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

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This project develops time-domain solvers for wave propagation problemsin two and three dimensions, with fundamentally improved properties,namely, significantly reduced sampling requirements and, at the sametime, significantly higher accuracy. Such solvers would allow modelingof linear and, eventually, nonlinear wave propagation in domains thatare thousands of characteristic wavelengths in size, with interfaces andvariable coefficients. This approach would also provide improved basesfor solving nonlinear advection-diffusion problems. The solvers arebuilt upon two new techniques, namely, optimal quadratures to representbandlimited functions, and a numerical generalization of separation ofvariables to accelerate applying higher-dimensional operators. Eachtechnique contains the potential to significantly advance computationalscience across a wide range of applications. Together they provide anew paradigm that efficiently organizes the information contained inoperators governing physical phenomena. This project develops thesetechniques further, and develops multiresolution representations foroperators and functions, based on the optimal quadratures.Any computational modeling of natural phenomena requires discretizationof the underlying mathematical equations. This project addressesquestions of optimality and efficiency of such discretizations, and oforganization of information for two important modeling areas, wavepropagation and geophysical fluid dynamics. This research aims togenerate a much wider use of efficient techniques for representinginformation in scientific modeling, and to increase the speed andaccuracy of simulations by up to two orders of magnitude, withforeseeable benefits to such areas as seismology, remote sensing,acoustics, optics, geophysical fluid dynamics, and quantum chemistry. Itwould reduce the computational cost of obtaining high accuracy, which isnecessary to describe phenomena that are highly sensitive to changes inphysical parameters, and that often cause technological bottlenecks.This Collaborative Research is led by the Department of AppliedMathematics at the University of Colorado at Boulder, and includes theDepartment of Meteorology at the University of Maryland at CollegePark. It will fund one research associate, two visiting scientists fromUMD and Ohio University, and one graduate student in a diverse researchgroup that includes a research associate, two postdoctoral researchers,a graduate student, and four undergraduate students.