Numerical Methods for Waves: Nonlocal, Nonlinear, and Multiscale Systems

对复杂物理现象的可靠模拟可以造福社会，并在无数方面满足人类的好奇心。例子的范围从非常小的，如纳米级设备的设计和量子系统的新兴应用，到非常大的，如地震和海啸引起的自然灾害，以及我们对宇宙动力学和进化的理解。尽管随着对百亿亿级系统的推动，计算能力正在增加，但计算机硬件本身正变得更加异构，难以有效使用，人们希望解决的模型所带来的挑战也在迅速增长。如果要实现新计算技术的巨大前景，就迫切需要开发和部署更好的算法。该研究计划的重点是发明新的快速和准确的方法来解决物理系统的综合模型，其中波传播起着核心作用，应用于上述问题的所有范围。模拟波的一个主要障碍是大多数应用问题的多尺度性质。一方面，波的定义特征是它们相对于其波长传播长距离的能力，这有效地导致了在无界域上提出的问题。另一方面，波与可能在波长尺度上或以下变化的介质相互作用。这些模型的一个中心主题是非局部算子的出现；该项目的一个主要目标是构建快速、准确和内存高效的算法来评估它们。这包括(i)一般波传播问题的有效和数学上合理的域截断算法的发展，目前仅适用于有限类别的系统；（ii）在材料特性存在亚波长变化的情况下，数值构建波传播的精确降阶模型的方法，能够有效地处理工程材料，而不需要假设尺度分离或周期性；（iii）分数和其他算子函数的低内存算法。该项目的第二个目标是将鲁棒和有效的离散化方案扩展到由作用原理导出的二阶非线性波动方程，特别是产生求解广义相对论和规范理论中出现的方程的新方法。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The reliable simulation of complex physical phenomena can benefit society and help satisfy human curiosity in countless ways. Examples range from the very small, such as the design of nanoscale devices and emerging applications of quantum systems, to the very large, such as natural disasters caused by earthquakes and tsunamis, as well as our understanding of the dynamics and evolution of the cosmos. Although computational capabilities are increasing, with a push towards exascale systems, the computer hardware itself is becoming more heterogeneous and difficult to use efficiently, and the challenges posed by the models one wishes to solve are also rapidly growing. The need to develop and deploy better algorithms is urgent if the tremendous promise of the new computing technologies is to be realized. This research program is focused on the invention of new fast and accurate methods for solving comprehensive models of physical systems where wave propagation plays a central role, with applications throughout the range of problems outlined above.A primary obstacle to simulating waves is the multiscale nature of most applied problems. On the one hand, the defining feature of waves is their ability to propagate long distances relative to their wavelength, effectively leading to problems posed on unbounded domains. On the other, waves interact with media that may vary at or below the wavelength scale. A central theme in such models is the appearance of nonlocal operators; one main goal of the project is the construction of fast, accurate, and memory-efficient algorithms to evaluate them. This includes the development of (i) effective and mathematically justified domain truncation algorithms for general wave propagation problems, which at present are only available for a limited class of systems, (ii) methods for numerically constructing accurate reduced-order models of wave propagation in the presence of subwavelength variations in material properties, capable of efficiently treating engineered materials without assumptions of scale separation or periodicity, and (iii) low-memory algorithms for fractional and other operator functions. The second goal of this project is the extension of robust and efficient discretization schemes to second-order nonlinear wave equations derived from action principles, yielding, in particular, new methods to solve equations arising in general relativity and gauge theories.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.