The Fast Sweeping Method and Its Applications

快速扫掠方法及其应用

• 批准号: 0811254
• 负责人: Hong-Kai Zhao
• 金额: $ 15.33万
• 依托单位: University of California-Irvine
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-07-01 至 2011-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0811254&HistoricalAwards=false
• 关键词: Fast; Sweeping; Method; Its; Applications

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.

最近发展的快速扫描方法已被证明是一类 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。将寻求该方法在其他问题中的应用。特别将完成以下工作：（1）理解与控制/博弈论的联系.(2)在迭代法的一般框架下研究快速扫描算法的收缩性质。(3)数值解及其导数的误差分析。(4)开发其他类型双曲偏微分方程的快速扫描方法，例如双曲守恒定律和辐射输运方程。Hamilton-Jacobi 方程是非线性偏微分方程，在经典力学、最优控制、地球物理学、几何光学和图像处理中有许多应用。这类问题的非线性给数学分析和开发高效的数值算法带来了巨大的挑战。快速扫描法是一种简单的迭代方法。它可以有效地解决一类困难的非线性问题，这是非常了不起的，值得进一步理解和发展。该项目更广泛的影响不仅是为科学和工程的实际应用提供有效的数值方法，而且为构建其他非线性问题的迭代方法。此外，还将设计与不同层次教育的结合。