Novel Approaches to Wave Propagation in Random Media

该项目解决了复杂环境中与成像和传感相关的数学挑战。遥感数据可用于对大气污染进行环境监测，评价森林和作物的健康状况，探测地球表面的细微细节，以及跟踪卫星和其他飞行物体。随着分辨率越来越高的数据变得可用，更好地了解这些数据如何受到环境中杂乱的影响是很重要的。在设计通过湍流介质（例如在湍流（杂乱）大气中发现的类型）进行成像和通信的方案以及生物医学成像的背景下，也需要这种理解。这种技术的进一步发展需要深入了解探测波场如何受到局部介质或组织结构（杂波）的影响。该项目还涉及更好地理解和建模金融和经济时间序列的技术，这对监测和稳定市场很有用。重要的问题是，当市场的结构，局部杂乱统计数据发生变化时，我们正在进入一个新的市场体制，如何设计早期检测方案。类似的挑战与互联网流量分析有关，例如检测由入侵者引起的正常流量的变化。该项目旨在开发基本的数学框架，使人们能够处理这些重要的挑战，此外，在检测织物变化时使用与机器学习相关的现代工具。研究生将通过参与研究得到训练。从物理和数学的角度来看，严格表征通过多尺度介质和界面传播的激光束的挑战是至关重要的。该项目研究的新方法允许人们在复杂的非经典介质中模拟波的传播，特别是通过所谓的非柯尔莫哥洛夫湍流。在大气传播和生物医学成像的背景下，这种建模变得越来越重要。分析将涉及由随机系数偏微分方程控制的场的统计矩的渐近分析，并利用系统中的尺度分离。新的成像技术越来越多地利用波谱信息，或波场本身偏振模式或其强度的协方差结构信息，即散斑相关。为了评估这种基于波谱信息的成像技术，人们需要高阶矩的信息来评估图像稳定性或噪声水平，并优化方案。该项目的一个主要目标是进一步发展描述这种统计结构的理论，并且在经典和深度学习成像模式的背景下使用它。具体来说，在生物医学成像的背景下，多尺度和复杂介质中波传播建模的新方法将用于成像方法的设计，这些成像方法可以检测局部组织结构的变化。研究者还将研究具有多重分形特征的复杂经济时间序列的建模和估计，并应用于检测市场制度变化。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。

This project addresses mathematical challenges associated with imaging and sensing in complicated environments. Remote sensing data can be used for environmental monitoring of atmospheric pollution, for evaluation of health of forests and crops, for detecting fine details on the earth's surface, and for tracking of satellites and other flying objects. As ever more, finer-resolution data becomes available, a better understanding of how these are affected by clutter in the environment is important. Such understanding is also needed in the context of designing schemes for imaging and communication through turbulent media, of the type found for instance in the turbulent (cluttered) atmosphere, as well as in the context of biomedical imaging. Further developments of such techniques require a deep understanding of how the probing wave field is affected by the local medium or tissue fabric, the clutter. The project also involves techniques for better understanding and modeling of financial and economic time series, which is useful for monitoring and stabilization of markets. Important questions relate then to how one can design schemes for early detection of when the fabric, the local clutter statistics, of the market is changing and we are entering a new market regime. Similar challenges relate to analysis of internet traffic, such as detecting changes to normal traffic flow caused by an intruder. The project aims to develop fundamental mathematical frameworks that allow one to deal with such important challenges, moreover, to use modern tools associated with machine learning in the detection of fabric changes. Graduate students will be trained through involvement in the research.The challenge of a rigorous characterization of laser beams propagating through multiscale media and interfaces is of fundamental importance both from the physical and mathematical perspectives. This project studies new approaches that allow one to model wave propagation in complicated non-classic media, particularly through so called non-Kolmogorov turbulence. Such modeling is becoming increasingly important in the context of propagation through the atmosphere and furthermore in biomedical imaging. The analysis will involve asymptotic analysis of statistical moments of fields governed by partial differential equations with random coefficients and exploit scale separations in the system. Novel imaging techniques increasingly make use of wave spectral information, or information in the covariance structure for the polarization modes of the wave field itself or its intensity, that is, speckle correlations. To evaluate such imaging techniques based on wave spectral information, one needs information about higher-order moments to assess image stability, or noise level, and to optimize the schemes. A main objective of the project is to further the theory that describes this statistical structure and, moreover, to use it in the context of classic and deep learning imaging modalities. Specifically, novel approaches for modeling wave propagation in multiscale and complex media will be used in the design of imaging approaches that can detect changes in the local tissue fabric in the context of biomedical imaging. The investigator will also study modeling and estimation in the context of complex economic time series with a multi-fractal character, with applications to detection of market regime changes.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.