  Nonequispaced discrete Fourier transformation (NDFT) is widely applied in all
aspects of computational science and engineering. The computational efficiency
and accuracy of NDFT has always been a critical issue in hindering its
comprehensive applications both in intensive and in extensive aspects of
scientific computing. In our previous work (2018, S.-C. Yang et al., Appl.
Comput. Harmon. Anal. 44, 273), a CUNFFT method was proposed and it shown
outstanding performance in handling NDFT at intermediate scale based on CUDA
(Compute Unified Device Architecture) technology. In the current work, we
further improved the computational efficiency of the CUNTTF method using an
efficient MPI-CUDA hybrid parallelization (HP) scheme of NFFT to achieve a
cutting-edge treatment of NDFT at super extended scale. Within this HP-NFFT
method, the spatial domain of NDFT is decomposed into several parts according
to the accumulative feature of NDFT and the detailed number of CPU and GPU
nodes. These decomposed NDFT subcells are independently calculated on different
CPU nodes using a MPI process-level parallelization mode, and on different GPU
nodes using a CUDA threadlevel parallelization mode and CUNFFT algorithm. A
massive benchmarking of the HP-NFFT method indicates that this method exhibit a
dramatic improvement in computational efficiency for handling NDFT at super
extended scale without loss of computational precision. Furthermore, the
HP-NFFT method is validated via the calculation of Madelung constant of
fluorite crystal structure, and thereafter verified that this method is robust
for the calculation of electrostatic interactions between charged ions in
molecular dynamics simulation systems.
