RTG: Randomized Numerical Analysis

North Carolina State University, Raleigh NC

Investigators

Abstract

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.