  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.
