Control Variates

Consider the Monte Carlo computation of a multidimensional integral (e.g., an ensemble average),

$$\langle A \rangle = \int d \xi g(\xi),$$ (355)

and assume that the integral

$$\int d \xi \tilde{g}(\xi) = A_0,$$ (356)

can be analytically computed for an approximate expression of the integrand $\tilde{g}(\xi) \approx g(\xi)$. The function $\tilde{g}(\xi)$ is called the control variate for $g(\xi)$.

The control variates method is an approach that exploits the information provided by Eq. (356) to reduce the variance of the Monte Carlo computation. The integral, introduced by Eq. (355), is written in two parts,

$$\langle A \rangle = A_0 + \int d \xi (g(\xi)-\tilde{g}(\xi)),$$ (357)

where the first term on the r.h.s. of Eq. (357) is analytically computed and the second term is computed by correlated sampling Monte Carlo integration. Note that since $\tilde{g}(\xi)$ mimcs $g(\xi)$ and usually absorbs most of its variation, the error in the Monte Carlo computation of the second term in the r.h.s. of Eq. (357) is usually appreciably smaller than those of a Monte Carlo evaluation of the integral introduced by Eq. (355).