SIAM/ASA Journal on Uncertainty Quantification, Volume 9, Issue 2, Page 354-383, January 2021.
We study an optimal control problem under uncertainty, where the target function is the solution of an elliptic partial differential equation with random coefficients, steered by a control function. The robust formulation of the optimization problem is stated as a high-dimensional integration problem over the stochastic variables. It is well known that carrying out a high-dimensional numerical integration of this kind using a Monte Carlo method has a notoriously slow convergence rate; meanwhile, a faster rate of convergence can potentially be obtained by using sparse grid quadratures, but these lead to discretized systems that are nonconvex due to the involvement of negative quadrature weights. In this paper, we analyze instead the application of a quasi-Monte Carlo method, which retains the desirable convexity structure of the system and has a faster convergence rate compared to ordinary Monte Carlo methods. In particular, we show that under moderate assumptions on the decay of the input random field, the error rate obtained by using a specially designed, randomly shifted rank-1 lattice quadrature rule is essentially inversely proportional to the number of quadrature nodes. The overall discretization error of the problem, consisting of the dimension truncation error, finite element discretization error, and quasi-Monte Carlo quadrature error, is derived in detail. We assess the theoretical findings in numerical experiments.

中文翻译：

---

不确定条件下最优控制的拟蒙特卡罗方法

SIAM / ASA不确定性量化期刊，第9卷，第2期，第354-383页，2021年1月。
我们研究了不确定性下的最优控制问题，其中目标函数是由控制函数控制的具有随机系数的椭圆型偏微分方程的解。优化问题的健壮表述表示为随机变量上的高维积分问题。众所周知，使用蒙特卡洛方法进行这种高维数值积分的收敛速度非常慢。同时，通过使用稀疏网格正交可以潜在地获得更快的收敛速度，但是由于负的正交权重，这些导致离散化的系统是非凸的。在本文中，我们改为分析准蒙特卡罗方法的应用，与普通的蒙特卡洛方法相比，它保留了系统所需的凸结构，并且具有更快的收敛速度。尤其是，我们表明，在对输入随机场的衰减进行适度假设的情况下，使用经过特殊设计的随机移位的rank-1格正交规则所获得的错误率基本上与正交节点的数量成反比。详细推导了该问题的整体离散误差，包括尺寸截断误差，有限元离散误差和准蒙特卡洛正交误差。我们评估数值实验中的理论发现。通过使用经过特殊设计的随机移位的rank-1格正交规则所获得的错误率基本上与正交节点的数量成反比。详细推导了该问题的整体离散误差，包括尺寸截断误差，有限元离散误差和准蒙特卡洛正交误差。我们评估数值实验中的理论发现。通过使用经过特殊设计的随机移位的rank-1格正交规则所获得的错误率基本上与正交节点的数量成反比。详细推导了该问题的整体离散误差，包括尺寸截断误差，有限元离散误差和准蒙特卡洛正交误差。我们评估数值实验中的理论发现。