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

本项目的目标是开发一种具有高度并行可扩展性的新型可扩展椭圆求解器，并为三维泊松或修正亥姆霍兹方程的求解提供了一种全新的方法。这些方程的解构成了许多计算工程问题的主要计算成本，如投影法的不可压缩流、分子生物学中的静电势问题、等离子体MHD模拟中磁场的无散度约束等。该方法将确定性局部边界积分方程方法与PDE解的随机布朗游走概率表示相结合，得到了椭圆型PDE的Dirichlet-to - Neumann映射的直接并行非迭代解，从而在FMM的帮助下得到了PDE的完全解。现在的高性能计算机使用数以十万计的核来并行实现。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解决来自现实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问题迈出的一步，例如计算生物学和等离子体中的流结构相互作用和静电。在该方法中同时使用随机和确定性方法的思想与传统的纯确定性方法（如多网格和领域分解方法）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重大影响。

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.