Inverse Source Problems, Splitting, and Uncertainty

逆源问题、分裂和不确定性

• 批准号: 1712525
• 负责人: John Sylvester
• 金额: $ 22.21万
• 依托单位: University of Washington
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2017
• 资助国家: 美国
• 起止时间: 2017-08-01 至 2022-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1712525&HistoricalAwards=false
• 关键词: Inverse; Source; Problems; Splitting; Uncertainty

This project studies fundamental problems which arise in the development of remote sensing technologies. Remote sensing systems are either active or passive. An underwater array of acoustic sensors that seeks to locate a submarine (the source) based on the noise radiated from the submarine's engine is a passive remote sensing system. A mathematical model for passive remote sensing is the Inverse Source Problem. A sonar array that transmits its own (incident) wave to locate the submarine based on the properties of the echo, is an active remote sensing system. In this case, the array, not the submarine, is the primary source and the submarine is referred to as the scatterer because it scatters the sound wave in new directions. A mathematical model for active remote sensing is the Inverse Scattering Problem. Scatterers are also called secondary sources or induced sources because the primary or incident wave causes them to radiate a secondary or scattered wave. This terminology emphasizes the close relationship between the inverse source problem and the inverse scattering problem. Every scattered wave is a wave radiated by an induced source, the source is different for different incident waves, but the support (size and location) is always the same. The principal investigator seeks to elucidate the fundamental limitations that physics places on these technologies, and to develop algorithms to extract all information within these limits.Rephrasing the uncertainty principle of Donoho and Stark in terms of inequalities for inner products and estimates on an underlying operator has made it possible to look for applications of these principles in new contexts, including the inverse source problem. Although the inverse source problem for the Helmholtz equation does not have a unique solution, the investigator has developed a well-defined notion of support for far fields radiated by unions of well-separated sources. The splitting operator decomposes a far field radiated by such a union of sources into the far fields radiated by each individual source. A major goal is to estimate the condition number of the splitting operator, and to develop algorithms to do the splitting and locate the individual sources. The condition number is estimated in terms of physically meaningful parameters, which are wavelength, diameters, and distance between sources. Another related objective is the development of norms and estimates for partial differential equations related to wave phenomena that, unlike many weighted norms, can be readily interpreted in physical units, and are therefore more easily accessible to the broader scientific community.

该项目研究遥感技术发展中出现的基本问题。遥感系统有主动式和被动式之分。水下声传感器阵列是一种被动遥感系统，它根据潜艇发动机辐射的噪声来定位潜艇（声源）。被动遥感的一个数学模型是源反问题。声纳基阵是一种主动遥感系统，它发射自己的（入射）波，根据回波的特性来定位潜艇。在这种情况下，阵列，而不是潜艇，是主要来源，潜艇被称为散射体，因为它在新的方向上散射声波。反散射问题是主动遥感的一个数学模型。散射体也被称为二次源或感应源，因为一次波或入射波使它们辐射二次波或散射波。这个术语强调了源逆问题和逆散射问题之间的密切关系。每一个散射波都是由一个感应源辐射的波，对于不同的入射波，源是不同的，但支撑（大小和位置）总是相同的。主要研究者试图阐明物理学对这些技术的基本限制，并开发算法来提取这些限制内的所有信息。将Donoho和Stark的不确定性原理重新表述为内积的不等式和对基础算子的估计，这使得在新的背景下寻找这些原理的应用成为可能，包括逆源问题。虽然亥姆霍兹方程的逆源问题没有一个唯一的解决方案，调查人员已经制定了一个定义明确的概念，支持远场辐射的工会以及分离的来源。分裂算子将由这样的源的联合辐射的远场分解为由每个单独的源辐射的远场。一个主要的目标是估计分裂算子的条件数，并开发算法来进行分裂和定位各个源。条件数是根据有物理意义的参数来估计的，这些参数是波长、直径和源之间的距离。另一个相关的目标是为与波动现象有关的偏微分方程制定规范和估计数，与许多加权规范不同，这些规范和估计数可以很容易地用物理单位解释，因此更容易为更广泛的科学界所用。

项目成果

Uncertainty Principles for Three-Dimensional Inverse Source Problems

• DOI: 10.1137/17m111287x
• 发表时间: 2017-11
• 期刊: SIAM J. Appl. Math.
• 作者: Roland Griesmaier;J. Sylvester

Uncertainty principles for inverse source problems for electromagnetic and elastic waves

• DOI: 10.1088/1361-6420/aab45c
• 发表时间: 2018-06
• 期刊: Inverse Problems
• 影响因子: 2.1
• 作者: Roland Griesmaier;J. Sylvester