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An unbiased Monte Carlo method to solve linear Volterra equations of the second kind

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Abstract

In previous works (Dimov and Maire in Adv Comput Math 45(3):1499–1519, 2019; Dimov et al. in Appl Math Model 39(15):4494–4510, https://doi.org/10.1016/j.apm.2014.12.018, 2015), we have developed two Monte Carlo algorithms to solve linear systems and Fredholm integral equations of the second kind. These algorithms rely on the computation of a score along a discrete or continuous homogeneous Markov chain until absorption. Here, we propose two approaches to extend the Fredholm algorithm to Volterra equations. The first one is based on a change in variable at each step of the Markov chain. The second one uses the indicator function to transform the Volterra equation into an appropriate form. The resulting Markov chains are inhomogeneous with an increasing absorption rate. The convergence is ensured as soon as the Volterra kernel is bounded. Numerical examples are given on basic reference problems and on high dimensional test cases up to 100 dimensions.

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References

  1. Abdou MA, Badr AA, Soliman MB (2011) On a method for solving a two-dimensional nonlinear integral equation of the second kind. J Comput Appl Math 235:3589–3598

    Article  MathSciNet  Google Scholar 

  2. Al-jarrah Y, Shatnawi TAM (2019) Sinc method for two-dimensional volterra integral equations of first and second kinds. Int J Differ Equ 14(2):195–206. https://doi.org/10.37622/IJDE/14.2.2019.195-206

    Article  Google Scholar 

  3. Arias MR, Benitez R, Bolos VJ (2019) Non-Lipschitz homogeneous Volterra integral equations. In: Theoretical aspects and numerical treatment. Modern Mathematics and Mechanics Fundamentals, Problems and Challenges. Springer, Berlin, pp 237–259

  4. Asady B, Hakimzadegan F, Nazarlue R (2014) Utilizing artificial neural network approach for solving two-dimensional integral equations. Math Sci 8:117. https://doi.org/10.1007/s40096-014-0117-6

    Article  MathSciNet  MATH  Google Scholar 

  5. Atkinson KE, Shampine LF Algorithme 876: solving Fredholm integral equations of the second kind in Matlab. ACM Trans Math Softw 34(4), Art. 21

  6. Bakhshi M, Asghari-Larimi M, Asghari-Larimi M (2012) Three-dimensional differential transform method for solving nonlinear three-dimensional Volterra integral equations. J Math Comput Sci 4(2):246–256

    Article  Google Scholar 

  7. Brunner H (2004) Collocation methods for volterra integral and related functional equations. Cambridge University Press, Cambridge

    Book  Google Scholar 

  8. Brunner H, Kauthen J-P (1989) The numerical solution of two-dimensional volterra integral equations by collocation and iterated collocation. IMA J Numer Anal 9(1):47–59. https://doi.org/10.1093/imanum/9.1.47

    Article  MathSciNet  MATH  Google Scholar 

  9. Cristina Guerreiro BB, Foltescu V, de Leeuw F (2014) Air quality status and trends in Europe. Atmos Environ 98:376–384. https://doi.org/10.1016/j.atmosenv.2014.09.017

    Article  Google Scholar 

  10. de Assis LS, de Jurair RP, Tarrataca L, Haddad DB (2019) Efficient Volterra systems identification using hierarchical genetic algorithms. Appl Soft Comput 85:105745. https://doi.org/10.1016/j.asoc.2019.105745

    Article  Google Scholar 

  11. Dimov IT, Maire S (2019) A new unbiased stochastic algorithm for solving linear Fredholm equations of the second kind. Adv Comput Math 45(3):1499–1519

    Article  MathSciNet  Google Scholar 

  12. Dimov IT, Maire S, Sellier JM (2015) A new walk on equations Monte Carlo method for solving systems of linear algebraic equations. Appl Math Model 39(15):4494–4510. https://doi.org/10.1016/j.apm.2014.12.018

