论文标题
部分可观测时空混沌系统的无模型预测
Fast operator splitting methods for obstacle problems
论文作者
论文摘要
障碍问题是一类自由边界问题，这些问题在许多学科中找到了应用程序，例如多孔媒体，金融数学和最佳控制。在本文中，我们提出了两种操作员分解方法，以解决线性和非线性障碍物问题。所提出的方法具有三种成分：（i）利用指标函数将约束问题作为不受约束的问题进行公式化，并将其与初始值问题相关联。然后将障碍问题转换为求解初始值问题的稳态解决方案。 （ii）运算符分拆策略，以分散初始值问题。分裂后，解决了带有障碍物的热方程式，其他子问题具有明确的解决方案，或者可以有效地解决。 （iii）一种新的约束交流方向显式方法，一种完全显式的方法，可以用障碍物解决热方程。提出的方法易于实现，不需要求解任何线性系统，并且比现有的数值方法更有效，同时保持相似的精度。还讨论了提出的方法扩展到相关的自由边界问题。
The obstacle problem is a class of free boundary problems which finds applications in many disciplines such as porous media, financial mathematics and optimal control. In this paper, we propose two operator-splitting methods to solve the linear and nonlinear obstacle problems. The proposed methods have three ingredients: (i) Utilize an indicator function to formularize the constrained problem as an unconstrained problem, and associate it to an initial value problem. The obstacle problem is then converted to solving for the steady state solution of an initial value problem. (ii) An operator-splitting strategy to time discretize the initial value problem. After splitting, a heat equation with obstacles is solved and other subproblems either have explicit solutions or can be solved efficiently. (iii) A new constrained alternating direction explicit method, a fully explicit method, to solve heat equations with obstacles. The proposed methods are easy to implement, do not require to solve any linear systems and are more efficient than existing numerical methods while keeping similar accuracy. Extensions of the proposed methods to related free boundary problems are also discussed.
