CAREER: Time Reversal and Inverse Problems in Wave and Particle Propagation

巴尔和他的学生研究了波和粒子在非均匀介质中的传播以及相关的反问题。更准确地说，他们分析了时间反转波的重聚焦和稳定特性，这可用于无损检测和无线通信。他们还分析了光子在人体组织中的传播，以了解如何通过光子强度测量来成像这些组织的特性。开发了用于模拟上述正反问题的数值工具和软件。研究人员还开设了一门关于理论和数值反问题的课程。在许多应用中，需要对波和粒子在复杂介质中的传播有很好的了解：地震产生的地震波传播，破坏肾结石的声波传播(碎石术)，无线通信中的微波传播，或光学层析成像中的光子传播，这是一种非常强大的成像技术，可以检测脑瘤和手指关节炎。研究人员设计了理论和数值工具来分析波和粒子的传播，并了解如何使用它来解决实际问题，例如在特定环境中可以获得的最大通信容量，或者可以在光学断层扫描中成像的最小尺寸的肿瘤。这也是研究人员的主要目标，向新一代本科生和研究生传授最近在应用数学界出现的解决这些问题的新技术。

Bal The investigator and his students study wave and particlepropagation in heterogeneous media and related inverse problems.More precisely, they analyze the refocusing and stabilityproperties of time reversed waves, which can be used innon-destructive testing and wireless communication. They alsoanalyze photon propagation in human tissues to understand how theproperties of these tissues can be imaged from photon intensitymeasurements. Numerical tools and software are developed tosimulate the aforementioned forward and inverse problems. Theinvestigator also develops a course on theoretical and numericalinverse problems. In many applications a good understanding of wave andparticle propagation in complicated media is necessary: seismicwave propagation generated by earthquakes, acoustic wavepropagation to destroy kidney stones (lithotripsy), microwavepropagation in wireless communication, or photon propagation inoptical tomography, a quite powerful imaging technique to detect,for instance, brain tumors and finger arthritis. The investigatordevises theoretical and numerical tools to analyze wave andparticle propagation and to understand how it can be used tosolve practical problems, such as the maximal capacity ofcommunication that can be obtained in a specific environment, orthe minimal size tumor that can be imaged in optical tomography.It is also a primary goal of the investigator to teach cominggenerations of undergraduate and graduate students the noveltechniques that have emerged recently in the applied mathematicalcommunity to address these problems.