    Article  MathSciNet  MATH  Google Scholar 

  13. Effati S, Buzhabadi R (2012) A neural network approach for solving Fredholm integral equations of the second kind. Neural Comput Appl 21:843–852. https://doi.org/10.1007/s00521-010-0489-y

    Article  Google Scholar 

  14. El Filali Ech-Chafiq Z, Lelong J, Reghai A (2021) Automatic control variates for option pricing using neural networks. Monte Carlo Methods Appl 27(2):91–104. https://doi.org/10.1515/mcma-2020-2081

    Article  MathSciNet  MATH  Google Scholar 

  15. Evaluation of Progress under the EU National Emission Ceilings Directive. Progress Towards EU Air Quality Objectives. European Environment Agency, Copenhagen. EEA Technical Report No 14/2012

  16. Farnoosh R, Ebrahimi M (2008) Monte Carlo method for solving Fredholm integral equations of the second kind. Appl Math Comput 195:309–315

    MathSciNet  MATH  Google Scholar 

  17. Haroonabadi H, Haghifam MR (2011) Generation reliability assessment in power markets using Monte Carlo simulation and soft computing. Appl Soft Comput 11(8):5292–5298

    Article  Google Scholar 

  18. https://ec.europa.eu/clima/policies/ozone/regulation_en

  19. https://www.euro.who.int/data/assets/pdf_file/0005/78647/E91843.pdf

  20. Jafarian A, Measoomy Nia SA, Golmankhaneh AK et al (2013) Numerical solution of linear integral equations system using the Bernstein collocation method. Adv Differ Equ 2013:123

    Article  MathSciNet  Google Scholar 

  21. Jafarian A, Measoomy S, Abbasbandy S (2015) Artificial neural networks based modeling for solving Volterra integral equations system. Appl Soft Comput 27:391–398

    Article  Google Scholar 

  22. Jasra A, Doucet A (2009) Sequential Monte Carlo methods for diffusion processes. Proc R Soc A Math Phys Eng Sci 465(2112):3709–3727

    MathSciNet  MATH  Google Scholar 

  23. Khezerloo M, Hajighasemi S (2012) Existence and uniqueness of solution of Volterra integral equations. Int J Ind Math 4:69–76

    Google Scholar 

  24. Kress R (1999) Linear integral equations, Springer, 2nd edn. 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, New York

  25. Kudelic R (2016) Monte-Carlo randomized algorithm for minimal feedback arc set problem. Appl Soft Comput 41:235–246. https://doi.org/10.1016/j.asoc.2015.12.018

    Article  Google Scholar 

  26. Lapeyre B, Pardoux E, Sentis R (2003) Introduction to Monte-Carlo methods for transport and diffusion equations. Oxford University Press, Oxford, New York

    MATH  Google Scholar 

  27. Lecot C, El Khettabi F (1999) Quasi-Monte Carlo simulation of diffusion. J Complex 15(3):342–359. https://doi.org/10.1006/jcom.1999.0509

    Article  MathSciNet  MATH  Google Scholar 

  28. Ma Y, Huang J, Wang C et al (2016) Sinc Nyström method for a class of nonlinear Volterra integral equations of the first kind. Adv Differ Equ 2016:151. https://doi.org/10.1186/s13662-016-0849-8

    Article  MATH  Google Scholar 

  29. Maleknejad K, Rashidinia J, Eftekhari T (2018) Numerical solution of three-dimensional Volterra–Fredholm integral equations of the first and second kinds based on Bernsteins approximation. Appl Math Comput 339:272–285. https://doi.org/10.1016/j.amc.2018.07.021

    Article  MathSciNet  MATH  Google Scholar 

  30. Nuvolone D, Petri D, Voller F (2018) The effects of ozone on human health. Environ Sci Pollut Res 25(9):8074–8088

    Article  Google Scholar 

  31. Ozan K (2015) A novel hybrid learning algorithm for full Bayesian approach of artificial neural networks. Appl Soft Comput 35:52–65. https://doi.org/10.1016/j.asoc.2015.06.003

