Microlocal analysis for waves and inverse problems

波和反问题的微局域分析

• 批准号: 1361432
• 负责人: Andras Vasy
• 金额: $ 36万
• 依托单位: Stanford University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-07-01 至 2018-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1361432&HistoricalAwards=false
• 关键词: Microlocal; analysis; waves; inverse; problems

The planned research develops and applies tools of the field of microlocal analysis. Roughly speaking, this field keeps track of the position and frequency, or momentum, of waves (or more generally, functions) simultaneously. The planned applications are to wave propagation and other related phenomena, as well as inverse problems for determining a function from integrals along curves (X-ray transform) and related problems for determining the structure of a material from boundary measurements. Although the proposal concerns their mathematical theory, these problems are closely connected to the physical world. Wave propagation is ubiquitous in nature, with electromagnetic waves, such as light, being one of the most prevalent examples; the theory of general relativity being another important physical example. Scattering theory of quantum particles (such as protons and electrons) is another subject governed by microlocal analysis: these aspects enter both into the description of quantum waves at large distances, and into semiclassical phenomena, i.e. when Planck's constant can be regarded as small, which happens often in chemistry. The inverse problems under study are also of broad significance: an application of the theory developed here is the determination of an unknown variable sound speed in an object via the measurement of travel times of waves, which for instance is relevant to imaging to interior of Earth using the travel times of earthquake waves.Some of the proposed projects describe the long-time or far field behavior of waves on curved space-times. Physically these arise in general relativity, including electromagnetic waves on a curved background. However, there are also examples of purely mathematical origin, such as asymptotically complex hyperbolic (ACH) space and asymptotically Anti de Sitter spaces (AdS); the latter does have a different connection with physics via string theory. The microlocal approach to analysis on these spaces has made breakthroughs possible in the author's work on linear problems asymptotically (real) hyperbolic (AH) spaces as well as Kerr-de Sitter space. The projects here aim to extend these tools to the ACH, AdS-type spaces, improve the understanding of Lorentzian scattering spaces (which include asymptotically Minkowski spaces), as well as to quasilinear PDE. Other projects concern the behavior of waves at edges, concretely the diffraction of the Rayleigh (surface) waves of elasticity and a refined, second-microlocal, description of diffraction of scalar or electromagnetic waves. Yet another main area is inverse problems, where the author, together with Uhlmann, has introduced new tools for spatially localized inversion of the geodesic X-ray transform. The projects aim to extend this to tensors, and to investigate boundary rigidity, i.e. recovering a Riemannian metric from its boundary distance function.

计划的研究开发并应用了微局部分析领域的工具。粗略地说，该领域同时跟踪波（或更一般的功能）的位置和频率或动量。计划的应用是为了浪潮传播和其他相关现象，以及沿曲线积分（X射线变换）的函数的逆问题，以及与边界测量结果确定材料结构的相关问题。尽管该提议涉及其数学理论，但这些问题与物理世界紧密相关。波传播本质上是普遍存在的，电磁波（例如光）是最普遍的例子之一。一般相对论的理论是另一个重要的物理例子。量子颗粒的散射理论（例如质子和电子）是另一个受微观分析控制的受试者：这些方面既进入了大距离的量子波的描述，又进入了半经典现象，即当普朗克的常数可以被视为小时，通常发生在化学中。研究中的反问题也具有广泛的意义：此处开发的理论的应用是通过测量波浪的行进时间来确定对象中未知的可变声速，例如，这与地震波浪的旅行时间与地球内部的成像相关。提议的项目的某些项目描述了长期或远距离的领域，描述了在弯曲的空间中的波浪行为。从物理上讲，这些是在一般相对论中出现的，包括弯曲背景上的电磁波。然而，也有纯粹数学起源的例子，例如渐近复杂的双曲线（ACH）空间和渐近的抗De Sitter空间（ADS）；后者确实通过字符串理论与物理有不同的联系。在这些空间上进行分析的微局部方法使作者在线性问题上的突破性（真实）（真）（AH）空间以及Kerr-DE Sitter空间成为可能。这里的项目旨在将这些工具扩展到ACH，ADS型空间，以提高对Lorentzian散射空间（包括渐近Minkowski空间）的理解，以及对准PDE的理解。其他项目涉及边缘的波的行为，具体地，雷利（表面）弹性的衍射和精制的，第二微胶质的，标量或电磁波的衍射描述。另一个主要领域是逆问题，作者与Uhlmann一起引入了新的工具，用于在空间定位的地球X射线变换上进行空间局部倒置。这些项目旨在将其扩展到张量，并研究边界刚度，即从其边界距离函数中恢复Riemannian指标。