Exact Error Estimates and Optimal Randomized Algorithms for Integration

  • Ivan T. Dimov
  • Emanouil Atanassov
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4310)


Exact error estimates for evaluating multi-dimensional integrals are considered. An estimate is called exact if the rates of convergence for the low- and upper-bound estimate coincide. The algorithm with such an exact rate is called optimal. Such an algorithm has an unimprovable rate of convergence.

The problem of existing exact estimates and optimal algorithms is discussed for some functional spaces that define the regularity of the integrand. Important for practical computations data classes are considered: classes of functions with bounded derivatives and Hölder type conditions.

The aim of the paper is to analyze the performance of two optimal classes of algorithms: deterministic and randomized for computing multi-dimensional integrals. It is also shown how the smoothness of the integrand can be exploited to construct better randomized algorithms.


Functional Space Deterministic Algorithm Monte Carlo Algorithm Integration Problem Bulgarian Academy 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Atanassov, E., Dimov, I.: A new Monte Carlo method for calculating integrals of smooth functions. Monte Carlo Methods and Applications 5(2), 149–167 (1999)zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Bachvalov, N.S.: On the approximate computation of multiple integrals. Vestnik Moscow State University, Ser. Mat., Mech., vol. 4, pp. 3–18 (1959)Google Scholar
  3. 3.
    Dimov, I.T.: Minimization of the probable error for some Monte Carlo methods. In: Proc. of the Summer School on Mathematical Modelling and Scientific Computations, pp. 159–170. House of the Bulg. Acad. Sci, Sofia (1991)URL, 159Google Scholar
  4. 4.
    Dimov, I.: Efficient and Overconvergent Monte Carlo Methods. Parallel algorithms. In: Dimov, I., Tonev, O. (eds.) Advances in Parallel Algorithms, pp. 100–111. IOS Press, Amsterdam (1994)Google Scholar
  5. 5.
    Dimov, I., et al.: Monte Carlo Algorithms for Elliptic Differential Equations. Data Parallel Functional Approach. Journal of Parallel Algorithms and Applications 9, 39–65 (1996)zbMATHGoogle Scholar
  6. 6.
    Dimov, I., Tonev, O.: Monte Carlo algorithms: performance analysis for some computer architectures. J. of Computational and Applied Mathematics 48, 253–277 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Halton, J.H., Handscome, D.C.: A method for increasing the efficiency of Monte Carlo integrations. J. Assoc. comput. machinery 4(3), 329–340 (1957)Google Scholar
  8. 8.
    Kalos, M.H.: Importance sampling in Monte Carlo calculations of thick shield penetration. Nuclear Sci. and Eng. 2(1), 34–35 (1959)Google Scholar
  9. 9.
    Karaivanova, A., Dimov, I.: Error analysis of an Adaptive Monte Carlo Method for Numerical Integration. Mathematics and Computers in Simulation 47, 201–213 (1998)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Ko, K.-I.: Computational Complexity of Real Functions. Birkhauser, Boston (1991)Google Scholar
  11. 11.
    Kolmogorov, A.N.: Foundations of the Theory of Probability, 2nd engl. edn. Chelsea Publishing Company, New York (1956)zbMATHGoogle Scholar
  12. 12.
    Novak, E., Ritter, K.: Optimal stochstic quadrature farmulas for convex functions. BIT 34, 288–294 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Novak, E., Ritter, K.: High Dimensional Integration of Smooth Functions over cubes. In: Numerische Mathematik, pp. 1–19 (1996)Google Scholar
  14. 14.
    Sendov, B., Andreev, A., Kjurkchiev, N.: Numerical Solution of Polynomial Equations, Handbook of Numerical Analysis. In: Ciarlet, P.G., Lions, J.L. (eds.) Handbook of Numerical Analysis, Solution of Equations in R n (Part 2), North-Holland, Amsterdam (1994)Google Scholar
  15. 15.
    Smolyak, S.A.: Quadrature and inperpolation formulas for tensor products of certain classes of functions. Soviet Math. Dokl. 4, 240–243 (1963)Google Scholar
  16. 16.
    Sobol, I.M.: Monte Carlo numerical methods. Nauka, Moscow (1973)Google Scholar
  17. 17.
    Sobol, I.M.: On Quadratic Formulas for Functions of Several Variables Satisfying a General Lipschitz Condition. USSR Comput. Math. and Math. Phys. 29(6), 936–941 (1989)MathSciNetGoogle Scholar
  18. 18.
    Traub, J.F., Wasilkowski, G.W., Wozniakowski, H.: Information-Based Complexity. Acad. Press, INC., New York (1988)zbMATHGoogle Scholar

Copyright information

© Springer Berlin Heidelberg 2007

Authors and Affiliations

  • Ivan T. Dimov
    • 1
  • Emanouil Atanassov
    • 2
  1. 1.Institute for Parallel Processing, Bulgarian Academy of Sciences Acad. G. Bonchev Str., bl. 25 A, 1113 Sofia, Bulgaria and ACET Centre, University of Reading Whiteknights, PO Box 217, Reading, RG6 6AHUK
  2. 2.Institute for Parallel Processing, Bulgarian Academy of Sciences, Acad. G. Bonchev Str., bl. 25 A, 1113 SofiaBulgaria

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