ITR: Collaborative Research: Solving PDEs Using Low Separation-Rank Representations and Optimal Quadratures for Exponentials

University Of Maryland, College Park, College Park MD

Investigators

Abstract

This project develops time-domain solvers for wave propagation problems in two and three dimensions, with fundamentally improved properties, namely, significantly reduced sampling requirements and, at the same time, significantly higher accuracy. Such solvers would allow modeling of linear and, eventually, nonlinear wave propagation in domains that are thousands of characteristic wavelengths in size, with interfaces and variable coefficients. This approach would also provide improved bases for solving nonlinear advection-diffusion problems. The solvers are built upon two new techniques, namely, optimal quadratures to represent bandlimited functions, and a numerical generalization of separation of variables to accelerate applying higher-dimensional operators. Each technique contains the potential to significantly advance computational science across a wide range of applications. Together they provide a new paradigm that efficiently organizes the information contained in operators governing physical phenomena. This project develops these techniques further, and develops multiresolution representations for operators and functions, based on the optimal quadratures. Any computational modeling of natural phenomena requires discretization of the underlying mathematical equations. This project addresses questions of optimality and efficiency of such discretizations, and of organization of information for two important modeling areas, wave propagation and geophysical fluid dynamics. This research aims to generate a much wider use of efficient techniques for representing information in scientific modeling, and to increase the speed and accuracy of simulations by up to two orders of magnitude, with foreseeable benefits to such areas as seismology, remote sensing, acoustics, optics, geophysical fluid dynamics, and quantum chemistry. It would reduce the computational cost of obtaining high accuracy, which is necessary to describe phenomena that are highly sensitive to changes in physical parameters, and that often cause technological bottlenecks. This Collaborative Research is led by the Department of Applied Mathematics at the University of Colorado at Boulder, and includes the Department of Meteorology at the University of Maryland at College Park. It will fund one research associate, two visiting scientists from UMD and Ohio University, and one graduate student in a diverse research group that includes a research associate, two postdoctoral researchers, a graduate student, and four undergraduate students.