Theory and Algorithms for Feedback Particle Filter

在许多工程应用中，为复杂的优化问题和其他相关的具有挑战性的数学问题找到准确的解是非常重要的。一些示例应用包括：目标跟踪和监视，其中使用多个传感器测量来跟踪目标，空中交通管理来跟踪飞机，天气监视来跟踪飓风，地面测绘，地球物理测量，遥感，自主导航和机器人。最先进的解决这些问题的方法包括卡尔曼滤波算法及其许多扩展。然而，在实践中，由于与动力学和不确定性的复杂性相关的技术问题，这种方法可能产生不准确和错误的解决方案。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“反馈粒子滤波”。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，并为上述跟踪应用的软件工具奠定基础。该项目还包括几项旨在鼓励本科生创业的教育举措。研究的一个主要目标是基于最优运输理论和平均场博弈形式的反馈粒子滤波器的最优控制公式的发展。理论研究与计算算法工作紧密结合。算法目标涉及泊松方程的数值解，有限多粒子粒子系统的收敛性分析，以及与基于重要采样的算法的比较。可交付成果包括将在软件中实现和演示的有效数值方案。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值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.