Approximation of boundary element matrices using GPGPUs and nested cross approximation

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 general-purpose graphics processing units (GPGPUs) to handle vectorizable portions of the computation: near-field computations are ideally suited for vectorization and can therefore be handled very well by GPGPUs. Modern far-field compression schemes can be split into a small adaptive portion that exhibits divergent control flows, and should therefore be handled by the CPU, and a vectorizable portion that can again be sent to GPGPUs. We propose a hybrid algorithm that splits the computation into tasks for CPUs and GPGPUs. Our method presented in this article is able to reduce 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.