Variance reduction by means of deterministic computation: Collision estimate

In a recent paper, Heinrich (1995), a new variance reduction technique was introduced for the Monte Carlo solution of Fredholm integral equations. The idea, based on work in complexity theory, Heinrich and Mathé (1993), Heinrich (1994), consists in constructing a new equation sufficiently close to the original one and then applying standard schemes to both equations simultaneously. So the approach is a special case of the separation of main part (also called control variate) technique. As shown in Heinrich (1995), neighboring equations can be constructed by exploiting the system and the approximate solution of deterministic schemes of solving the equation. The gain in variance reduction can be controlled by the discretization error. Hence, by applying the Monte Carlo method with n samples, the overall error is essentially the deterministic discretization error multiplied by the classical Monte Carlo rate n−1/2 (see Heinrich and Mathé (1993), Heinrich (1994) for a theoretical foundation of this statement). Considerable improvements are possible this way as experiments in Heinrich (1995) showed.