    Article  Google Scholar 

  32. Sato T (1953) Sur léquation intégrale non linéaire de volterra. Compos Math 11:271–290

    MATH  Google Scholar 

  33. Simonov NA (1997) Monte Carlo methods for connective diffusion equations. Russ J Numer Anal Math Model 12(l):67–81

    MATH  Google Scholar 

  34. Tang Q, Waxman D (2003) An integral equation describing an asexual population in a changing environment. Nonlinear Anal 53:683–699

    Article  MathSciNet  Google Scholar 

  35. Tari A, Rahimi MY, Shahmorad S, Talati F (2009) Solving a class of two-dimensional linear and nonlinear Volterra integral equations by the differential transform method. J Comput Appl Math 228:70–76

    Article  MathSciNet  Google Scholar 

  36. Tomasiello S, Macías-Díaz JE, Khastan A et al (2019) New sinusoidal basis functions and a neural network approach to solve nonlinear Volterra–Fredholm integral equations. Neural Comput Appl 31:4865–4878. https://doi.org/10.1007/s00521-018-03984-y

    Article  Google Scholar 

  37. Umar M, Sabir Z, Raja MAZ (2019) Intelligent computing for numerical treatment of nonlinear prey-predator models. Appl Soft Comput 80:506–524. https://doi.org/10.1016/j.asoc.2019.04.022

    Article  Google Scholar 

  38. van Zelm R et al (2008) European characterization factors for human health damage of PM10 and ozone in life cycle impact assessment. Atmos Environ 42(3):441–453

    Article  Google Scholar 

  39. Wang F-K, Chang K-K, Hsiao Y-Y (2013) Implementing a diffusion model optimized by a hybrid evolutionary algorithm to forecast notebook shipments. Appl Soft Comput 13(2):1147–1151. https://doi.org/10.1016/j.asoc.2012.11.004

    Article  Google Scholar 

  40. Ziqan A, Armiti S, Suwan I (2016) Solving three-dimensional Volterra integral equation by the reduced differential transform method. Int J Appl Math Res 5(2):103–106

    Article  Google Scholar 

Download references

Acknowledgements

Venelin Todorov is supported by the Bulgarian National Science Fund under Projects KP-06-M32/2-17.12.2019 ”Advanced Stochastic and Deterministic Approaches for Large-Scale Problems of Computational Mathematics” and by the National Scientific Program ”Information and Communication Technologies for a Single Digital Market in Science, Education and Security” (ICTinSES), contract No. D01-205/23.11.2018, financed by the Ministry of Education and Science. The work is supported by the Bulgarian National Science Fund under Project DN 12/5-2017 ”Efficient Stochastic Methods and Algorithms for Large-Scale Problems” and KP-06-Russia/17 “New Highly Efficient Stochastic Simulation Methods and Applications” funded by the National Science Fund-Bulgaria.

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Authors and Affiliations

  1. Department of Parallel Algorithms, Institute of Information and Communication Technologies, Bulgarian Academy of Sciences, Acad. G. Bonchev Str., Block 25 A, 1113, Sofia, Bulgaria

    Ivan Dimov & Venelin Todorov

  2. CNRS, LIS, Toulon, France, UMR 7020, Aix Marseille Université, Université de Toulon, 83957, La Garde, France

    Sylvain Maire

  3. Information Modeling Department, Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. Georgi Bonchev Str., Block 8, 1113, Sofia, Bulgaria

    Venelin Todorov

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  1. Ivan Dimov
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Correspondence to Venelin Todorov.

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Dimov, I., Maire, S. & Todorov, V. An unbiased Monte Carlo method to solve linear Volterra equations of the second kind. Neural Comput & Applic 34, 1527–1540 (2022). https://doi.org/10.1007/s00521-021-06417-5

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