Mathematics of Revealing Inaccessible Objects Using Linear and Nonlinear Waves

该项目涉及反问题的数学理论。在许多医学和地震成像应用以及勘探、地球物理和非破坏性评价中都会出现逆问题，在这些应用中，人们感兴趣的是通过在外部进行的测量产生介质内部不可接近的图像。通常，为了揭示内部结构，人们会测量介质在被不同种类的波（从电磁波到x射线）探测时的反应。最近，人们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。同样，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的对比度通常大于线性参数。该项目致力于发展显著突出的数学方法，其中波的非线性相互作用用于承担来自应用的具有挑战性的逆问题。这些新颖的数学技术可能会带来重大进步，特别是在医学和地震成像方面。该项目的一个具体重点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部件无损检测的成像方式。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将通过沿一小部分边界进行的测量来确定。尽管这类逆问题具有重要意义，并且在应用中无处不在，但它们是该领域最基本的开放问题之一。本文的目标是在研究者和合作者的最新进展的基础上，解决这些重要的非线性PDE问题，包括拟线性各向异性电导率方程，并努力解决这些线性环境下的开放性问题。第二个主题是致力于各向异性卡尔德隆问题的几何版本，其中一个试图确定在紧致黎曼流形薛定谔方程的势从边界测量。通过引入非线性，目标是求解几何设置下非线性薛定谔方程的各向异性Calderon问题，其对应的逆问题在线性情况下仍然是开放的。第三个主题涉及物理和几何中基本系统的逆问题，如各向异性麦克斯韦和杨-米尔系统。第四个主题是关于有边界流形上的非线性双曲偏微分方程的逆问题，旨在利用非线性作为求解线性对应物开时的一些逆问题的工具。该奖项反映了美国国家科学基金会的法定使命，并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.