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

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Abstract

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.

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References

  1. Acebrón, J.A., Busico, M.P., Lanucara, P., Spigler, R.: Domain decomposition solution of elliptic problems via probabilistic methods. SIAM J. Sci. Comput. 27, 440–457 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bernal, F., Acebrón, J.A.: A comparison of higher-order weak numerical schemes for stopped stochastic differential equations in bounded domains. Submitted (2015)

  3. Buchmann, F.M.: Simulation of stopped diffusions. J. Comput. Phys. 202(2), 446–462 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chatterjee, S., Hadi, A.S.: Regression Analysis by Example. Wiley, London (2012)

    MATH  Google Scholar 

  5. Freidlin, M.: Functional Integration and Partial Differential Equations. Princeton University Press, Princeton (1985)

  6. Giles, M.B.: Multi-level Monte Carlo path simulation. Oper. Res. 56(3), 607–617 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  7. Gobet, E., Menozzi, S.: Exact approximation rate of killed hypoelliptic diffusions using the discrete Euler scheme. Stoch. Proc. Appl. 112(2), 201–223 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  8. Gobet, E., Menozzi, S.: Stopped diffusion processes: overshoots and boundary correction. Stoch. Proc. Appl. 120, 130–162 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  9. Higham, D.J., Mao, X., Roj, M., Song, Q., Yin, G.: Mean exit times and the multilevel Monte Carlo method. SIAM/ASA J. Uncertain. Quantif. 1(1), 2–18 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  10. Jansons, K.M., Lythe, G.D.: Exponential timestepping with boundary test for stochastic differential equations. SIAM J. Sci. Comput. 24, 1809–1822 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  11. Kloeden, P.E., Platten, E.: Numerical Solution of Stochastic Differential Equations. Springer, Applications of Mathematics, vol. 23 (1999)

  12. Mancini, S. : Monte Carlo approximations of boundary value problems: an efficient algorithm. Master thesis, Università degli Studi di Milano (2013)

  13. Mannella, R.: Absorbing boundaries and optimal stopping in a stochastic differential equation. Phys. Lett. A 254, 257–62 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  14. Milstein, G.N., Tretyakov, M.V.: Stochastic Numerics for Mathematical Physics. Springer, Berlin (2004)

    Book  MATH  Google Scholar 

  15. Neter, J., Kutner, M., Wasserman, W., Nachtsheim, C.: Applied Linear Statistical Models, 4th edn. Irwin (1996)

  16. Nocedal, J., Wright, S.J.: Numerical Optimization. Springer, Berlin (2000)

    Google Scholar 

  17. Sabelfeld, K.K.: Monte Carlo Methods in Boundary Value Problems. Series in Computational Physics. Springer, Berlin (1991)

  18. Talay, D., Tubaro, L.: Expansion of the global error for numerical schemes solving stochastic differential equations. Stoch. Anal. Appl. 8(4), 94–120 (1990)

    Article  MathSciNet  Google Scholar 

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Acknowledgments

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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Authors and Affiliations

  1. Dipartimento di Matematica ’Federigo Enriques’, Università degli Studi di Milano, Via Cesare Saldini 50, 20133, Milan, Italy

    Sara Mancini

  2. Department of Mathematics, Center for Mathematics and its Applications (CEMAT), Instituto Superior Técnico, Av. Rovisco Pais, 1049-001, Lisbon, Portugal

    Francisco Bernal

  3. Departamento de Ciências e Tecnologias de Informação, ISCTE - Instituto Universitário de Lisboa, Av. das Forças Armadas, 1649-026, Lisbon, Portugal

    Juan A. Acebrón

  4. INESC-ID\ IST, TU Lisbon, Rua Alves Redol 9, 1000-029, Lisbon, Portugal

    Juan A. Acebrón

Authors
  1. Sara Mancini
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  2. Francisco Bernal
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  3. Juan A. Acebrón
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Correspondence to Juan A. Acebrón.

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Mancini, S., Bernal, F. & Acebrón, J.A. An Efficient Algorithm for Accelerating Monte Carlo Approximations of the Solution to Boundary Value Problems. J Sci Comput 66, 577–597 (2016). https://doi.org/10.1007/s10915-015-0033-4

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  • DOI: https://doi.org/10.1007/s10915-015-0033-4

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