Theory and Algorithms for Feedback Particle Filter

在许多工程应用中，寻找复杂优化问题和其他具有挑战性的数学问题的精确解是非常重要的。一些示例应用包括：使用多个传感器测量来跟踪目标的目标跟踪和监视、用于跟踪飞机的空中交通管理、用于跟踪飓风的天气监视、地面测绘、地球物理调查、遥感、自主导航和机器人。解决这些问题的最先进的方法包括卡尔曼滤波算法及其许多扩展。然而，在实践中，这种方法可能会产生不准确和错误的解决方案，因为与动力学和不确定性的复杂性有关的技术问题。在过去的十年中，出现了一类新的算法解决这些问题的方法，称为反馈粒子过滤器。反馈粒子滤波器可以更好地处理与这种复杂的动力学和不确定性相关的技术问题。这项研究将推进反馈粒子滤波算法的理论发展和验证，并为在上述跟踪应用中有用的软件工具奠定基础。该项目还包括几项旨在让本科生参与创业的教育倡议。该研究的一个主要目的是基于最优传输理论和平均场博弈理论建立反馈粒子滤波的最优控制公式。理论研究与计算算法的工作紧密结合在一起。算法目标涉及Poisson方程的数值解，有限粒子粒子系统的收敛分析，以及与基于重要性抽样的算法的比较。可交付成果包括将在软件中实施和演示的高效数值方案。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Finding accurate solutions for complex optimization problems and other related challenging mathematical problems is very important in many engineering applications. Some example applications include: target tracking and surveillance where multiple sensor measurements are used to track targets, air traffic management to track airplanes, weather surveillance to track hurricanes, ground mapping, geophysical surveys, remote sensing, autonomous navigation, and robotics. State-of-the-art solution approaches to these problems include the Kalman filter algorithm and its many extensions. However, in practice, such approaches can yield inaccurate and erroneous solutions because of technical issues related to complexity in dynamics and uncertainty. In the past decade, a new class of algorithmic solution approaches to these problems has emerged referred to as the "Feedback Particle Filter". The Feedback Particle Filter can better handle the technical issues related to such complex dynamics and uncertainty. This research will advance the theoretical development and verification of the Feedback Particle Filter algorithm, and lay the groundwork for software tools that will be useful in tracking applications noted above. The project also includes several educational initiatives that seek to engage undergraduate students in entrepreneurship. A major objective of the research concerns the development of optimal control formulations of the feedback particle filter based on optimal transportation theory and mean-field games formalisms. The theoretical research is closely integrated with the work on computational algorithms. The algorithmic objectives pertain to numerical solution of the Poisson equation, convergence analysis of the particle system with finitely many particles, and comparisons with importance sampling-based algorithms. The deliverables include efficient numerical schemes which will be implemented and demonstrated in software.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.