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

该项目开发了二维和三维波传播问题的时域解算器，具有根本改进的特性，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区域中对线性以及最终的非线性波传播进行建模。这种方法还将为解决非线性对流-扩散问题提供更好的基础。求解器建立在两种新技术的基础上，即表示带宽受限函数的最优求积，以及加速应用高维算子的分离变量的数值推广。每一种技术都有可能在广泛的应用领域显著推动计算科学的发展。它们共同提供了一种新的范式，有效地组织了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，并基于最优二次曲面开发了运算符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基本的数学方程进行离散化。该项目解决了这种离散化的最优化和效率问题，以及波传播和地球物理流体动力学这两个重要建模领域的信息组织问题。这项研究的目的是在科学建模中更广泛地使用高效的信息表示技术，并将模拟的速度和精度提高高达两个数量级，从而为地震学、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。这将减少获得高精度所需的计算成本，而高精度是描述对物理参数变化高度敏感且经常导致技术瓶颈的现象所必需的。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，包括马里兰大学帕克分校的气象系。它将资助一名研究助理，两名来自UMD和俄亥俄大学的访问科学家，以及一个多元化研究小组中的一名研究生，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 same time,significantly higher accuracy. Such solvers would allow modeling of linear and, eventually, nonlinearwave propagation in domains that are thousands of characteristic wavelengthsin size, with interfaces and variable coefficients.This approach would also provide improved bases for solvingnonlinear advection-diffusion problems. The solvers are built upon twonew techniques, namely, optimal quadratures to represent bandlimited functions, and a numerical generalization of separation of variables 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 these techniquesfurther, and develops multiresolution representations for operators andfunctions, based on the optimal quadratures.Any computational modeling of natural phenomena requires discretizationof the underlying mathematical equations. This project addresses questions ofoptimality and efficiency of such discretizations, and of organizationo information for two important modeling areas, wave propagation andgeophysical fluid dynamics. This research aims to generate a much wider use ofefficient techniques for representing information in scientific modeling, and toincrease 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 wouldreduce the computational cost of obtaining high accuracy, which is necessary todescribe phenomena that are highly sensitive to changes in physicalparameters, 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 the Department ofMeteorology at the University of Maryland at College Park. It will fundone research associate, two visiting scientists from UMD and OhioUniversity, and one graduate student in a diverse research group that includes aresearch associate, two postdoctoral researchers, a graduate student, and fourundergraduate students.