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，从根本上改善了性能，即显著减少了采样要求，同时显著提高了精度。这样的求解器将允许在具有界面和可变系数的数千个特征波长的域中对线性和最终的非线性波传播进行建模。这种方法也将为解决非线性平流扩散问题提供改进的基础。求解器建立在两种新技术上，即最优正交来表示带宽有限的函数，以及变量分离的数值推广以加速高维算子的应用。每一种技术都有潜力在广泛的应用中显著推进计算科学。它们共同提供了一种新的范式，可以有效地组织控制物理现象的运算符中包含的信息。该项目进一步发展了这些技术，并基于最优正交开发了算子和函数的多分辨率表示。任何自然现象的计算模型都需要对基础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，并组织了两个重要的建模领域，波传播和地球物理流体动力学的信息。这项研究的目的是在科学建模中更广泛地使用有效的技术来表示信息，并将模拟的速度和准确性提高两个数量级，在地震学、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，这对于描述对物理参数变化高度敏感的现象是必要的，并且经常导致技术瓶颈。这项合作研究由科罗拉多大学博尔德分校应用数学系领导，马里兰大学帕克分校气象系也参与其中。它将资助一名研究助理、两名来自马里兰大学和俄亥俄大学的访问科学家，以及一个多元化研究小组的一名研究生，该小组包括一名研究助理、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.