FFT、FMM或Multigrid？单位立方体中均匀网格和非均匀网格Poisson求解器的比较研究
FFT, FMM, or Multigrid? A comparative Study of State-Of-the-Art Poisson Solvers for Uniform and Nonuniform Grids in the Unit Cube
摘要：
在这项工作中，我们基准测试和讨论的泊松问题，在实践中广泛使用的可扩展方法的性能：快速傅里叶变换（FFT），快速多极方法（FMM），几何多重网格（GMG），代数多重网格（AMG）。我们总共比较了五种不同的代码，其中三种是在我们组中开发的。我们的FFT，GMG和FMM是并行求解器，使用高阶近似方案的泊松问题与连续强制功能（源或右手边）。我们检查和报告结果弱缩放，强缩放，解决方案的时间均匀和高度精细的网格。我们目前的结果踩踏系统在得克萨斯州高级计算中心和泰坦系统在橡树岭国家实验室。在我们最大的测试案例中，我们解决了泰坦229，379个核心上的6000亿个未知数。总的来说，所有的方法都能很好地适应这些问题的大小。我们已经用不同的源函数测试了所有的方法（泊松问题的右边）。我们的研究结果表明，FFT是平滑源函数，需要统一的分辨率的选择方法。然而，当源函数具有高度局部化的特征（如内部尖锐层）时，FFT失去了其性能优势。对于这些情况，FMM和GMG的性能大大优于FFT。FMM和GMG之间的区别不太明显，并且对底层实现的质量（从性能的角度来看）很敏感。高阶精度版本的GMG和FMM显着优于其低阶精度的同行。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
