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.

本项目的目标是开发一种具有高并行可伸缩性的新型可伸缩椭圆求解器，并提出了一种求解三维Poisson方程或修正的Helmholtz方程的全新方法。这些方程的求解构成了许多计算工程问题的主要计算代价，如不可压缩流的投影方法、分子生物学中的静电势问题、等离子体MHD模拟中的磁场无散度约束等。该方法基于确定性局部边界积分方程法和随机布朗游动概率表示法相结合，得到椭圆型偏微分方程组的Dirichlet-to Neumann映射的直接并行非迭代求解器，从而在FMM的帮助下得到偏微分方程组的完整解。如今的高性能计算机使用了许多为并行实现而设计的数十万个内核。算法设计者面临的挑战是开发高度可扩展的并行方法来求解来自现实世界科学和工程问题表示的数学方程。这个项目中提出的算法的发展是朝着实现计算生物学和等离子体中的流-结构相互作用和静电学等问题的可伸缩性和并行性迈出的一步。该方法同时使用随机方法和确定性方法的思想与传统的单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，并且有可能在极端尺度的科学和工程计算领域产生巨大的影响。