Theory, Methods for Diffusive Optical Imaging, Graph Based Fokker-Planck Equations and Mass Transportations

这项研究项目涉及逆源问题和质量输运方程的理论和计算方面。逆源问题在科学、工程、医学等领域有着广泛的应用。在不同类型的逆源问题中，扩散光学成像尤其重要，如荧光分子层析成像(FMT)。FMT使用无害的红外线，而不是X射线，来捕捉生物组织中的分子特定包裹体。它在癌症早期检测和药物监测方面具有巨大的潜力。然而，由于光在组织中传播的扩散特性，扩散光学成像需要大规模的计算和仔细的病态处理。该项目旨在开发一种新的高效和健壮的计算策略来解决逆源问题，以提高图像分辨率并显著加快计算速度。最优质量传输理论在物流、交通、物理、化学等领域有着重要的应用，在研究社交媒体上的信息传播方面也具有潜在的应用前景。尽管近年来这一理论在不断的背景下取得了显著的发展，但关于图或网络上的问题却知之甚少。该项目将对基于图形的质量传输问题进行理论和数值分析，设计有效和准确的模拟方法，并将其应用于数据分析。研究活动将通过研讨会和课程与本科生、研究生和博士后的教育和培训相结合。本项目包括两个领域的研究：逆源问题的数值方法，以及基于图形的福克-普朗克方程和质量传输问题的理论和数值分析。对于逆源问题，本研究旨在开发一种新的两阶段方法，称为正交解和核校正算法，以分离常规正则化方法中对正则性和边界数据保真度的竞争要求，从而更有效地解决这两个要求。该方法利用自适应多尺度基、有限元方法和谱方法来获得显著的计算速度和分辨率改善。该方法可以集成到FMT数据采集设备中。福克-普朗克方程和最优质量传输理论在许多重要的应用中起着至关重要的作用，近年来，人们对它们的理解在连续的环境中取得了显著的发展，但当人们考虑图或网络上的问题时，人们所知的要少得多。该项目将对基于图形的福克-普朗克方程和质量传输问题进行理论和数值分析，设计高效和准确的模拟方法，并将其应用于数据分析，旨在理解和优化策略，以处理隐藏在大规模、高维数据集中的信息。

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.