Algorithms and software for the hamilton-jacobi equation and applications in control, robotics and verification

我建议开发一个哈密顿-雅可比工作坊(HJW)，它是一个算法和实现的集合，用于逼近哈密顿-雅可比(HJ)偏微分方程(PDE)的时间依赖和定常版本的解。该方程出现在流体和燃烧模拟、图形和动画、图像处理和计算机视觉、金融数学、机器人和最优控制以及验证等多个领域；因此，它受到了相当大的研究关注。我的研究议程与数字分析师对这个方程的研究截然不同，并与之相辅相成。虽然该社区已经开发了强大、高效和准确的数值方法，但来自其他领域的研究人员和从业者发现很难在各种方案中进行选择和实施，而HJ PDE的新手必须复制巨大的软件基础设施来展示他们自己的贡献。因此，我的研究将集中在开发易于使用和可扩展的实现，在准确性和功能的前沿，但因式分解的方式，以提供优雅的灵活性，以尝试各种替代方案。在必要的情况下，代码可能不会达到最佳的执行效率，但通过访问所有源代码，一旦用户确定了针对其特定应用程序的最佳方案，他们就可以调整或重新实现。这种技术的集合不能孤立地开发，这项研究将从HJW用户的反馈和应用领域的个人研究中获得信息，例如车辆的可证明安全的控制综合和过滤，动态运动约束和多目标下的路径规划，以及连续和混合系统中的自动可达性工具。

I propose development of a Hamilton-Jacobi Workshop (HJW), which is a collection of algorithms and implementations for approximating the solution of both the time dependent and stationary versions of the Hamilton-Jacobi (HJ) partial differential equation (PDE). This equation shows up in such diverse fields as fluid and combustion simulation, graphics and animation, image processing and computer vision, financial mathematics, robotics and optimal control, and verification; for this reason it has received considerable research attention. My research agenda is distinct from and complementary to the study of this equation undertaken by numerical analysts. While that community has developed powerful, efficient and accurate numerical methods, researchers and practitioners from other fields find it difficult to choose among and implement the various schemes, and students new to HJ PDEs must replicate an enormous software infrastructure to demonstrate their own contributions. Therefore, my research will focus on developing easy to use and extensible implementations at the cutting edge of accuracy and power, yet factorized in such a manner as to provide graceful flexibility for trying various alternative schemes. Where necessary the code may not achieve optimal execution efficiency, but with access to all of the source code users can tune or reimplement once they have determined the best schemes for their particular application. Such a collection of techniques cannot be developed in isolation, and this research will be informed by feedback from HJW users and by personal study in application fields, such as provably safe control synthesis and filtering for vehicles, path planning under dynamic motion constraints and multiple objectives, and automated tools for reachability in continuous and hybrid systems.