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A new unbiased stochastic algorithm for solving linear Fredholm equations of the second kind

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Abstract

In this paper, we propose and analyse a new unbiased stochastic approach for solving a class of integral equations. We study and compare the proposed unbiased approach against the known biased Monte Carlo method based on evaluation of truncated Liouville-Neumann series. We also compare the proposed algorithm against the deterministic Nystrom method. Extensions of the unbiased method for the weak and global solutions are described. Extensive numerical experiments have been performed to support the theoretical studies regarding the convergence of the unbiased algorithms. The results are compared to the best known biased Monte Carlo algorithms for numerical integration done in our previous studies. Conclusions about the applicability and efficiency of the proposed algorithms have been drawn.

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Funding

This work has been supported by the EC FP7 Project AComIn (FP7-REGPOT-2012-2013-1), by SeaTech, Université de Toulon, as well as by the Bulgarian NSF Grant DN 12/5-2017 ”Efficient Stochastic Methods and Algorithms for Large-scale Problems”.

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Authors and Affiliations

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

    I. T. Dimov

  2. Laboratoire LIS, UMR 7020 Equipe Signal et Image, SeaTech - Université de Toulon, Avenue de l’Université 83130, La Garde, France

    S. Maire

Authors
  1. I. T. Dimov
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  2. S. Maire
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Correspondence to I. T. Dimov.

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Communicated by: Alexander Barnett

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Dimov, I.T., Maire, S. A new unbiased stochastic algorithm for solving linear Fredholm equations of the second kind. Adv Comput Math 45, 1499–1519 (2019). https://doi.org/10.1007/s10444-019-09676-y

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  • DOI: https://doi.org/10.1007/s10444-019-09676-y

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