CBMS Conference: Algorithms for solving elliptic PDEs on modern computers---fast direct solvers, randomized methods, and high order discretizations,

该奖项支持将于2014年6月在达特茅斯学院举行的NSF/CBMS区域会议，主题是所谓的线性偏微分方程（PDEs）数值求解的“快速方法”。这一研究领域最近取得了非常迅速的进展，会议的目的是向初级研究人员介绍最新的研究成果，并使他们参与当前的研究。特别是，会议将描述(1)新的“直接”（相对于“迭代”）求解器，它可以在一次扫描中构造所考虑的方程的近似解，(2)加速某些线性代数计算的新的随机化方法，以及(3)直接求解器和高阶离散化技术之间的相互作用，这些技术可以使偏微分方程的求解精度达到十位数或更多，甚至在复杂的几何上。会议期间讨论的研究的更广泛动机是物理现象的计算模拟在科学和工程中的重要性日益增加。能够进行快速的虚拟实验而不是真实的（昂贵的）实验是技术进步的关键驱动力。虽然更强大的计算机很重要，但开发在现代硬件上运行良好的高效数值算法同样至关重要。支持的CBMS会议将传播有关新开发的强大算法的知识，这些算法从头开始设计，以最大限度地减少通信，这正在成为最重要的计算瓶颈。除了传统的讲座外，会议还将为参与者提供使用教程式软件进行动手计算实验的机会。讲座将被录下来，并与讲座幻灯片和教程软件一起发布在网上。将编写一本专著，以创造一套综合资源，最大限度地扩大会议的影响和范围。该项目将由数学科学部基础设施项目和计算数学项目共同支持。

The award supports an NSF/CBMS regional conference that will be held at Dartmouth College in June of 2014 on the topic of so called "fast methods" for numerically solving linear partial differential equations (PDEs). This field of research has recently seen very rapid progress, and the purpose of the conference is to introduce recent results to junior researchers, and to engage them with current research. In particular, the conference will describe (1) new "direct" (as opposed to "iterative") solvers that in a single sweep construct an approximate solution to the equation under consideration, (2) new randomized methods for accelerating certain linear algebraic computations, and (3) the interplay between direct solvers and high-order discretization techniques that allow the solution of PDEs to ten digits of accuracy or more, even on complicated geometries.The broader motivation for the research to be discussed during the conference is the growing importance of computational simulations of physical phenomena in science and engineering. The ability to perform rapid virtual experiments rather than real (expensive) ones is a key driver of technological progress. While more powerful computers are important, development of efficient numerical algorithms that run well on modern hardware is equally crucial. The CBMS conference supported will disseminate knowledge about newly developed powerful algorithms that are designed from the ground up to minimize communication, which is emerging as the most important computational bottleneck. The conference will in addition to traditional lectures also give participants an opportunity for hands-on computational experimentation using tutorial style software. The lectures will be recorded and will be posted online along with the lecture slides, and the tutorial software. A monograph will be produced to create an integrated set of resources that will maximize the impact and reach of the conference. This project will be jointly supported by the Infrastructure Program and the Computational Mathematics Program within the Division of Mathematical Sciences.