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”.
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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”.
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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
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