The numerical approximation of boundary value problems by means of a probabilistic representations often has the drawback that the Monte Carlo estimate of the solution is substantially biased due to the presence of the domain boundary. We introduce a scheme, which we have called the leading-term Monte Carlo regression, which seeks to remove that bias by replacing a ’cloud’ of Monte Carlo estimates –carried out at dierent discretization levels– for the usual single Monte Carlo estimate. The practical result of our scheme is an acceleration of the Monte Carlo method. Theoretical analysis of the proposed scheme, confirmed by numerical experiments, shows that the achieved speedup can be well over 100.