Hyperbolic Inverse Problems in Random Environments

随机环境中的双曲反问题

• 批准号: 1510429
• 负责人: Liliana Borcea
• 金额: $ 23.34万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-09-01 至 2019-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1510429&HistoricalAwards=false
• 关键词: Hyperbolic; Inverse; Problems; Random; Environments

The objective of this project is to obtain a rigorous mathematical theory of imaging with waves in complex (cluttered) environments. The topic lies at the interface where mathematics meets physics, probability, numerical simulations, and signal processing. The research is driven by challenges in application areas such as ground penetrating radar, satellite imaging and tracking through atmospheric turbulence, nondestructive ultrasonic testing of materials such as aging concrete, imaging in shallow water, and underground exploration. Complex media are ubiquitous in such applications and pose a serious impediment to the imaging process, which is largely ignored by the present imaging technology. Moreover, computer modeling of wave propagation in complex media is faced with formidable computational challenges. Mathematical analysis is needed to unravel the complicated scattering effects of such media so that the present imaging technology can be advanced. This project seeks to analyze long-range propagation of sound and electromagnetic waves in complex media that may also vary in time, develop novel adaptive imaging methodologies that mitigate the medium scattering effects, and propose new measurement setups that can improve the imaging process. Complex environments are naturally modeled with random processes, and the wave equation has random coefficients and boundaries. The propagation of uncertainty from these random processes to the uncertainty of the waves measured in imaging applications is a highly non-trivial problem. One goal of the project is to develop a better understanding of this problem for the acoustic wave equation and Maxwell's system of equations. The mathematics is a combination of asymptotic stochastic analysis and invariant imbedding for studying transmission and reflection operators. The project considers mixing random processes that are static or may vary in time. The time variations are studied in various setups, with both rapid and slow time changes with respect to the duration of pulses emitted by sensors that probe the complex environment. Another goal of the project is to develop a novel robust and adaptive imaging methodology in complex environments. The methods should be able to detect the loss of coherence of the measured waves due to scattering in the environment and to enhance the signal to noise ratio by filtering the components of the fields that are useless in imaging. The analysis of the imaging methods seeks a resolution theory that quantifies the focusing of images and their statistical stability.

该项目的目的是在复杂（混乱）环境中获得具有波浪的严格数学理论。该主题位于数学符合物理，概率，数值模拟和信号处理的界面。这项研究是由应用区域（例如地面穿透性雷达），卫星成像和通过大气湍流，对材料的非破坏性超声测试（例如老化混凝土，浅水成像和地下探索）进行的挑战所驱动的。复杂的媒体在这种应用中无处不在，并对成像过程造成了严重的障碍，这在当前的成像技术中在很大程度上被忽略了。此外，复杂媒体中波传播的计算机建模面临着巨大的计算挑战。需要数学分析来揭示此类介质的复杂散射效应，以便可以推进当前的成像技术。该项目旨在分析复杂介质中声音和电磁波的长距离传播，这些介质也可能随着时间的流逝而变化，开发新型的自适应成像方法，从而减轻中等散射效应，并提出新的测量设置，以改善成像过程。复杂的环境自然而然地使用随机过程对其进行建模，并且波动方程具有随机的系数和边界。这些随机过程到成像应用中测得的波的不确定性的不确定性传播是一个高度不平凡的问题。该项目的一个目标是为声波方程和麦克斯韦的方程系统更好地理解此问题。该数学是渐近随机分析和不变的嵌入式的组合，用于研究传播和反射操作员。该项目考虑混合静态或可能随时间变化的随机过程。在各种设置中研究了时间变化，随着探测复杂环境的传感器发射的脉冲持续时间，快速和缓慢的时间变化。该项目的另一个目标是在复杂的环境中开发一种新颖的健壮和自适应成像方法。这些方法应能够检测由于环境中散射而导致的测量波的相干性丧失，并通过过滤成像中无用的磁场的组件来增强信号与噪声比。对成像方法的分析寻求一种解决理论，该理论量化了图像的聚焦及其统计稳定性。