Mathematical Analysis of Finite Difference Method Applied to Hamilton-Jacobi Equations: A Stochastic and Variational Approach
Mathematical Analysis of Finite Difference Method Applied to Hamilton-Jacobi Equations: A Stochastic and Variational Approach
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应用于 Hamilton-Jacobi 方程的有限差分法的数学分析：一种随机变分方法
DOI：
10.11540/bjsiam.32.3_127
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发表时间：
2022
期刊：
影响因子：
--
通讯作者：
曽我 幸平
中科院分区：
文献类型：
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作者：
Masanori Adachi;Nanao Kita;Miguel Cardona and Diego A. Mejia;曽我 幸平
这项研究解释了一种随机和变分的有限差分方法，该方法适用于由Tonelli类的哈密顿生成的一阶Hamilton-Jacobi方程。第一节阐述了哈密顿-雅可比方程与哈密顿动力学之间的联系，这意味着引入弱解的必要性。在第二节中，利用Tonelli变分法将特征线法得到的局部经典解的表示式推广到整体定义的函数。该函数在最优控制理论中称为值函数，是粘性解意义下的Hamilton-Jacobi方程的弱解。在第三节中，我们用最基本的有限差分方法对离散的Hamilton-Jacobi方程进行了类似于上一节的推导，其中由于数值粘性而出现了随机特征。用随机和变分技术证明了逼近的收敛，同时得到了粘性解及其导数和特征曲线。这种方法在很大程度上应用于弱Kolmogorov-Arnold-Moser(KAM)理论。
This study explains a stochastic and variational approach to the finite difference method applied to first order Hamilton-Jacobi equations generated by Hamiltonians of the Tonelli class. In Section 1, the connection between Hamilton-Jacobi equations and Hamiltonian dynamics is stated, which implies the necessity to introduce weak solutions. In Section 2, the representation formula of a local classical solution obtained via the method of characteristics is extended to a globally defined function employing Tonelli’s calculus of variations. This function is called a value function in optimal control theory, and it is a weak solution of the Hamilton-Jacobi equation in the sense of viscosity solutions. In Section 3, reasoning similar to the previous section is demonstrated for the discretized Hamilton-Jacobi equations with the most elementary finite difference method, where a stochastic feature appears due to numerical viscosity. The convergence of approximation is proven in terms of stochastic and variational techniques, yielding a viscosity solution, its derivative and characteristic curves all at once. This approach is substantially applied to weak Kolmogorov-Arnold-Moser (KAM) theory.