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Local-global model reduction method for stochastic optimal control problems constrained by partial differential equations

机译：偏微分方程约束的随机最优控制问题的局部-全局模型简化方法

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In this paper, a local-global model reduction method is presented to solve stochastic optimal control problems constrained by stochastic partial differential equations (stochastic PDEs). If the optimal control problems involve uncertainty, we need to use a few random variables to parameterize the uncertainty. The stochastic optimal control problems require solving coupled optimality system for a large number of samples in the stochastic space to quantify the statistics of the system response and explore the uncertainty quantification. Thus the computation is prohibitively expensive. To overcome the difficulty, model reduction is necessary to significantly reduce the computation complexity. We exploit the advantages from both reduced basis method and Generalized Multiscale Finite Element Method (GMsFEM) and develop the local-global model reduction method for stochastic optimal control problems with stochastic PDE constraints. This local-global model reduction can achieve much more computation efficiency than using only local model reduction approach and only global model reduction approach. We recast the stochastic optimal problems in the framework of saddle-point problems and analyze the existence and uniqueness of the optimal solutions of the reduced model. In the local-global approach, most of computation steps are independent of each other. This is very desirable for scientific computation. Moreover, the online computation for each random sample is very fast via the proposed model reduction method. This allows us to compute the optimality system for a large number of samples. To demonstrate the performance of the local-global model reduction method, a few numerical examples are provided for different stochastic optimal control problems. (C) 2018 Elsevier B.V. All rights reserved.
机译：本文提出了一种局部-全局模型约简方法来解决由随机偏微分方程（PDEs）约束的随机最优控制问题。如果最优控制问题涉及不确定性，我们需要使用一些随机变量来参数化不确定性。随机最优控制问题需要解决随机空间中大量样本的耦合最优系统，以量化系统响应的统计量并探索不确定性量化。因此，计算非常昂贵。为了克服该困难，必须进行模型简化以显着降低计算复杂度。我们利用简化基方法和广义多尺度有限元方法（GMsFEM）的优势，针对具有随机PDE约束的随机最优控制问题，开发了局部全局模型简化方法。与仅使用局部模型简化方法和仅使用全局模型简化方法相比，这种局部全局模型简化可以实现更高的计算效率。我们在鞍点问题的框架内重铸了随机最优问题，并分析了简化模型最优解的存在性和唯一性。在局部全局方法中，大多数计算步骤彼此独立。这对于科学计算是非常理想的。此外，通过提出的模型约简方法，每个随机样本的在线计算都非常快。这使我们能够为大量样本计算最优系统。为了证明局部-全局模型简化方法的性能，提供了一些数值示例来说明不同的随机最优控制问题。 （C）2018 Elsevier B.V.保留所有权利。

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