RTG: Randomized Numerical Analysis

RTG：随机数值分析

基本信息

批准号：
1745654
负责人：
Ilse C.F. Ipsen
金额：
$ 214万
依托单位：
North Carolina State University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-08-15 至 2024-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1745654&HistoricalAwards=false
关键词：
RTG Randomized Numerical Analysis

项目摘要

Classical methods from scientific computing were designed with the natural goal of finding exact answers to exact questions. Such methods cannot address most of today's large and complex computational models. This research training group focuses on the development of methods which aim instead at providing approximate answers to approximate questions thereby opening up the door for new generations of numerical tools well adapted to 21st century problems. The program provides training opportunities for undergraduate, graduate, and postdoctoral participants who benefit from their integration in vertically structured working groups. It is important for Science, Technology Engineering and Math (STEM) students to have professional skills extending beyond technical expertise; they must be able to communicate their results to non-technical audiences in clear, compelling and engaging ways. Through its emphasis on multi-layered working groups, the program offers a prime training ground for its participants to gain the communication skills necessary to bridge disciplinary divides, as is required for work addressing most of society's grand challenges. The program also involves the development of new course material, both online and on campus, that reflects and addresses challenges in present-day scientific computing.The paradigm of numerical analysis as the study of algorithms for the problems of continuous mathematics needs to be updated. An increasing number of data intensive applications are better described through discrete mathematics in terms of graphs or networks rather than through the smooth manifolds of continuous mathematics. Additionally, current computational models are often neither well-posed nor well-conditioned; new approaches are needed. The program addresses this pressing need by using randomization as the key scientific tool. The research is organized around three complementary thrusts in numerical linear algebra, nonlinear solvers and global sensitivity analysis. By analyzing the effect on numerical solutions of perturbations caused by randomization, or corrupted data, the first two thrusts fill a critical gap in the theoretical foundation to numerical analysis under large perturbations and low accuracy: even the notion of numerical solution has to be revisited. The third thrust aims at reducing model complexity through novel sensitivity analysis methods and the use of surrogate models; this thrust both capitalizes on and contributes to the previous two.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

科学计算的经典方法的设计自然目标是找到确切问题的确切答案。此类方法无法解决当今大多数大型且复杂的计算模型。该研究培训小组专注于开发方法，旨在为近似问题提供近似答案，从而为适应 21 世纪问题的新一代数值工具打开大门。该计划为本科生、研究生和博士后参与者提供培训机会，他们可以从垂直结构工作组的整合中受益。对于科学、技术、工程和数学 (STEM) 学生来说，拥有超越技术专业知识的专业技能非常重要；他们必须能够以清晰、引人注目和引人入胜的方式向非技术受众传达他们的结果。通过强调多层次的工作组，该计划为其参与者提供了一个主要的培训场地，以获得弥合学科分歧所需的沟通技能，这是解决大多数社会重大挑战的工作所需要的。该计划还涉及在线和校园新课程材料的开发，以反映和解决当今科学计算中的挑战。作为连续数学问题的算法研究的数值分析范式需要更新。越来越多的数据密集型应用程序可以通过图形或网络的离散数学更好地描述，而不是通过连续数学的平滑流形。此外，当前的计算模型通常既不适定也不适定。需要新的方法。该计划通过使用随机化作为关键的科学工具来解决这一迫切需求。该研究围绕数值线性代数、非线性求解器和全局敏感性分析三个互补的主旨进行组织。通过分析随机化或损坏数据引起的扰动对数值解的影响，前两个推力填补了大扰动和低精度下数值分析理论基础的关键空白：甚至必须重新审视数值解的概念。第三个重点旨在通过新颖的敏感性分析方法和替代模型的使用来降低模型的复杂性；这一推动力既利用了前两个奖项，又为前两个奖项做出了贡献。该奖项反映了 NSF 的法定使命，并且通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。