A new approximation for effective Hamiltonians

Hamilton-Jacobi方程是一般不存在经典解的非线性双曲型偏微分方程解。必须定义适当的弱解，即粘性解。因此，传统的基于渐近展开和正则性假设的均匀化技术不起作用。对于非线性问题的齐次化，无论是数学理论还是数值方法都是远远不够的。目前，哈密顿-雅可比方程的齐次化是通过定义每个动量变量的胞元问题来实现的。因此，许多电池问题必须得到解决。这项研究的主要动机是PI和他的合作者提出的一个新的公式，它将有效哈密顿量与一个合适的有效方程联系起来。该公式的主要优点是只需解一个辅助方程即可计算出所有动量变量的有效哈密顿量。此外，我们的公式中的有效方程是一个标准的带边值的Hamilton-Jacobi方程，对于该方程有许多有效的数值算法。哈密顿-雅可比方程在经典力学、动力学系统、最优控制、地球物理、几何光学、燃烧和图像处理等领域有着重要的应用。对于许多应用，相应的哈密顿量可以有多个尺度，例如经典力学中的振动势或前沿传播中的脉动速度场。在这个项目中，PI提出了一种新的公式来研究一类Hamilton-Jacobi方程的齐次化，并开发了计算相应有效哈密顿量的有效数值算法。该项目的研究成果将为科学和工程中的许多应用提供重要的数学基础和有效的数值方法。此外，还将设计与不同级别的教育相结合。与拟议研究相关的受监督的研究项目和研讨会将提供给初级/高级本科生和毕业生。

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.