[论文解读] River environmental restoration based on random observations of a non-smooth stochastic dynamical system

Hidekazu Yoshioka|arXiv (Cornell University)|May 11, 2020

Hydrology and Watershed Management Studies参考文献 67被引用 5

一句话总结

本文提出了一种随机控制模型，通过随机、离散观测非光滑、分段确定性的泥沙动力学，实现成本效益最优的河流环境修复。该模型推导出退化椭圆形式的最优性方程，并为折现、遍历和完全信息情形下的最优补沙政策提供了解析解与数值解。

ABSTRACT

Earth and soils are indispensable elements of river environment. Dam-downstream environment and ecosystems have been severely affected by reduced or even stopped sediment supply from the upstream. Replenishing earth and soils from outside the river has been considered as an effective way to mitigate this issue. However, its cost-effective implementation has not been considered from a theoretical side. This paper presents a tractable new stochastic control model to deal with this issue. The sediment dynamics in the river environment follow non-smooth and continuous-time piecewise deterministic dynamics. The model assumes that the observation of the sediment dynamics is carried out only randomly and discretely, and that the sediment can be replenished at each observation time with cost. This partial observation assumption is consistent with the fact that continuously obtaining the environmental information is difficult in applications. The performance index to penalize the sediment depletion has a non-smooth term as well. We demonstrate that these non-smoothness factors harmonize with a dynamic programming principle, and derive the optimality equation in a degenerate elliptic form governing the most cost-efficient sediment replenishment policy. We analytically derive and verify an exact solution under a simplified condition for a discounted case, an Ergodic case, and a complete information case. A more realistic case is handled using a high-resolution finite difference scheme. We then provide the optimal sediment replenishment policy numerically.

研究动机与目标

• 开发一种适用于通过泥沙补沙实现河流环境修复的可处理随机控制框架。
• 解决在环境观测仅限于随机、离散时间点时，实现成本效益最优泥沙补沙的挑战。
• 在泥沙演化的非光滑动力学与性能指标中的非光滑惩罚项中引入非光滑性。
• 推导并求解在部分观测下决定最经济补沙策略的最优性方程。
• 在简化条件下提供解析解，在更现实的情景下提供数值解。

提出的方法

• 构建一个连续时间、分段确定性的动力系统，以建模河流中的泥沙动力学。
• 引入泥沙水平的随机、离散观测时间，反映现实世界中数据稀缺的状况。
• 将泥沙补沙建模为在每次观测时间施加的控制动作，并关联相应成本。
• 在性能指标中引入非光滑惩罚项，以惩罚泥沙减少。
• 应用动态规划原理，推导出退化椭圆形式的最优性方程。
• 采用高分辨率有限差分格式，在复杂情形下数值求解最优性方程。

实验结果

研究问题

• RQ1 当仅能获得河流泥沙水平的随机、离散观测时，如何确定最优泥沙补沙策略？
• RQ2 在非光滑动力学与非光滑性能惩罚下，最优控制策略的结构是怎样的？
• RQ3 能否为折现与遍历性能准则的简化情形推导出解析解？
• RQ4 在部分信息下，模型在现实、非简化的条件下表现如何？
• RQ5 何种数值方法能确保最优补沙策略的精确与稳定计算？

主要发现

• 在简化条件下，推导出折现情形下最优补沙策略的解析解。
• 遍历情形也存在精确的解析解，证实了长期成本效益。
• 完全信息情形下得到闭式解，验证了模型的一致性。
• 在现实情景中，高分辨率有限差分格式可实现最优策略的精确数值计算。
• 动力学与性能指标中的非光滑性与动态规划原理相容，支持解的推导。
• 所得最优策略在维持泥沙水平高于临界阈值的同时，最小化了长期成本。