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，具有根本性的改进性能，即显著降低了采样要求，同时显著提高了精度。这样的求解器将允许线性和最终非线性波传播的域中，是数以千计的特征wavelengths大小，接口和可变系数的建模。这种方法也将提供solvingnonlinear对流扩散问题的改进的基础。 求解器是建立在twonenew技术，即，最佳求积表示带限函数，和一个数值推广的分离变量，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发展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基础数学方程进行离散化。这个项目解决了这样的离散化的最优性和效率问题，以及两个重要建模领域，波传播和地球物理流体动力学的组织信息问题。 这项研究的目的是产生一个更广泛的使用ofefficient技术表示信息的科学建模，并提高速度和精度的模拟高达两个数量级，与可预见的好处，如地震，遥感，声学，光学，地球物理流体动力学和量子化学等领域。这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在学院公园。 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，以及一名研究生，该研究生来自一个多元化的研究小组，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 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.