This paper considers the problem of optimizing the average tracking error for an elliptic partial differential equation with an uncertain lognormal diffusion coefficient. In particular, the application of the multilevel quasi-Monte Carlo (MLQMC) method to the estimation of the gradient is investigated, with a circulant embedding method used to sample the stochastic field. A novel regularity analysis of the adjoint variable is essential for the MLQMC estimation of the gradient in combination with the samples generated using the CE method. A rigorous cost and error analysis shows that a randomly shifted quasi-Monte Carlo method leads to a faster rate of decay in the root mean square error of the gradient than the ordinary Monte Carlo method, while considering multiple levels substantially reduces the computational effort. Numerical experiments confirm the improved rate of convergence and show that the MLQMC method outperforms the multilevel Monte Carlo method and the single level quasi-Monte Carlo method.

翻译：本文考虑了优化对流部分差异方程式的平均跟踪错误以及不确定的正数传播系数的问题。特别是,对多级准蒙卡罗(MLQMC)方法应用于估计梯度进行了调查,并采用了循环嵌入法对随机偏差系数进行取样。对联合变量进行新颖的定期分析对于MLQMC结合使用CE方法生成的样本估算梯度至关重要。严格的成本和误差分析表明,随机转移的准蒙卡罗方法导致梯度根正方形错误的衰减速度高于普通蒙特卡洛方法,同时考虑多级,大大降低了计算努力。数字实验证实了已改进的趋同率,并表明MLQMC方法超越了多级蒙特卡洛方法和单级准蒙卡罗方法。