Multiscale Algorithms for Wave Propagation

大多数科学过程及其相关的数学模型在广泛的时间和长度尺度上都具有重要的特征。与波的计算有关的典型例子有高频波的传播和散射以及波场与复杂介质的相互作用。直接在最精细的尺度上离散这些问题，并用标准的数值算法求解得到的系统，不可避免地会导致巨大的计算问题，具有不可接受的长计算时间和大内存需求。基于我们以前在多尺度算法方面的经验，我们计划为高频波传播和相关领域的问题设计、实施和分析新的算法。这些问题是具有挑战性的，因为许多众所周知的技术，如多重网格和标准快速多极子方法，效率有限。我们将集中于以下三个主题：(1)高频声波和电磁散射，(2)高频波和薛定谔方程的高斯束方法，以及(3)具有多个重复尺度的复杂介质的均匀化。拟议的研究将在几个应用领域产生直接影响。这些算法将帮助我们解决数千个波长尺度上的大规模复杂散射问题，就像在无线通信天线设计中一样。我们还将通过在勘探地震学中的应用，更好地理解波在复合材料中的传播。在教育方面，现代数值算法和软件的发展要求研究人员理解计算数学的不同方面。我们计划用这笔助学金资助两名研究生。这不仅有助于培养掌握算法设计、数学分析和软件开发的新一代研究人员，而且有助于提高本科生和弱势群体对计算数学的意识和兴趣。我们将与来自工业和政府实验室的研究人员合作，传播想法并提供具有挑战性的应用程序的操作软件。

Most scientific processes and their related mathematical models have important features in a wide range of time and length scales. Some typical examples related to the computation of waves are propagation and scattering of high frequency waves and interaction of the wave field with complex media. Discretizing these problems directly at the finest scale and solving the resulting systems with standard numerical algorithms inevitably leads to an enormous computational problem with unacceptable long computation times and large memory requirements. Building on our previous experience in multiscale algorithms, we plan to design, implement, and analyze novel algorithms for problems in high frequency wave propagation and related fields. Such problems are challenging since many well-known techniques, such as multigrid and standard fast multipole methods have limited efficiency. We will focus on the following three topics: (1) high frequency acoustic and electromagnetic scattering, (2) Gaussian beam methods for high frequency wave and Schrodinger equations, and (3) homogenization of complex media with multiple reiterated scales.The overarching theme of the research presented in this proposal is to exploit the geometric structures and asymptotic features of a variety of multiscale problems. The proposed research will have direct impact in several application fields. These algorithms will help us to solve large scale complicated scattering problems on the scales of thousands of wavelengths as in antenna design for wireless communication. We will also better understand wave propagation in composite materials with applications in exploration seismology. On the education side, the development of modern numerical algorithms and softwares requires researchers to understand different aspects of computational mathematics. We plan to use this grant to support two graduate students. This will not only help training a new generation of researchers who master algorithmic design, mathematical analysis, and software development, but also promote the awareness and interests in computational mathematics among undergraduates and underrepresented groups. We will work with researchers from industrial and government laboratories to disseminate ideas and deliver operational softwares for challenging applications.