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地区会议，该会议的主题是数值求解线性偏微分方程(PDE)的所谓“快速方法”。这一领域的研究最近取得了很快的进展，会议的目的是向初级研究人员介绍最新成果，并让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”)解算器，它在一次扫描中构造所考虑方程的近似解，(2)加速某些线性代数计算的新的随机化方法，以及(3)直接解算器和高阶离散化技术之间的相互作用，它使偏微分方程组的解的精度达到十位数或更高，甚至在复杂的几何形状上也是如此。会议期间讨论的研究的更广泛的动机是物理现象的计算模拟在科学和工程中日益重要。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也同样关键。所支持的CBMS会议将传播关于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，通信正在成为最重要的计算瓶颈。这次会议除了传统的讲座外，还将为与会者提供使用教程风格的软件进行动手计算实验的机会。讲座将被录制下来，并将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，以创建一套综合资源，最大限度地扩大会议的影响和影响。该项目将由数学科学部的基础设施计划和计算数学计划共同支持。

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.