Numerical Methods for Inverse Problems of the Linear Boltzmann Transport Equation

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。本项目的目标是为许多应用领域中出现的线性和非线性逆输运问题开发快速和鲁棒的数值重建方法。提出的主要方法是将先验信息合并到过去开发的基于优化的算法中。先验信息既包括待重构未知量的知识，也包括前向运输问题数值方法的知识。更准确地说，建议的研究包括：(1)开发利用先验信息减少重建中未知数量的方法，并重建感兴趣对象中的特征；(2)通过分析前向运输问题的结构，加快重建过程，使更大的数据集可以用于重建过程；(3)开发基于传输的成像问题中不确定性量化的高效贝叶斯计算方法。所提出的研究介于数值数学和应用之间。从计算的角度来看，开发快速、鲁棒和精确的基于输运的病态反问题重构算法是非常具有挑战性的，这不仅是因为它涉及到先进的数值优化、数值偏微分方程技术，而且还因为它需要对病态反问题的理论有深刻的理解。在这个建议中发展的许多思想应该在其他基于模型的反问题的数值解中有直接的应用。从应用的角度来看，逆输运问题在医学成像（主要是光学层析成像和光学分子成像）、随机介质检测和成像以及大气光学等各个领域都有应用。该研究可能对这些应用领域产生长期影响，提高这些成像方法的稳定性和准确性。在这个项目中开发的一些想法和技术将被纳入研究生水平的反问题数值方法课程，这可能会使那些对应用数学和计算技术解决现实世界问题感兴趣的毕业生受益。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).The objective of this project is to develop fast and robust numerical reconstruction methods for linear and nonlinear inverse transport problems appeared in many application areas. The main approach proposed is to incorporate a priori information into optimization-based algorithms that have been developed in the past. The a priori information includes both knowledge on the unknowns to be reconstructed and knowledge on numerical methods for forward transport problems. More precisely, the proposed research include: (1) to develop methods that utilize a priori information to reduce the number of unknowns in the reconstruction, and to reconstruct features in the object of interests; (2) to accelerate the reconstruction process by analyzing the structure of the forward transport problem so that larger set of data can be used in the reconstruction process; and (3) to develop efficient Bayesian computational methods for uncertainty quantification in transport-based imaging problems. The proposed research lies between numerical mathematics and applications. From computational point of view, developing fast, robust and accurate reconstruction algorithms for ill-posed transport-based inverse problems is very challenging, not only because of the advanced numerical optimization, numerical partial differential equations techniques it involves, but also because of the fact that a deep understanding of the theory of ill-posed inverse problems is required. Many ideas developed in this proposal should have straightforward applications in numerical solutions of other model-based inverse problems. From application point of view, inverse transport problems find applications in various areas such as medical imaging (mainly optical tomography and optical molecular imaging), detection and imaging in random media, and atmospheric optics. The proposed research can potentially has long-term impacts in those application areas, improving both the stability and the accuracy of those imaging methods. Some of the ideas and techniques developed in this project will be incorporated into a graduate level class on numerical methods for inverse problems which presumably will benefit graduates who are interested in applying mathematical and computational techniques to solve real world problems.