show that for stencil based methods such parallelization can be done very efficiently, while for some unstructured matrices the parallelization to multiple GPUs is severely limited by the throughput of the PCIe bus.
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