A Unified Framework for Large Scale Scientific Computing

本研究为椭圆型偏微分方程的可扩展、高效求解建立了统一的计算框架。研究人员开发了一种新的高阶多尺度多网格计算方法，该方法将高精度计算和快速计算方法无缝结合。这项研究工作可能会影响许多计算科学和工程以及工业建模和仿真应用。随着美国高科技产业从基于实验的设计和开发转向计算机辅助设计和开发，高性能的数值方法和更快的计算机模拟技术将使美国工业受益，使设计和开发工程师能够在进行昂贵的实验之前在计算机上进行快速验证，以测试他们的新想法。这些技术对于美国工业在竞争激烈的世界市场上保持其领导地位至关重要。研究生，包括来自代表性不足群体的成员，被训练成为具有扎实科学计算技能的下一代研究人员和教育工作者。该方法同时在两个方面推进了偏微分方程的数值解。一种是利用高阶离散化方法计算高精度解，另一种是利用最快的稀疏线性系统求解器在最少的计算机时间内计算离散解。这个统一的框架通过融合多尺度离散化和多网格计算的思想和优势，共同推进了这两个前沿，以达到以最小的计算机成本计算精确数值解的最终目标。它是许多研究人员多年来在几个不同领域的研究工作的汇集。该计算框架具有高精度、高速度、高可扩展性，为椭圆型偏微分方程数值解的计算提供了最佳效率。

This research builds a unified computational framework for scalable and high efficiency solution of elliptic partial differential equations. The investigators develop a novel high-order multiscale multigrid computation methodology, which combines high accuracy computation and fast computing methods in a seamless way. This research work may impact many computational science and engineering and industry modeling and simulation applications. As U.S. high-tech industry moves from experiment-based design and development to computer-assisted design and development, higher performance numerical methods and faster computer simulation techniques will benefit U.S. industry by enabling design and development engineers to conduct quick verification to test their new ideas on computers, before committing to expensive experiments. These technologies are essential for the U.S. industry to maintain its leadership position in the competitive world market. Graduate students, including members from underrepresented groups, are trained to become the next generation researchers and educators with solid scientific computing skills. The technique simultaneously advances the numerical solution of partial differential equations in two fronts. One is to compute high accuracy solution by using high-order discretization methods, another is to compute the discrete solution in a minimum amount of computer time by using the fastest sparse linear system solvers. This unified framework advances the two fronts collectively by fusing the ideas and advantages of multiscale discretization and multigrid computations, to achieve the ultimate goal of computing accurate numerical solution at the minimum computer costs. It is the convergence of years of research work by many researchers in several different areas. This computational framework possesses high accuracy, high speed, high scalability, and delivers optimal efficiency for computing the numerical solution of elliptic partial differential equations.