Novel Approaches to Wave Propagation in Random Media

该项目解决了与复杂环境中的成像和传感相关的数学挑战。遥感数据可用于大气污染的环境监测，评估森林和农作物的健康状况，探测地球表面的细微细节，以及跟踪卫星和其他飞行物体。随着更多、更精细的数据变得可用，更好地了解环境中的杂乱如何影响这些数据是很重要的。在设计用于通过湍流介质进行成像和通信的方案的背景下，例如在湍流(杂乱)大气中发现的类型，以及在生物医学成像的背景下，也需要这样的理解。这种技术的进一步发展需要对探测波场如何受到局部介质或组织结构、杂波的影响有深入的了解。该项目还涉及更好地理解金融和经济时间序列并对其进行建模的技术，这对监测和稳定市场很有用。因此，重要的问题涉及到如何设计方案来及早发现市场的面料和本地杂乱统计数据何时发生变化，我们正在进入一个新的市场制度。类似的挑战与互联网流量分析有关，例如检测入侵者导致的正常流量变化。该项目旨在开发基本的数学框架，使人们能够应对这样的重要挑战，此外，还将使用与机器学习相关的现代工具来检测织物变化。研究生将通过参与这项研究进行培训。从物理和数学的角度来看，严格描述激光在多尺度介质和界面中传播的挑战具有基本的重要性。该项目研究了一种新的方法，使人们能够模拟复杂的非经典介质中的波传播，特别是通过所谓的非科尔莫戈洛夫湍流。在大气传播的背景下，这种建模变得越来越重要，而且在生物医学成像中也变得越来越重要。该分析将涉及由随机系数偏微分方程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，或波场本身的偏振模式或其强度的协方差结构中的信息，即散斑相关。为了评估这种基于波谱信息的成像技术，人们需要关于高阶矩的信息来评估图像稳定性或噪声水平，并对方案进行优化。该项目的一个主要目标是进一步发展描述这一统计结构的理论，并将其用于经典和深度学习成像模式。具体地说，用于模拟波在多尺度和复杂介质中传播的新方法将被用于成像方法的设计，该成像方法可以在生物医学成像的背景下检测局部组织结构的变化。研究人员还将研究具有多重分形性的复杂经济时间序列的建模和估计，并将其应用于市场制度变化的检测。该奖项反映了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。

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.