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.
