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.

该项目的目的是开发具有高平行可伸缩性的新型可扩展椭圆求解器，并提出了一种在3-D中解决泊松或修改后的Helmholtz方程的新方法。这些方程式的解决方案构成了许多计算工程师问题的主要计算成本，例如通过投影方法，分子生物学中的静电潜在问题以及在等离子体MHD模拟中对磁场的无差约束等实施磁场的差异的方法等。所提出的方法是基于确定性的局部整体方程方法的方法，椭圆形PDE的Dirichlet到noumann映射的非文字求解器，因此在FMM的帮助下，PDE的完整解。 如今，高性能计算机使用许多用于并行实现的核心。算法设计师的挑战性是开发高度可扩展和平行的方法，以解决来自现实世界科学和工程问题表示的数学方程。该项目中提出的算法的开发是迈向实现问题的程度可扩展性和并行性的一步，例如计算生物学和等离子体中的流程结构相互作用和静电。在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粹确定性方法（例如多网格和域分解方法）根本不同，并且有可能在极端规模的科学和工程计算领域产生高影响。