  This paper defines an approximation scheme for a solution of the Poisson
equation of a geometrically ergodic Metropolis-Hastings chain $\Phi$. The
approximations give rise to a natural sequence of control variates for the
ergodic average $S_k(F)=(1/k)\sum_{i=1}^{k} F(\Phi_i)$, where $F$ is the force
function in the Poisson equation. The main result of the paper shows that the
sequence of the asymptotic variances (in the CLTs for the control-variate
estimators) converges to zero and gives a rate of this convergence. Numerical
examples in the case of a double-well potential are discussed.
