RTG: Randomized Numerical Analysis

科学计算的经典方法的设计自然是为了找到确切问题的确切答案。这些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方法的发展，而不是旨在提供近似的答案近似的问题，从而打开了大门，为新一代的数值工具，以及适应21世纪世纪的问题。该计划为本科生，研究生和博士后参与者提供培训机会，他们从垂直结构的工作组中受益。重要的是科学，技术工程和数学（STEM）的学生要有专业技能，超越技术专业知识;他们必须能够传达他们的结果，以明确的，引人注目的和吸引人的方式非技术观众。通过强调多层次的工作组，该计划为参与者提供了一个主要的培训场所，以获得弥合学科鸿沟所需的沟通技能，这是解决大多数社会重大挑战所需的工作。该计划还涉及新的课程材料的开发，无论是在网上和校园，反映和解决当今科学计算的挑战。数值分析的范式作为连续数学问题的算法研究需要更新。越来越多的数据密集型应用程序更好地描述通过离散数学的图形或网络，而不是通过连续数学的光滑流形。此外，目前的计算模型往往既不是适定的，也没有良好的条件，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。本研究围绕数值线性代数、非线性求解器和全局灵敏度分析三个互补的方向展开。通过分析随机化或损坏的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：甚至数值解的概念也必须重新审视。第三个目标是通过新颖的敏感性分析方法和使用替代模型来降低模型的复杂性;这个目标既利用了前两个目标，又对前两个目标做出了贡献。这个奖项反映了NSF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.