Acceleration of derivative calculations with application to radial basis function
Gordon Erlebacher, Érik Saule, Natasha Flyer, Evan F. Bollig · 2014
In this paper, we develop an efficient scheme for the cal- culation of derivatives within the context of Radial Ba- sis Function Finite-Difference (RBF-FD). RBF methods express functions as a linear combination of spherically symmetric basis functions on an arbitrary set of nodes. The Finite-Difference component expresses this combi- nation over a local set of nodes neighboring the point where the derivative is sought. The derivative at all points takes the form of a sparse matrix/vector multiplication (SpMV). In this paper, we consider the case of local stencils with a fixed number of nodes at each point and encode the sparse matrix in ELLPACK format. We increase the number of operations relative to memory bandwidth by interleaving the calculation of four derivatives of four different functions, or 16 different derivatives. We demonstrate a novel implementation on the Intel MIC archi- tecture, taking into account its advanced swizzling and channel interchange features. We present benchmarks on a real data set that show an almost sevenfold in- crease in speed compared to efficient implementations of a single derivative, reaching a performance of almost 140 Gflop/s in single precision. We explain the results through consideration of operation count versus memory bandwidth.