Homogenization of Metric Hamilton-Jacobi Equations
Homogenization of Metric Hamilton-Jacobi Equations
复制标题
度量 Hamilton-Jacobi 方程的齐次化
DOI：
10.1137/080743019
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发表时间：
2009
期刊：
影响因子：
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通讯作者：
A. Vladimirsky
中科院分区：
文献类型：
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作者：
Adam M. Oberman;Ryo Takei;A. Vladimirsky
在这项工作中，我们为一类静态 Hamilton-Jacobi (HJ) 方程提供了一种新颖的均质化方法，我们将其称为度量 HJ 方程。我们将 HJ 方程的解与相应黎曼或芬斯勒度量中的距离函数联系起来。度量方法使我们能够得出结论，均质方程也可以导出度量。该方法的优点是我们只需求解一个辅助方程即可恢复齐次哈密顿量 $\bar{H}(p)$。这是对现有方法的重大改进，现有方法需要针对每个 p 值求解单元问题（或变分问题）。当可用于分段恒定周期和随机速度函数时，给出计算结果并与分析结果进行比较。
In this work we provide a novel approach to homogenization for a class of static Hamilton–Jacobi (HJ) equations, which we call metric HJ equations. We relate the solutions of the HJ equations to the distance function in a corresponding Riemannian or Finslerian metric. The metric approach allows us to conclude that the homogenized equation also induces a metric. The advantage of the method is that we can solve just one auxiliary equation to recover the homogenized Hamiltonian $\bar{H}(p)$. This is a significant improvement over existing methods which require the solution of the cell problem (or a variational problem) for each value of p. Computational results are presented and compared with analytic results when available for piecewise constant periodic and random speed functions.