A new approximation for effective Hamiltonians

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Hamilton-Jacobi equations are nonlinear hyperbolic partial differential equation for which classical solution does not exist in general. Appropriate weak solution, the viscosity solution, has to be defined. Hence traditional homogenization techniques based on asymptotic expansion and assumption of regularity do not work. Both mathematical theory and numerical method for homogenization of nonlinear problem is far from adequate. Currently the homogenization of Hamilton-Jacobi equation is through the definition of a cell problem for each momentum variable. Hence many cell problems have to be solved. The key motivation of this study is a new formulation proposed by the PI and his collaborators that links the effective Hamiltonian to a suitable effective equation. The main advantage of this formulation is that only one auxiliary equation needs to be solved in order to compute the effective Hamiltonian for all momentum variables. Furthermore, the effective equation in our formulation is a standard Hamilton-Jacobi equation with boundary value for which many efficient numerical algorithms are available. Hamilton-Jacobi equations have many important applications in classical mechanics, dynamical systems, optimal control, geophysics, geometric optics, combustion and image processing. For many applications the corresponding Hamiltonians may have multiple scales, such as oscillatory potential in classical mechanics or fluctuating velocity field in front propagation. In this project the PI proposes a new formulation to study homogenization of a class of Hamilton-Jacobi equations and develop efficient numerical algorithms for computing the corresponding effective Hamiltonians. Results coming from this project will provide important mathematical foundation and efficient numerical methods for many applications in science and engineering. In addition integration with education at different levels will be designed. Supervised research projects and seminars related to the proposed research will be available to junior/senior undergraduates and graduates.