Inverse Boundary Value Problems with Incomplete Data

本文的目的是研究从在介质边界上获得的不完全测量数据中恢复介质内部参数和物理性质的数学理论。用偏微分方程来模拟物理情况，目的是通过对边界上解的测量来确定偏微分方程的系数。本研究的主题包括偏微分方程组的反问题（具有Yang-Mills势的Navier-Stokes方程和Schroedinger方程）和无界域上的反问题（无限平板和无限圆柱）。申请人追求三个具体目标：(1)证明确定结果的唯一性；(2)建立和完善稳定性评估；(3)建立解析重构公式。主要途径是在解决方案的构建、分析和控制上提出新的策略。它们在数学上具有挑战性，因为关于边界上解的可用信息较少，并且需要对微分算子和域的系统有深刻的理解。反边值问题包括通过边界测量恢复内部参数。这些问题及其解决方案对科学研究至关重要，因为在许多情况下，切开一个东西看看里面是什么是不可能或不可行的。一个人必须用间接的方法“看到”物体的内部，测量重要的现象，而不损害被分析的东西，而能够做到这一点的关键是逆问题。然而，在实践中，从整个边界收集数据有时要么是不可能的，要么是极其昂贵的。提出的研究为边界测量数据不完整时的反问题提供了新的见解。本建议所处理的问题出现在医学成像、石油勘探、人体血液液诊断等方面。提出的研究通过提供基本的数学分析，可以在这些应用领域产生潜在的长期影响。申请人将把本提案中提出的一些想法和技术纳入研究生水平课程，并指导研究生和本科生（少数民族）学生。

The objective of this proposal is to study the mathematical theory of recovering the internal parameters of a medium and physical properties from the incomplete measurements obtained on the boundary of the medium. The physical situation is modeled by partial differential equations and the goal is to determine the coefficients of the partial differential equations from some measurements of the solutions on the boundary. The topics of this research include the inverse problems for systems of partial differential equations (the Navier-Stokes equations and Schroedinger equations with Yang-Mills potentials) and the inverse problems on unbounded domains (an infinite slab and an infinite cylinder). The proposer pursues three specific aims: (1) to prove unique determination results; (2) to establish and improve stability estimates; (3) to develop analytical reconstruction formulas. The main approach is the new strategies on the construction, analyzing and control of the solutions. They are mathematically challenging because less information about the solutions on the boundary is available and a deep understanding of systems of differential operators and domains is required. Inverse boundary value problems consist of recovering internal parameters by boundary measurements. Such problems and their solutions are critical to scientific endeavors, because in many instances it simply is not possible or feasible to cut something open to see what is inside. One must use indirect methods to "see" inside objects and measure vital phenomena without harming the thing being analyzed, and the key to being able to that is inverse problems. However, collecting data from the whole boundary is sometimes either not possible or extremely expensive in practice. The proposed research provides new insights to inverse problems when the data obtained by boundary measurements is incomplete. The problems addressed in this proposal arise in medical imaging, oil exploration, diagnosis of human blood fluid, among others. The proposed research can potentially have long-term impacts in those application areas by providing the fundamentally mathematical analysis. The proposer will incorporate some of the ideas and techniques developed in this proposal into a graduate level class, and mentor graduate and undergraduate (minority) students.