Oscillatory Integral Operators, Inverse Problems and Non-Transformation Optics

振荡积分算子、反问题和非变换光学

• 批准号: 1362271
• 负责人: Allan Greenleaf
• 金额: $ 21.9万
• 依托单位: University of Rochester
• 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=1362271&HistoricalAwards=false
• 关键词: Oscillatory; Integral; Operators; Inverse; Problems

This proposal consists of four groups of problems. Three of them are concerned with the interaction between geometry and analysis, the study of functions and the properties of operators, which are transformations that turn functions into new functions. The operators studied in this project arise either when trying to understand how waves propagate (for example, acoustic waves in the Earth), or in basic mathematical tools that are used to study wave propagation, or in the study of geometric properties of large sets of points. Waves can be idealized as traveling along rays, such as light rays, and when the rays concentrate in a small region, new methods need to be developed to obtain accurate descriptions of the waves. Two parts of this project will continue the principal investigator's study of various operators and how their properties can be predicted from the structure of rays or more general underlying geometry. Possible applications include improved understanding of artifacts in seismic imaging. A third part concerns multilinear operators, which act on several functions at a time. These arise in the study of how two or more waves interact and also in the study of point clouds in discrete geometry. The fourth part is a new approach to the design of resonant devices, such as antennas. Recent progress in materials science, physics, and mathematics has led to rapid advances in the design of devices constructed from structured composite materials, also called metamaterials, which have radical effects on wave propagation. One particularly successful approach is based on transformation optics. The principal investigator will investigate a new, non-transformation optics design methodology. Two of the proposed projects concern developing new tools for dealing with degeneracies or singularities of smooth or real-analytic functions. In one, composition of degenerate Fourier integral operators with smooth phases leads to operators that have wave-front relations that are not smooth, and understanding how to associate Fourier-integral-operator-like operators to these geometries will expand the reach of microlocal analysis and help analyze certain inverse problems. In another, new techniques will need to be found to deal with oscillatory integral operators with real-analytic phases in one plus two or two plus two dimensions. The difficulties include understanding to what extent two or more functions can have their zero varieties simultaneously resolved. Some of the same techniques will also be applied to try to find an algorithmic description of jumping numbers, which are the subject of current interest in singularity theory and algebraic geometry. Doing so will strengthen the connection between analysis and these fields. Progress on the third part of the project will not only further develop the theory of multilinear operators within harmonic analysis, but it will also have immediate applications to geometric measure theory and discrete geometry. The final component will show how established ideas from linear partial differential equations can be useful in designing and rigorously verifying the properties of resonant structures such as antennas and will help contribute a new design methodology to the rapidly developing area of metamaterials.

该提案由四组问题组成。其中三个关注几何与分析之间的相互作用，功能的研究以及操作员的性质，这是将函数转化为新函数的转换。当试图了解波浪传播的方式（例如，地球上的声波）或用于研究波传播的基本数学工具或研究大量点的几何特性的基本数学工具中，就会出现研究该项目中研究的操作员。波浪可以被理想地沿着光线行驶，例如光线，当射线集中在小区域时，需要开发新的方法以获得对波浪的准确描述。该项目的两个部分将继续主要研究者对各种操作员的研究，以及如何从射线或更一般的基础几何形状的结构中预测其性质。可能的应用包括对地震成像中对伪影的理解。第三部分涉及多线性操作员，该操作员一次对多个功能起作用。这些是在研究如何相互作用以及离散几何形状中点云的研究中的研究。第四部分是一种设计共振设备（例如天线）的新方法。材料科学，物理和数学的最新进展导致了由结构化复合材料（也称为超材料）构建的设备设计的快速进步，这对波传播具有根本影响。一种特别成功的方法是基于转换光学器件。主要研究人员将研究一种新的非转化光学设计方法。拟议的两个项目涉及开发新工具，以处理流畅或实现功能的简单性或奇异性。在一个中，具有光滑相位的退化傅里叶积分运算符的组成会导致具有不平稳关系的波沿关系的运算符，并且了解如何将类似傅立叶的操作员将类似于傅立叶的操作员与这些几何相关联将扩大微局部分析的范围，并帮助分析某些反相反问题。另一方面，将需要发现新技术与具有实用相位阶段的振荡整体操作员在一个加上两个或两个加二维中的振荡阶段。困难包括了解两个或多个功能在多大程度上可以同时解决其零品种。一些相同的技术也将应用于尝试对跳跃数字的算法描述，这是当前在奇异理论和代数几何形状中兴趣的主题。这样做将加强分析与这些领域之间的联系。 项目的第三部分的进展不仅将进一步发展谐波分析中的多线性操作员的理论，而且还将立即应用于几何测量理论和离散的几何形状。 最终组件将显示来自线性偏微分方程的既定思想如何在设计和严格验证天线等共振结构的特性中有用，并将有助于为快速发展的超材料的快速发展区域贡献新的设计方法。