Imaging Electrical Conductivities from Their Induced Current and Network Tomography for Random Walks on Graphs

本项目着重于利用反问题领域方法的两个应用。第一部分涉及下一代医学成像方法的数学分析，并为当前基于密度的成像方法提供理论基础。成功的结果将使图像具有比当前医学成像方式更高的质量和准确性。这种高度精确的成像方法对于癌症的早期发现、诊断和治疗至关重要，并为风险和侵入性手术提供了替代方案，降低了治疗成本。这些方法也适用于具有规定电流的电气网络的分析，并将包括在图上开发随机游走模型来描述电导率的转变。图上的随机漫步出现在许多科学领域，提出的研究对计算机和社会网络、密码学、流行病学、统计物理学、经济学和生物学的分析有直接影响。项目的第一部分侧重于从内部感应电流密度矢量场和狄利克雷或诺伊曼边界条件的知识中恢复物体内部电导率的逆问题。这种混合反问题结合了高分辨率的磁共振成像（MRI）和高对比度的电阻抗断层成像（EIT），提供高分辨率和高对比度的图像。它与加权最小梯度问题和变系数1-拉普拉斯方程密切相关，需要进一步研究这类问题解的存在唯一性。第二部分包括从沿边缘的感应电流确定电导率矩阵的问题。它可以推广到根据随机漫步器沿边缘经过的期望净次数来确定图上随机漫步模型转移概率的反问题。这些基本数学问题的结果将有助于各种看似无关的具体现实世界应用，并可以扩展到许多其他领域。本项目运用了偏微分方程、变分法、最小曲面理论、几何测量理论和数值分析等多种数学工具。

This project focuses on two applications that utilize methods from the field of inverse problems. The first concerns the mathematical analysis of the next generation medical imaging methods and contributes to the theoretical foundation of current density based imaging methods. Successful results will allow images with significantly higher quality and accuracy than possible with current medical imaging modalities. Such highly accurate imaging methods are crucial for early detection, diagnoses, and treatment of cancer, and provide alternatives for risky and invasive procedures and reduce the cost of treatment. The methods also apply to the analysis of electrical networks with a prescribed current, and will include the development of random walk models on graphs to describe transitions in conductivity. Random walks on graphs arise in many areas of science and the proposed research has direct impact in the analysis of computer and social networks, cryptography, epidemiology, statistical physics, economics, and biology.The first part of the project focuses on the inverse problem of recovering the electrical conductivity inside a body from the knowledge of the induced current density vector field in the interior and Dirichlet or Neumann boundary conditions. This hybrid inverse problem combines high resolution of Magnetic Resonance Imaging (MRI) and high contrast of Electrical Impedance Tomography (EIT) to provide images with high resolutions and high contrast. It is closely related to the weighted least gradient problems and 1-Laplacian equations with variable coefficients, and more study is needed on the existence and uniqueness of solutions of such problems. The second part includes the problem of determining the conductivity matrix of an electrical network from the induced current along the edges. It can be generalized to the inverse problem of determining transition probabilities of random walk models on graphs from the knowledge of the expected net number of times the random walker passes along the edges. The results from these fundamental mathematical questions will contribute to various seemingly unrelated specific real world applications and can be extended to many other areas. This project brings to bear a variety of mathematical tools from partial differential equations, calculus of variations, theory of minimal surfaces, geometric measure theory, and numerical analysis.