Abstract
In this paper three possible approaches to compute linear functionals of the solution of the Fredholm integral equation of the second kind are under consideration: a biased Monte Carlo method based on evaluation of truncated Liouville-Neumann series; transformation of this problem into the problem of computing a finite number of integrals, and an unbiased stochastic approach. In the second case several Monte Carlo algorithms for numerical integration have been applied including optimized stochastic approaches developed in our previous studies. The unbiased stochastic approach has been applied to a multidimensional numerical example. A comprehensive analysis about the reliability and the efficiency of the algorithms has been done.
The presented work was supported by the Bulgarian National Science Fund under the Bilateral Project KP-06-Russia/17 “New Highly Efficient Stochastic Simulation Methods and Applications”, Project KP-06-N52/5 “Efficient methods for modeling, optimization and decision making” and Project KP-06-N62/6 “Machine learning through physics-informed neural networks”.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Bradley, P., Fox, B.: Algorithm 659: implementing Sobol’s quasi random sequence generator. ACM Trans. Math. Software 14(1), 88–100 (1988)
Curtiss, J.H.: Monte Carlo methods for the iteration of linear operators. J. Math Phys. 32, 209–232 (1954)
Dimov, I.: Monte Carlo methods for applied scientists., World Scientific, New Jersey, London, Singapore, World Scientific (2008). 291p., ISBN-10 981–02-2329-3
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, Mathematics and Statistics, vol. 45, pp. 137–156. Springer, New York (2013). https://doi.org/10.1007/978-1-4614-7172-1_8, ISBN: 978-1-4614-7171-4 (book chapter)
Dimov, I., Georgieva, R.: Monte Carlo method for numerical integration based on Sobol’s sequences. In: Dimov, I., Dimova, S., Kolkovska, N. (eds.) NMA 2010. LNCS, vol. 6046, pp. 50–59. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-18466-6_5
Dimov, I.T., Georgieva, R., Ostromsky, Tz., Zlatev, Z.: Advanced algorithms for multidimensional sensitivity studies of large-scale air pollution models based on Sobol sequences, Computers and Math. Appl. 65(3), 338–351 (2013). "Efficient Numerical Methods for Scientific Applications", Elsevier
Dimov, I.T., Maire, S.: A new unbiased stochastic algorithm for solving linear Fredholm equations of the second kind. Adv. Comput. Math. 45(3), 1499–1519 (2019). https://doi.org/10.1007/s10444-019-09676-y
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. Modelling (2014). https://doi.org/10.1016/j.apm.2014.12.018
Farnoosh, R., Ebrahimi, M.: Monte Carlo method for solving Fredholm integral equations of the second kind. Appl. Math. Comput. 195, 309–315 (2008)
Kalos, M.H., Whitlock, P.A.: Monte Carlo Methods. Wiley-VCH (2008). ISBN 978-3-527-40760-6
Niederreiter, H.: Low-discrepancy and low-dispersion sequences. J. Number Theory 30, 51–70 (1988)
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, pp. 607–622. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-74496-2_36
Sobol, I., Asotsky, D., Kreinin, A., Kucherenko, S.: Construction and comparison of high-dimensional Sobol’ generators. Wilmott J. 67–79 (2011)
Sobol, I.M.: Monte Carlo Numerical Methods. Nauka, Moscow (1973). (in Russian)
Sobol, I.M.: On quadratic formulas for functions of several variables satisfying a general Lipschitz condition. USSR Comput. Math. and Math. Phys. 29(6), 936–941 (1989)
Weyl, H.: Ueber die Gleichverteilung von Zahlen mod Eins. Math. Ann. 77(3), 313–352 (1916)
Acknowledgements
The authors thanks to Prof. Sylvain Maire for the useful discussion regarding the USA method. The presented work was supported by the Bulgarian National Science Fund under the Bilateral Project KP-06-Russia/17 “New Highly Efficient Stochastic Simulation Methods and Applications”. Venelin Todorov is supported by the BNSF under Projects KP-06-N52/5 “Efficient methods for modeling, optimization and decision making” and KP-06-N62/6 “Machine learning through physics-informed neural networks”.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this paper
Cite this paper
Todorov, V., Dimov, I., Georgieva, R., Ostromsky, T. (2023). Optimized Stochastic Approaches Based on Sobol Quasirandom Sequences for Fredholm Integral Equations of the Second Kind. In: Georgiev, I., Datcheva, M., Georgiev, K., Nikolov, G. (eds) Numerical Methods and Applications. NMA 2022. Lecture Notes in Computer Science, vol 13858. Springer, Cham. https://doi.org/10.1007/978-3-031-32412-3_27
Download citation
DOI: https://doi.org/10.1007/978-3-031-32412-3_27
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-32411-6
Online ISBN: 978-3-031-32412-3
eBook Packages: Computer ScienceComputer Science (R0)