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

该项目开发了二维和三维波传播问题的时域求解器，具有根本性的改进性能，即显著降低了采样要求，同时具有显著更高的精度。 这样的求解器将允许线性和最终非线性波传播的域中的建模，这些域的大小是数千个特征波长，具有界面和可变系数。 这种方法也将为求解非线性对流扩散问题提供改进的基础。 求解器是建立在两个新的技术，即，最佳的quadratures表示bandlimited功能，和一个数值推广的分离的变量，以加速应用高维算子。 每一种技术都有可能在广泛的应用中大大推进计算科学。 它们一起提供了一种新的范式，有效地组织了管理物理现象的操作符中包含的信息。 本项目进一步发展了这些技术，并在最优求积的基础上开发了算子和函数的多分辨率表示。任何自然现象的计算建模都需要对底层数学方程进行离散化。该项目解决了这种离散化的最优性和效率问题，以及波传播和地球物理流体动力学两个重要建模领域的信息组织问题。 这项研究旨在更广泛地使用有效的技术来表示科学建模中的信息，并将模拟的速度和准确性提高两个数量级，对地震学，遥感，声学，光学，地球物理流体动力学和量子化学等领域带来可预见的好处。它将减少获得高精度的计算成本，这是必要的描述现象，是高度敏感的物理参数的变化，往往会导致技术瓶颈。这项合作研究是由数学系在博尔德的科罗拉多大学，并包括气象学系在马里兰州大学在CollegePark。 它将资助一名研究助理，两名来自UMD和俄亥俄州大学的访问科学家，以及一名研究生，其中包括一名研究助理，两名博士后研究人员，一名研究生和四名本科生。

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.