Journal of Scientific Computing

, Volume 66, Issue 2, pp 577–597 | Cite as

An Efficient Algorithm for Accelerating Monte Carlo Approximations of the Solution to Boundary Value Problems

  • Sara Mancini
  • Francisco Bernal
  • Juan A. AcebrónEmail author


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 different 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.


Monte Carlo method Romberg extrapolation Bounded diffusion Feynman–Kac formula First exit time Parallel computing 



This work was supported by national funds through Fundação para a Ciência e a Tecnologia (FCT) under grants UID/CEC/50021/2013, and PTDC/EIA-CCO/098910/2008. FB also acknowledges FCT funding under grant SFRH/BPD/79986/2011.


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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Sara Mancini
    • 1
  • Francisco Bernal
    • 2
  • Juan A. Acebrón
    • 3
    • 4
    Email author
  1. 1.Dipartimento di Matematica ’Federigo Enriques’Università degli Studi di MilanoMilanItaly
  2. 2.Department of Mathematics, Center for Mathematics and its Applications (CEMAT)Instituto Superior TécnicoLisbonPortugal
  3. 3.Departamento de Ciências e Tecnologias de InformaçãoISCTE - Instituto Universitário de LisboaLisbonPortugal
  4. 4.INESC-ID\ IST, TU LisbonLisbonPortugal

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