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

该项目涉及一些成像、优化设计和反问题的计算研究，这些问题与动力学方程系统有关，这些方程组模拟了粒子在复杂非均质介质中的传播。目标是通过智能地利用模型方程和反问题的结构信息来开发快速和健壮的数值算法，这些信息是我们通过对它们的计算和数学研究获得的，有时是在简化的设置中。准确地说，提出的研究包括：(i)通过使用计算成本较低的模型来预处理计算，开发快速正演和反演算法；（ii）在涉及随机性的成像和逆问题中，为不确定性量化和方差减少开发有效的计算策略；（iii）开发和验证在异质介质中扩展目标成像的直接非迭代方法。这项研究的主要动机是将问题应用于(1)优化对癌症患者的放射治疗；（ii）设计高效的纳米级半导体器件；（三）在高散射随机环境中成像扩展目标。该研究预计将对用于癌症治疗的调强辐射治疗计划、用于能量收集的半导体太阳能电池设计以及用于国土安全应用的异构环境成像等实践产生长期影响。该项目包括一个重要的教育组成部分，旨在培养本科生和研究生。在这个项目中发展的思想和技术将被纳入成像和反问题的数值方法的研究生水平课程，这将有利于那些对应用先进的数学和计算技术来解决现实世界问题感兴趣的毕业生。这门研究生课程的课堂讲稿将通过PI所在机构的网页向公众开放。

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.