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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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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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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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