Abstract
In this paper we propose and analyse advanced biased stochastic methods for solving a class of integral equations—the second kind Fredholm integral equations. We study and compare innovative possible approaches to compute linear functionals of the integral under consideration: biased Monte Carlo method based on evaluation of truncated Liouville-Neumann series and transformation of this problem into the problem of computing a finite number of integrals. Advanced Monte Carlo algorithms for numerical integration based on modified Sobol sequence have been applied for computing linear functionals. Error balancing of both stochastic and systematic errors has been discussed and applied during the numerical implementation of the biased algorithms. We compare the results obtained by some of the best biased stochastic approaches with the results obtained by the standard Monte Carlo method for integral equations.
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Acknowledgements
Venelin Todorov is supported by the Bulgarian National Science Fund under Project KP-06-M32/2 - 17.12.2019 “Advanced Stochastic and Deterministic Approaches for Large-Scale Problems of Computational Mathematics”. The work is also supported by the Bulgarian National Science Fund under Project KP-06-N52/5 “Efficient methods for modeling, optimization and decision making” and by the Project KP-06-Russia/17 “New Highly Efficient Stochastic Simulation Methods and Applications”, funded by the National Science Fund – Bulgaria.
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Todorov, V., Dimov, I., Georgieva, R. (2023). Advanced Biased Stochastic Approaches Based on Modified Sobol Sequences for the Fredholm Integral Equation. In: Slavova, A. (eds) New Trends in the Applications of Differential Equations in Sciences. NTADES 2022. Springer Proceedings in Mathematics & Statistics, vol 412. Springer, Cham. https://doi.org/10.1007/978-3-031-21484-4_36
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