Collaborative Research: Novel Microlocal-Analysis and Domain-Decomposition Based Fast Algorithms for Elastic Wave Modeling and Inversion in Variable Media

波的传播是许多科学和工程学科中的重要现象。从雷达、声纳、地震成像、医学成像、潜艇探测、隐身技术、遥感和电子技术到显微镜和纳米技术，计算波传播已经成为一项基本的、蓬勃发展的技术。这些应用对石油工业、医学成像和材料科学尤其重要。如何高效、准确地进行大规模的高频波传播是计算波传播中最具挑战性的问题之一。该项目的研究人员将开发新的、快速的高频弹性波传播和反演算法。特别是，他们将专注于技术，包括微局部分析和基于区域分解的快速惠更斯扫描方法和快速多尺度高斯光束方法，以应对这一长期存在的挑战。研究生将参与并接受跨学科培训。该项目的动机是科学和工程应用，它将促进几个理论和计算方面的创新。目标是发展高效和准确的基于Hadamard-Babich展开的快速惠更斯扫描方法和基于多尺度高斯波包变换的快速多尺度高斯光束，用于在高频和焦散存在的可变介质中传播弹性波。几种研究途径将被推进。首先，提出的新方法将解决在存在焦散的情况下大规模高频弹性波建模和反演的挑战。其次，将在开发基于偏微分方程组的欧拉几何光学和计算波传播的基于Hadamard-Babich展开、区域分解和蝴蝶算法的快速惠更斯扫描方法方面取得进展。基于Hadamard-Babich展开和区域分解的快速惠更斯扫描法和快速多尺度高斯束方法都能够在高频区产生超越焦散的一致渐近解。第三，新方法将为非均匀介质中的许多与弹性波相关的应用提供以前没有使用过的有效工具，如地震成像和反演以及医学成像和反演。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Wave propagation is an important phenomenon in many science and engineering disciplines. Computational wave propagation has become a fundamental, vigorously growing technology in diverse disciplines, ranging from radar, sonar, seismic imaging, medical imaging, submarine detection, stealth technology, remote sensing and electronics to microscopy and nanotechnology. These applications are important in particular for the petroleum industry, medical imaging, and material sciences. One of the most challenging problems in computational wave propagation is how to carry out large-scale high frequency wave propagation efficiently and accurately. The investigators in this project will develop novel, fast algorithms for high frequency elastic wave propagation and inversion. In particular, they will focus on techniques including microlocal-analysis and domain-decomposition based fast Huygens sweeping methods and fast multiscale Gaussian beam methods to tackle this long-standing challenge. Graduate students will be involved and receive interdisciplinary training. The project is motivated by science and engineering applications, and it will foster innovations in several theoretical and computational aspects. The goal is to develop efficient and accurate Hadamard-Babich expansion based fast Huygens sweeping methods and multiscale Gaussian wavepacket transform based fast multiscale Gaussian beams for elastic wave propagation in variable media in the high frequency regime and in the presence of caustics. Several avenues of research will be pusrued. First, the proposed new methods will address the challenges in large-scale high-frequency elastic wave modeling and inversion in the presence of caustics. Second, advances will be made in developing novel Hadamard-Babich expansion, domain decomposition, and butterfly-algorithm based fast Huygens sweeping methods for partial differential equation-based Eulerian geometrical optics and computational wave propagation. Both the Hadamard-Babich expansion and domain-decomposition based fast Huygens sweeping method and the fast multiscale Gaussian beam method are capable of producing uniform asymptotic solutions beyond caustics for wave propagation in the high-frequency regime. Third, the new methods will provide efficient tools not used before for many elastic wave-related applications in inhomogeneous media, such as seismic imaging and inversion, and medical imaging and inversion.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.