A parallel Poisson/Helmholtz solver using local boundary integral equation and random walk methods

使用局部边界积分方程和随机游走方法的并行泊松/亥姆霍兹求解器

• 批准号: 1315128
• 负责人: Wei Cai
• 金额: $ 14万
• 依托单位: University of North Carolina at Charlotte
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-09-01 至 2015-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1315128&HistoricalAwards=false
• 关键词: parallel; Poisson; Helmholtz; solver; using

The objective of this project is to develop a new type of scalable elliptic solvers with high parallel scalability, and a fundamentally new approach in solving Poisson or modified Helmholtz equations in 3-D is proposed. The solution of these equations constitutes a major computational cost for many computational engineer problems such as incompressible flows by projection methods, electrostatic potential problems in molecular biology, and enforcing divergence free constraints of magnetic field in plasma MHD simulations, etc. The approach proposed is based on combining deterministic local boundary integral equation methods and random Brownian walk probabilistic representations of PDE solutions, resulting in straightforward parallel non-iterative solvers for the Dirichlet-to Neumann mappings of the elliptic PDEs, thus the complete solutions of the PDEs with the help of the FMM. The high performance computers nowadays use many cores in the order of hundreds of thousands designed for parallel implementations. The challenging for algorithms designers is to develop highly scalable and parallel methods to solve the mathematical equations coming from the representations of real world science and engineering problems. The development of the proposed algorithm in this project is a step toward to achieving such a degree of scalability and parallelism for problems, such as flow-structure interactions and electrostatics in computational biology and plasmas. The idea of using both random and deterministic methods in the proposed method is fundamentally different from traditional purely deterministic methods such as multi-grid and domain decomposition methods, and has the potential to produce high impact in the field of scientific and engineering computing at extreme scale.

该项目的目标是开发一种新型的可扩展椭圆解算器，具有高度的并行可扩展性，并提出了一种从根本上解决泊松或修改的亥姆霍兹方程在3-D的新方法。这些方程的解构成了许多计算工程问题的主要计算成本，例如通过投影方法的不可压缩流动，分子生物学中的静电势问题，以及等离子体MHD模拟中磁场的强制发散自由约束，所提出的方法是基于结合确定性局部边界积分方程方法和随机布朗行走概率表示的偏微分方程的解，从而直接并行非迭代求解椭圆偏微分方程的Dirichlet-to-Neumann映射，从而完全解决方案的帮助下的FMM。 当今的高性能计算机使用许多核，数量级为数十万，设计用于并行实现。算法设计者面临的挑战是开发高度可扩展和并行的方法来求解来自真实的世界科学和工程问题的数学方程。在这个项目中提出的算法的发展是朝着实现这种程度的可扩展性和并行性的问题，如计算生物学和等离子体中的流-结构相互作用和静电。在所提出的方法中同时使用随机和确定性方法的想法与传统的纯确定性方法（如多重网格和区域分解方法）有着根本的不同，并且有可能在极端规模的科学和工程计算领域产生很大的影响。