A new approximation for effective Hamiltonians

Hamilton-Jacobi方程是一类非线性双曲型偏微分方程，一般不存在经典解。必须定义适当的弱解，即粘度解。因此，传统的均匀化技术的基础上渐近展开和假设的规则性不工作。非线性问题均匀化的数学理论和数值方法还远远不够。目前，Hamilton-Jacobi方程的均匀化是通过为每个动量变量定义一个胞元问题来实现的。因此，许多细胞问题需要解决。这项研究的主要动机是PI及其合作者提出的一种新的公式，该公式将有效哈密顿量与合适的有效方程联系起来。该公式的主要优点是，只需求解一个辅助方程，即可计算所有动量变量的有效哈密顿量。此外，在我们制定的有效方程是一个标准的Hamilton-Jacobi方程的边界值，许多有效的数值算法。Hamilton-Jacobi方程在经典力学、动力系统、最优控制、微物理、几何光学、燃烧和图像处理等领域有着重要的应用。在许多应用中，相应的哈密顿量可能具有多个尺度，如经典力学中的振荡势或波前传播中的脉动速度场。在这个项目中，PI提出了一个新的配方来研究一类Hamilton-Jacobi方程的均匀化，并开发了有效的数值算法来计算相应的有效哈密顿量。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，还将设计与各级教育的融合。与拟议研究有关的监督研究项目和研讨会将提供给大三/大四本科生和毕业生。

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.