The Fast Sweeping Method and Its Applications

近年来发展起来的快速扫描方法是求解一类Hamilton-Jacobi方程的有效迭代方法。在本项目中，将对快速扫描方法进行持续的研究和分析。将继续把这种方法应用于其他问题。特别是以下工作需要完成：(1)理解与控制/博弈论的联系。(2)研究了迭代法一般框架下快速扫描算法的收缩特性。(3)数值解及其导数的误差分析。(4)发展其他类型双曲型偏微分方程的快速扫描方法，如双曲守恒定律和辐射输运方程。Hamilton-Jacobi方程是一类非线性偏微分方程，在经典力学、最优控制、地球物理、几何光学和图像处理等领域有着广泛的应用。这类问题的非线性对数学分析和开发有效的数值算法提出了巨大的挑战。快速扫描法是一种简单的迭代法。它能有效地解决一类复杂的非线性问题，值得我们进一步认识和发展。该项目的广泛影响不仅是为科学和工程中的实际应用提供了有效的数值方法，而且为构建其他非线性问题的迭代方法提供了见解。此外，还将设计与不同层次的教育相结合。

The recently developed fast sweeping method has been proved to be an efficient iterative method for a class of Hamilton-Jacobi equations. In this project, a continuing study and analysis of the fast sweeping method will be carried out.Applications of the method to other problems will be pursued.In particular the following work will be done:(1) Understand the connection to control/game theory.(2) Study the contraction property of the fast sweeping algorithm in the general framework of iterative method.(3) Error analysis for the numerical solution and its derivative.(4) Develop fast sweeping method for other type of hyperbolic partial differential equations, such as hyperbolic conservation laws and radiative transport equation.Hamilton-Jacobi equations are nonlinear partial differential equations that have many applications in classical mechanics, optimal control, geophysics, geometric optics and image processing. The nonlinearity of this type of problem poses great challenges for mathematical analysis and for developing efficient numerical algorithms. The fast sweeping method is a simple iterative method. It can work efficiently for a class of difficult nonlinear problem, which is quite remarkable and worth further understanding and development.The broader impact of this project is not only to provide efficient numerical methods for real applications in science and engineering but also to shed insight for constructing iterative methods for other nonlinear problems.In addition integration with education at different levels will also be designed.