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Approximation of BEM matrices using GPGPUs

Steffen Borm, Sven Christophersen
Universitat Kiel
arXiv:1510.07244 [cs.MS], (25 Oct 2015)

@article{borm2015approximation,

   title={Approximation of BEM matrices using GPGPUs},

   author={Borm, Steffen and Christophersen, Sven},

   year={2015},

   month={oct},

   archivePrefix={"arXiv"},

   primaryClass={cs.MS}

}

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The efficiency of boundary element methods depends crucially on the time required for setting up the stiffness matrix. The far-field part of the matrix can be approximated by compression schemes like the fast multipole method or $mathcal{H}$-matrix techniques. The near-field part is typically approximated by special quadrature rules like the Sauter-Schwab technique that can handle the singular integrals appearing in the diagonal and near-diagonal matrix elements. Since computing one element of the matrix requires only a small amount of data but a fairly large number of operations, we propose to use GPUs to handle vectorizable portions of the computation: near-field computations are ideally suited for vectorization and can therefore be handled very well by GPUs. Modern far-field compression schemes can be split into a small adaptive portion that exhibits divergent control flows and is handled by the CPU and a vectorizable portion that can again be sent to GPUs. We propose a hybrid algorithm that splits the computation into tasks for CPUs and GPUs. Our method presented in this article is able to speedup the setup time of boundary integral operators by a significant factor of 19-30 for both the Laplace and the Helmholtz equation in 3D when using two consumer GPGPUs compared to a quad-core CPU.
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