Fast evaluation of Helmholtz potential on graphics processor units ( GPUs )

Non-uniform grid method (NGM) is a fast algorithm to accelerate the integral equation based method of static and dynamic field evaluation in various areas such as electromagnetics, optics, magnetics etc. The NG method reduce the computational complexity of direct evaluation of interaction between N unknowns from 2 ( ) O N to ( ) O N in static and low-frequency regime and ( log ) O N N in the high-frequency regime. Compare with other fast algorithm like FMM, the NGM also provides adaptivity to sparse problems, mixed frequency problems and problems with reduced dimension or non-uniform geometries. Graphic processing units (GPU) as an emerging hardware platform to computational science community provide researchers cluster-level computational power at an exceptionally cheap price and remarkable flexibility. As the NGM relies on spatial integral kernel smoothing and interpolations combined with hierarchal domain decomposition to provide its remarkable speed and adaptivity, it is not readily to be migrated on to GPU platform. The NGM code should be parallelized and sometimes rewritten completely from another perspective in order to utilize all computational resources and detour the weak points of GPU which are originally expected to handle graphic applications. After significant parallelization and modification are made, our GPU implementation illustrated in this paper can handle problem with up to 16 millions of unknowns and under various accuracy up to 120 times faster than CPU version of NGM and a few million times faster compared with direct computation.