RTG: Randomized Numerical Analysis

科学计算的经典方法设计的自然目标是为精确的问题找到精确的答案。这样的方法不能解决当今大多数大型和复杂的计算模型。这个研究培训小组的重点是开发方法，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。该项目为本科生、研究生和博士后参与者提供培训机会，这些参与者受益于他们在垂直结构的工作组中的整合。对于科学、技术、工程和数学（STEM）专业的学生来说，拥有超越技术专长的专业技能是很重要的；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小组，该计划为参与者提供了一个主要的培训场地，以获得必要的沟通技巧，以弥合学科分歧，这是解决大多数社会重大挑战所需要的工作。该计划还包括开发新的在线和校园课程材料，以反映和解决当今科学计算中的挑战。作为连续数学问题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最好通过离散数学的图或网络来描述，而不是通过连续数学的光滑流形来描述。此外，当前的计算模型通常既不具备良好的定姿，也不具备良好的条件；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、非线性求解和全局灵敏度分析三个互补的方向进行。通过分析随机化或数据损坏引起的扰动对数值解的影响，前两次推力填补了大扰动和低精度下数值分析理论基础的一个关键空白：甚至数值解的概念也必须重新审视。第三个目标是通过新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；这一推力既利用了前两种推力，也促进了前两种推力。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.