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Balancing of Systematic and Stochastic Errors in Monte Carlo Algorithms for Integral Equations

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Numerical Methods and Applications (NMA 2014)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8962))

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Abstract

The problem of balancing of both systematic and stochastic error is very important when Monte Carlo algorithms are used. A Monte Carlo method for integral equations based on balancing of systematic and stochastic errors is presented. An approach to the problem of controlling the error in non- deterministics methods is presented. The problem of obtaining an optimal ratio between the number of realizations \(N\) of the random variable and the mean value \(k\) of the number of steps in each random trajectory is discussed. Lower bounds for \(N\) and \(k\) are provided once a preliminary given error is given. Meaningful numerical examples and experiments are presented and discussed. Experimental and theoretical relative errors are presented. Monte Carlo algorithms with various initial and transition probabilities are compared. An almost optimal Monte Carlo algorithm is discussed and it is proven that it gives more reliable results.

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Acknowledgments

This work was supported by the Bulgarian National Science Fund under the grant FNI I 02/20 Efficient Parallel Algorithms for Large-Scale Computational Problems.

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Authors and Affiliations

  1. IICT, Bulgarian Academy of Sciences, Acad. G. Bonchev 25 A, 1113, Sofia, Bulgaria

    Ivan Dimov, Rayna Georgieva & Venelin Todorov

Authors
  1. Ivan Dimov
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  2. Rayna Georgieva
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  3. Venelin Todorov
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Corresponding author

Correspondence to Venelin Todorov .

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Editors and Affiliations

  1. Institute of Information and Communication Technologies, Bulgarian Academy of Sciences, Sofia, Bulgaria

    Ivan Dimov

  2. Institute of Information and Communication Technologies, Bulgarian Academy of Sciences, Sofia, Bulgaria

    Stefka Fidanova

  3. Institute of Information and Communication Technologies, Bulgarian Academy of Sciences, Sofia, Bulgaria

    Ivan Lirkov

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Dimov, I., Georgieva, R., Todorov, V. (2015). Balancing of Systematic and Stochastic Errors in Monte Carlo Algorithms for Integral Equations. In: Dimov, I., Fidanova, S., Lirkov, I. (eds) Numerical Methods and Applications. NMA 2014. Lecture Notes in Computer Science(), vol 8962. Springer, Cham. https://doi.org/10.1007/978-3-319-15585-2_5

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  • DOI: https://doi.org/10.1007/978-3-319-15585-2_5

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-15584-5

  • Online ISBN: 978-3-319-15585-2

  • eBook Packages: Computer ScienceComputer Science (R0)

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