Mathematics of Revealing Inaccessible Objects Using Linear and Nonlinear Waves

该项目涉及反问题的数学理论。逆问题出现在许多医学和地震成像应用以及勘探地球物理学和非破坏性评估中，其中人们感兴趣的是从外部执行的测量产生介质的不可接近内部的图像。通常情况下，为了揭示内部结构，当用不同种类的波（从电磁波到X射线）探测时，测量介质的响应。最近，人们已经观察到，非线性地震反应可以提供有关地球内部结构的额外信息。类似地，医学成像中的非线性超声技术可以提供更好的图像，因为非线性介质参数中的对比度通常大于线性参数中的对比度。该项目致力于开发显着突出的数学方法，其中波的非线性相互作用用于承担来自应用程序的具有挑战性的逆问题。这些新的数学技术可能会导致显着的进步，特别是在医疗和地震成像。该项目的具体重点之一是非线性各向异性介质的电阻抗断层成像问题，这是一种应用于生物医学成像和机械零件无损检测的成像模式。该项目的一个组成部分是关于研究生的教育培训。该项目包括四个研究课题。第一个主题涉及非线性椭圆型偏微分方程（PDE）的部分数据反问题，其中未知介质的非线性参数是由沿沿着一小部分边界进行的测量确定的。尽管这些反问题的重要意义及其在应用中的普遍性，它们是该领域中一些最基本的开放问题。这里的目标是解决这些问题的重要非线性偏微分方程，包括准线性各向异性电导率方程，建立在研究人员和合作者的最新进展，并努力解决这些开放的问题，在线性设置。第二个主题是致力于各向异性卡尔德龙问题的几何版本，其中一个旨在确定潜在的薛定谔方程上的一个紧凑的黎曼流形从边界测量。通过引入一个非线性，目标是解决各向异性Calderon问题的非线性薛定谔方程的几何设置，相应的反问题的线性情况下仍然是开放的。第三个主题是讨论物理学和几何学中基本系统的反问题，如各向异性的麦克斯韦和杨-密尔系统。第四个课题是关于带边界流形上的非线性双曲偏微分方程的反问题，目的是在线性方程组开放的情况下，利用非线性作为解决其中一些问题的工具。该奖项反映了NSF的法定使命，通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The project is concerned with the mathematical theory of inverse problems. Inverse problems arise in numerous medical and seismic imaging applications as well as in exploration geophysics and non-destructive evaluation, where one is interested in producing images of an inaccessible interior of a medium from measurements performed in the exterior. Typically, in order to reveal the internal structure, one measures the response of the medium when probed with different kinds of waves, ranging from electromagnetic waves to X-rays. Recently, it has been observed that nonlinear seismic responses may give additional information concerning the interior structure of the Earth. Similarly, nonlinear ultrasound techniques in medical imaging may provide better images since the contrast in nonlinear media parameters is usually larger than that in the linear parameters. The project strives to develop significantly prominent mathematical methods where the nonlinear interaction of waves is used to bear on challenging inverse problems coming from applications. These novel mathematical techniques may lead to significant advances, in particular in medical and seismic imaging. One of the specific focuses of the project is the Electrical Impedance Tomography problem for nonlinear anisotropic media, an imaging modality with applications in biomedical imaging and non-destructive testing of mechanical parts. An integral part of the project is concerned with the educational training of graduate students.The project consists of four research topics. The first topic deals with partial data inverse problems for nonlinear elliptic partial differential equations (PDE), where nonlinear parameters of an unknown medium are to be determined from measurements performed along a small portion of the boundary. Despite the great significance of such inverse problems and their ubiquity in applications, they are among some of the most fundamental open questions in the field. The goal here is to solve such problems for important nonlinear PDE, including the quasilinear anisotropic conductivity equation, building upon the recent advances by the investigator and collaborators, and to work towards the solution of these open problems in the linear setting. The second topic is devoted to the geometric version of the anisotropic Calderon problem, where one seeks to determine potential in the Schrodinger equation on a compact Riemannian manifold from boundary measurements. By introducing a nonlinearity, the goal is to solve the anisotropic Calderon problem for the nonlinear Schrodinger equation in geometric settings for which the corresponding inverse problem in the linear case is still open. The third topic deals with inverse problems for the fundamental systems in physics and geometry, such as the anisotropic Maxwell and Yang-Mill’s systems. The fourth topic is concerned with inverse problems for nonlinear hyperbolic PDE on manifolds with boundary, aiming to exploit the nonlinearity as a tool to solve some of them in cases when the linear counterpart is open.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.