FFT, FMM, or Multigrid? A comparative Study of State-Of-the-Art Poisson   Solvers for Uniform and Nonuniform Grids in the Unit Cube
  In this work, we benchmark and discuss the performance of the scalable methods for the Poisson problem which are used widely in practice: the fast Fourier transform (FFT), the fast multipole method (FMM), the geometric multigrid (GMG), and algebraic multigrid (AMG). In total we compare five different codes, three of which are developed in our group. Our FFT, GMG, and FMM are parallel solvers that use high-order approximation schemes for Poisson problems with continuous forcing functions (the source or right-hand side). We examine and report results for weak scaling, strong scaling, and time to solution for uniform and highly refined grids. We present results on the Stampede system at the Texas Advanced Computing Center and on the Titan system at the Oak Ridge National Laboratory. In our largest test case, we solved a problem with 600 billion unknowns on 229,379 cores of Titan. Overall, all methods scale quite well to these problem sizes. We have tested all of the methods with different source functions (the right-hand side in the Poisson problem). Our results indicate that FFT is the method of choice for smooth source functions that require uniform resolution. However, FFT loses its performance advantage when the source function has highly localized features like internal sharp layers. FMM and GMG considerably outperform FFT for those cases. The distinction between FMM and GMG is less pronounced and is sensitive to the quality (from a performance point of view) of the underlying implementations. The high-order accurate versions of GMG and FMM significantly outperform their low-order accurate counterparts. 
Amir Gholami, Dhairya Malhotra, Hari Sundar, George Biros
math.NA cs.NA