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A HJB-POD approach for the control of nonlinear PDEs on a tree structure
Applied Numerical Mathematics ( IF 2.8 ) Pub Date : 2020-09-01 , DOI: 10.1016/j.apnum.2019.11.023
Alessandro Alla , Luca Saluzzi

The Dynamic Programming approach allows to compute a feedback control for nonlinear problems, but suffers from the curse of dimensionality. The computation of the control relies on the resolution of a nonlinear PDE, the Hamilton-Jacobi-Bellman equation, with the same dimension of the original problem. Recently, a new numerical method to compute the value function on a tree structure has been introduced. The method allows to work without a structured grid and avoids any interpolation. Here, we aim to test the algorithm for nonlinear two dimensional PDEs. We apply model order reduction to decrease the computational complexity since the tree structure algorithm requires to solve many PDEs. Furthermore, we prove an error estimate which guarantees the convergence of the proposed method. Finally, we show efficiency of the method through numerical tests.

中文翻译:

用于控制树结构上非线性偏微分方程的 HJB-POD 方法

动态规划方法允许计算非线性问题的反馈控制,但受到维数灾难的影响。控制的计算依赖于非线性偏微分方程(Hamilton-Jacobi-Bellman 方程)的分辨率,与原始问题的维度相同。最近,引入了一种新的数值方法来计算树结构上的值函数。该方法允许在没有结构化网格的情况下工作并避免任何插值。在这里,我们旨在测试非线性二维偏微分方程的算法。我们应用模型降阶来降低计算复杂度,因为树结构算法需要解决许多 PDE。此外,我们证明了保证所提出方法收敛的误差估计。最后,我们通过数值测试证明了该方法的有效性。
更新日期:2020-09-01
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