Computational Algorithms for Imaging, Design and Inverse Problems of Particle Propagation in Heterogeneous Media

该项目涉及与模拟粒子在复杂非均匀介质中传播的动力学方程组相关的一些成像、优化设计和逆问题的计算研究。我们的目标是开发快速和强大的数值算法，智能地利用信息的结构模型方程和反问题，我们通过计算和数学研究，有时在简化的设置。准确地说，所提出的研究包括：（i）通过用计算成本较低的模型对计算进行预处理来开发快速正演和反演算法;（ii）开发用于成像和逆问题中的不确定性量化和方差减小的有效计算策略，所述成像和逆问题涉及随机性;以及（iii）开发和验证用于在非均匀介质中的扩展目标中成像的直接非迭代方法。该研究的主要动机是应用问题（i）优化癌症患者的放射治疗;（ii）设计高效的纳米级半导体器件;（iii）在高散射随机环境中成像扩展目标。预计拟议的研究将对癌症治疗的强度调制辐射治疗计划，能量收集的半导体太阳能电池设计以及国土安全应用的异构环境中的成像实践产生长期影响。该项目包括一个重要的教育部分，旨在培训本科生和研究生。在这个项目中开发的想法和技术将被纳入研究生水平的成像和逆问题的数值方法，这将有利于毕业生谁有兴趣应用先进的数学和计算技术来解决现实世界的问题。这门研究生课程的课堂讲稿将通过PI所在机构主办的PI网页向公众开放。

This project is concerned with the computational study of some imaging, optimal design and inverse problems related to systems of kinetic equations that model the propagation of particles in complex heterogeneous media. The objective is to develop fast and robust numerical algorithms by intelligently utilizing information on the structures of both the model equations and the inverse problems that we obtained through computational and mathematical studies of them, sometimes in simplified settings. Precisely, the proposed research include (i) to develop fast forward and inversion algorithms by preconditioning the computation with computationally less expensive models; (ii) to develop efficient computational strategies for uncertainty quantification and variance reduction in the imaging and inverse problems involve randomness; and (iii) to develop and validate direct non-iterative methods for imaging in extended targets in heterogeneous media. The main motivation for the study is the application of the problems in (i) optimizing the delivery of radiation therapy to caner patients; (ii) designing efficient nano-scale semiconductor devices; and (iii) imaging extended targets in highly-scattering random environments. The proposed research is expected to have long-term impacts in the practices of intensity-modulated radiation treatment planning for cancer treatment, semiconductor solar cell design for energy harvesting, and imaging in heterogeneous environment for homeland security applications. The project involves a significant education component that aims at training both undergraduate and graduate students. The ideas and techniques developed in this project will be incorporated into a graduate level class on numerical methods for imaging and inverse problems which will benefit graduates who are interested in applying advanced mathematical and computational techniques to solve real-world problems. The lecture notes for this graduate class will be made accessible to the general public through the PI's webpage hosted by the PI's institution.