Advances in Computational Mathematics

, Volume 45, Issue 3, pp 1499–1519 | Cite as

A new unbiased stochastic algorithm for solving linear Fredholm equations of the second kind

  • I. T. DimovEmail author
  • S. Maire


In this paper, we propose and analyse a new unbiased stochastic approach for solving a class of integral equations. We study and compare the proposed unbiased approach against the known biased Monte Carlo method based on evaluation of truncated Liouville-Neumann series. We also compare the proposed algorithm against the deterministic Nystrom method. Extensions of the unbiased method for the weak and global solutions are described. Extensive numerical experiments have been performed to support the theoretical studies regarding the convergence of the unbiased algorithms. The results are compared to the best known biased Monte Carlo algorithms for numerical integration done in our previous studies. Conclusions about the applicability and efficiency of the proposed algorithms have been drawn.


Integral equations Algorithm Unbiased approach 

Mathematics Subject Classification (2010)

45Bxx 65Bxx 65Cxx 65Rxx 


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Funding information

This work has been supported by the EC FP7 Project AComIn (FP7-REGPOT-2012-2013-1), by SeaTech, Université de Toulon, as well as by the Bulgarian NSF Grant DN 12/5-2017 ”Efficient Stochastic Methods and Algorithms for Large-scale Problems”.


  1. 1.
    Arnold, I.: Ordinary differential equations, The MIT Press. ISBN 0-262-51018-9 (1978)Google Scholar
  2. 2.
    Atkinson, K.E., Shampine, L.F.: Algorithme 876: Solving Fredholm integral equations of the second kind in Matlab. ACM Trans. Math. Software 34(4), 21 (2007)zbMATHGoogle Scholar
  3. 3.
    Bratley, P., Fox, B.: Algorithm 659: Implementing Sobol’s quasi random sequence generator. ACM Trans. Math. Soft. 14(1), 88–100 (1988)CrossRefzbMATHGoogle Scholar
  4. 4.
    Curtiss, J.H.: Monte Carlo methods for the iteration of linear operators. J. Math Phys. 32, 209–232 (1954)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Dimov, I.: Efficient and overconvergent Monte Carlo methods. In: Dimov, I., Tonev, O. (eds.) Parallel algorithms., advances in parallel algorithms, pp 100–111. IOS Press, Amsterdam (1994)Google Scholar
  6. 6.
    Dimov, I.: Monte Carlo Methods for Applied Scientists, p 291. World Scientific, New Jersey (2008). ISBN-10 981-02-2329-3 (monograph)zbMATHGoogle Scholar
  7. 7.
    Dimov, I.T., Georgieva, R.: Multidimensional sensitivity analysis of large-scale mathematical models. In: Iliev, O.P., et al. (eds.) Numerical solution of partial differential equations: Theory, algorithms, and their applications, Springer Proceedings in Mathematics and Statistics, 45. ISBN: 978-1-4614-7171-4 (book chapter), pp 137–156. Springer Science+Business Media, New York (2013)Google Scholar
  8. 8.
    Dimov, I.T., Georgieva, R.: Monte Carlo method for numerical integration based on Sobol’ sequences, in: LNCS 6046, Springer, 50–59 (2011)Google Scholar
  9. 9.
    Dimov, I.T., Georgieva, R., Ostromsky, T.Z., Zlatev, Z.: Advanced algorithms for multidimensional sensitivity studies of large-scale air pollution models based on Sobol sequences, Computers and Mathematics with Applications 65 (3), ”Efficient Numerical Methods for Scientific Applications”, Elsevier, pp. 338-351. ISSN: 0898-1221 (2013)Google Scholar
  10. 10.
    Dimov, I.T., Maire, S., Sellier, J.M.: A new walk on equations monte carlo method for solving systems of linear algebraic equations. Appl. Math. Model. 39(15), 4494–4510 (2015). See, also the final version published online, MathSciNetCrossRefGoogle Scholar
  11. 11.
    Ermakov, S.M., Mikhailov, G.A.: Statistical Modeling. Moscow, Nauka (1982). (in Russian)Google Scholar
  12. 12.
    Ermakov, S.M., Sipin, A.S.: Monte Carlo method and parametric separation of algorithms, Publishing house of Sankt-Petersburg University. (in Russian) (2014)Google Scholar
  13. 13.
    Farnoosh, R., Ebrahimi, M.: Monte Carlo method for solving fredholm integral equations of the second kind. Appl. Math. Comput. 195, 309–315 (2008)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Georgieva, R., Ivanovska, S.: Importance separation for solving integral equations. In: Proceedings of large-scale scientific computing 2003 (I. Lirkov, S. Margenov, J. Wasniewski, and P. Yalamov - Eds.), LNCS 2907, pp 144–152. Springer (2004)Google Scholar
  15. 15.
    Kalos, M.H., Whitlock, P.A.: Monte Carlo Methods. Wiley-VCH. ISBN 978-3-527-40760-6 (2008)Google Scholar
  16. 16.
    Kress, R.: Linear integral equations, Springer, 2nd ed. ISBN 978-1-4612-6817-8 ISBN 978-1-4612-0559-3 (eBook) DOI 10.1007/978-1-4612-0559-3 1. Integral equations. 1. Title. II. Series: Applied mathematical sciences (Springer-Verlag New York Inc.) (1999)Google Scholar
  17. 17.
    Sabelfeld, K.: Algorithms of the Method Monte Carlo for Solving Boundary Value Problems. Moscow, Nauka (1989). (in Russian)Google Scholar
  18. 18.
    Saito, M., Matsumoto, M.: SIMD-oriented fast Mersenne Twister: a 128-bit pseudorandom number generator. In: Keller, A., Heinrich, S., Niederreiter, H (eds.) Monte Carlo and quasi-monte Carlo methods 2006, Springer, pp 607–622 (2008)Google Scholar
  19. 19.
    Sobol, I.M.: Monte Carlo Numerical Methods. Moscow, Nauka (1973). (in Russian)zbMATHGoogle Scholar
  20. 20.

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Authors and Affiliations

  1. 1.Institute of Information and Communication TechnologiesBulgarian Academy of SciencesSofiaBulgaria
  2. 2.Laboratoire LIS, UMR 7020 Equipe Signal et ImageSeaTech - Université de ToulonLa GardeFrance

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