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

这个项目的重点是两个应用程序，利用反问题领域的方法。第一个关注下一代医学成像方法的数学分析，并有助于基于电流密度的成像方法的理论基础。成功的结果将使图像具有比当前医学成像模式更高的质量和准确性。这种高度准确的成像方法对于癌症的早期检测、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。这些方法也适用于分析具有规定电流的电网络，并将包括在图上开发随机游走模型来描述电导率的转变。图上的随机游走出现在许多科学领域，拟议的研究对计算机和社交网络、密码学、流行病学、统计物理学、经济学、本项目的第一部分重点是从内部感应电流密度矢量场和Dirichlet或Neumann边界的知识中恢复体内电导率的逆问题条件该混合逆问题结合了磁共振成像（MRI）的高分辨率和电阻抗断层成像（EIT）的高对比度，以提供具有高分辨率和高对比度的图像。它与变系数的加权最小梯度问题和1-Laplacian方程有着密切的联系，这类问题解的存在性和唯一性还有待进一步的研究。第二部分包括从沿沿着边的感应电流确定电网络的电导率矩阵的问题。它可以推广到确定转移概率的随机行走模型的图从预期的净次数的随机步行者通过沿着边的知识的逆问题。这些基本数学问题的结果将有助于各种看似无关的具体真实的世界的应用，并可以扩展到许多其他领域。这个项目带来承担各种数学工具，从偏微分方程，变分法，极小曲面理论，几何测量理论和数值分析。

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.