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.
