  Control variates are variance reduction techniques for Monte Carlo
estimators. They can reduce the cost of the estimation of integrals involving
computationally expensive scientific models. We propose an extension of control
variates, multilevel control functional (MLCF), which uses non-parametric
Stein-based control variates and multifidelity models with lower cost to gain
better performance. MLCF is widely applicable. We show that when the integrand
and the density are smooth, and when the dimensionality is not very high, MLCF
enjoys a fast convergence rate. We provide both theoretical analysis and
empirical assessments on differential equation examples, including a Bayesian
inference for ecological model example, to demonstrate the effectiveness of our
proposed approach.
