Accurate evaluation of local averages on GPGPUs

We discuss fast and accurate evaluation of local averages on GPGPU. This work was motivated by the need to calculate reference fluid densities in the classical density functional theory (DFT) of electrolytes proposed in Gillespie et al. (2002). In Knepley et al. (2010) we developed efficient algorithms for the minimization of three-dimensional DFT models of biological ion channel permeation and selectivity. One of the essential bottlenecks of 3D DFT models is the evaluation of local screening averages of the chemical species’ densities. But the problem has wider applicability and fast evaluation of averages over the local spherical screening neighborhood of every grid point are typically inaccurate due to the use of collocation approximations of densities on Cartesian grids. Accurate evaluations based spectral quadrature were proposed and used in Knepley et al. (2010), but they are significantly more computationally expensive because of their nonlocal nature in the Fourier space. Here we show that the passage to the Fourier space can, in fact, make the averaging calculation much more amenable to efficient implementation on GPGPU architectures. This allows us to take advantage of both improved accuracy and hardware acceleration to arrive at a fast and accurate screening calculations.