This research project concerns theoretical and computational aspects of inverse source problems and mass transport equations. Inverse source problems have a wide range of applications in science, engineering, and medicine. Among different types of inverse source problems, diffusive optical imaging, such as Fluorescence Molecular Tomography (FMT), is particularly important. FMT uses harmless infrared light, instead of X-ray, to capture molecular specific inclusions in biological tissues. It offers great potential in early cancer detection and drug monitoring. However, diffusive optical imaging demands large scale computation and careful treatment of the ill-posedness due to the diffusive nature of light propagation in tissues. This project aims to develop a new efficient and robust computation strategy for inverse source problems to improve image resolution and speed up computation significantly. Optimal mass transport theory plays a crucial role in many important applications, such as logistics, transportation, physics, and chemistry, and the theory also has potential application to study information propagation on social media. Despite remarkable development in the theory in continuous settings in recent years, much less is known concerning the problems on graphs or networks. This project will conduct theoretical and numerical analysis of graph-based mass transport problems, to design efficient and accurate simulation methodologies and to apply them to data analytics. The research activities will be integrated with education and training of undergraduates, graduate students, and postdocs through seminars and courses.
This project includes research in two areas: numerical methods for inverse source problems, and theoretical and numerical analyses for graph-based Fokker-Planck equations and mass transport problems. For inverse source problems, the research aims to develop a new 2-stage methodology, called orthogonal solution and kernel correction algorithm, to separate the competing requirements on regularity and boundary data fidelity in the common regularization approach, so that both requirements can be addressed more effectively. The method leverages an adaptive multiscale basis, the finite element method, and the spectral method to gain significant computation speed up and resolution improvement. The method can be integrated into FMT data acquisition equipment. Understanding of the Fokker-Planck equation and optimal mass transport theory that play crucial roles in many important applications has undergone remarkable development in continuous settings in recent years, but much less is known when one considers the problems on graphs or networks. This project will conduct theoretical and numerical analysis of graph based Fokker-Planck equations and mass transport problems, to design efficient and accurate simulation methodologies and to apply them to data analytics, which aim to understand and to optimize strategies to handle information hidden in large scale, high dimension data sets.
