SIAM Journal on Scientific Computing, Volume 42, Issue 5, Page A2750-A2772, January 2020.
Models incorporating uncertain inputs, such as random forces or material parameters, have been of increasing interest in PDE-constrained optimization. In this paper, we focus on the efficient numerical minimization of a convex and smooth tracking-type functional subject to a linear partial differential equation with random coefficients and box constraints. The approach we take is based on stochastic approximation where, in place of a true gradient, a stochastic gradient is chosen using one sample from a known probability distribution. Feasibility is maintained by performing a projection at each iteration. In the application of this method to PDE-constrained optimization under uncertainty, new challenges arise. We observe the discretization error made by approximating the stochastic gradient using finite elements. Analyzing the interplay between PDE discretization and stochastic error, we develop a mesh refinement strategy coupled with decreasing step sizes. Additionally, we develop a mesh refinement strategy for the modified algorithm using iterate averaging and larger step sizes. The effectiveness of the approach is demonstrated numerically for different random field choices.

中文翻译：

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不确定条件下PDE约束优化的网格细化随机梯度方法

SIAM科学计算杂志，第42卷，第5期，第A2750-A2772页，2020年1月。
在PDE约束优化中，包含不确定输入（例如随机力或材料参数）的模型越来越引起人们的关注。在本文中，我们将重点放在具有随机系数和框约束的线性偏微分方程的凸光滑跟踪型函数的有效数值最小化上。我们采用的方法是基于随机近似的，其中使用真实已知概率分布中的一个样本来选择随机梯度来代替真实梯度。通过在每次迭代中执行投影来保持可行性。在不确定性下将这种方法应用于PDE约束优化时，出现了新的挑战。我们观察到通过使用有限元近似随机梯度而产生的离散化误差。通过分析PDE离散化和随机误差之间的相互作用，我们开发了一种网格细化策略，同时减小了步长。此外，我们使用迭代平均和较大步长为改进算法开发了网格细化策略。通过数值证明了该方法对于不同随机场选择的有效性。