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Monte-Carlo and Quasi-Monte Carlo Methods 1998 pp 114–136Cite as

Efficiency of Quasi-Monte Carlo Algorithms for High Dimensional Integrals

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Abstract

This paper reports recent progress and presents a few new results on the efficiency of quasi-Monte Carlo algorithms that use n function values for approximation of multivariate integration of high dimension d. We consider the worst case error of a quasi-Monte Carlo algorithm over the unit ball of a normed space of functions of d variables. We indicate for which spaces of functions there exist quasi-Monte Carlo algorithms whose worst case errors go to zero polynomially in n -1 and are independent of d or polynomially dependent on d.

The author was partially supported by the National Science Foundation. This work was done while the author was a member of the Mathematical Sciences Research Institute at Berkeley in Fall 1998.

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References

  1. N. Aronszajn, Theory of reproducing kernels,Trans. Amer. Math. Soc.68 (1950) 337–404

    Article  MATH  MathSciNet  Google Scholar 

  2. N. S. Bakhvalov, On approximate calculation of integrals (in Russian),Vestnik MGU, Ser. Math. Mech. Astron. Phys. Chem.4 (1959) 3–18

    Google Scholar 

  3. Z. Ciesielski, On Levy’s Brownian motion with several-dimensional time, inLecture Notes in Mathematics, A. Dold and B. Eckman, eds., Springer Verlag, New York 472 (1975) 29–56

    Google Scholar 

  4. R. E. Caflisch and W. J. Morokoff, Quasi-Monte Carlo computation of a finance problem, in “Workshop on Quasi-Monte Carlo Methods and Their Applications”, ed. K. T. Fang and F. J. Hickernell, Hong Kong Baptist University (1996) 15–30

    Google Scholar 

  5. R. E. Caflisch, W. J. Morokoff and A. B. Owen, Valuation of mortgage backed securities using Brownian bridges to reduce effective dimension,UCLA CAM Report(1996) 96–16

    Google Scholar 

  6. M. Drmota and R. Tichy, Sequences, discrepancies, and applications,Lecture Notes in Mathematics, Berlin 1651 1997

    Google Scholar 

  7. K. Frank and S. Heinrich, Computing discrepancy of Smolyak quadrature rules,J. of Complexity12 (1996) 287–314

    Article  MATH  MathSciNet  Google Scholar 

  8. S. Heinrich, Efficient algorithms for computing theL 2-discrepancy,Math. Comput.65 (1995) 1621–1633

    Article  MathSciNet  Google Scholar 

  9. F. J. Hickernell, A generalized discrepancy and quadrature error bound,Math. Comput.67 (1997) 299–322

    Article  MathSciNet  Google Scholar 

  10. F. J. Hickernell and H. Woźniakowski, Integration and approximation in arbitrary dimensions. Submitted for publication (1998)

    Google Scholar 

  11. S. Joe, Formulas for the computation of the weighted L2 discrepancy. In preparation (1997)

    Google Scholar 

  12. C. Joy, P. P. Boyle and K. S. Tan, Quasi-Monte Carlo methods in numerical finance, Working Paper, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1 (1995)

    Google Scholar 

  13. J. Matoušek, The exponent of discrepancy is at least 1.0669,J. Complexity14 (1998) 448–453

    Article  MATH  MathSciNet  Google Scholar 

  14. G. M. Molchan, On some problems concerning Brownian motion in Levy’s sense,Theory of Prob. Appi12 (1967) 682–690

    Article  MATH  Google Scholar 

  15. W. Morokoff and R. E. Caflisch, Quasi-Monte Carlo integration,J. Comp. Phys.122 (1995) 218–230

    Article  MATH  MathSciNet  Google Scholar 

  16. B. Moskowitz and R.E. Caflisch, Smoothness and dimension reduction in Quasi- Monte Carlo methods,J. Math. Comp. Modeling23 (1996) 37–54

    Article  MATH  MathSciNet  Google Scholar 

  17. H. Niederreiter, Random Number Generation and Quasi-Monte Carlo Methods,CBMS-NSF, SIAM, Philadelphia (1992)

    Google Scholar 

  18. S. Ninomiya and S. Tezuka, Toward real-time pricing of complex financial derivatives,Applied Math. Finance3 (1996) 1–20

    Article  MATH  Google Scholar 

  19. E. Novak, Deterministic and Stochastic Error Bounds in Numerical Analysis,Lecture Notes in Mathematics, Springer Verlag, Berlin 1349 (1988)

    Google Scholar 

  20. E. Novak, Intractability results for positive cubature formulas and extremal problems for trigonometric polynomials. Submitted for publication (1998)

    Google Scholar 

  21. E. Novak, I. H. Sloan and H. Woźniakowski, Tractability of tensor product linear operators,J. of Complexity13 (1997) 387–418

    Article  MATH  Google Scholar 

  22. A. Papageorgiou and J. F. Traub, Beating Monte Carlo,Risk9: 6 (1996) 63–65

    Google Scholar 

  23. A. Papageorgiou and J. F. Traub, Faster evaluation of multidimensional integrals,Computers in Physics11 (1997) 574–579

    Article  Google Scholar 

  24. S. H. Paskov, New methodologies for valuing derivatives, in “Mathematics of Derivative Securities”, ed. by S. Pliska and M. Dempster, Isaac Newton Institute, Cambridge University Press, Cambridge, UK (1997)

    Google Scholar 

  25. S. H. Paskov and J. F. Traub, Faster valuation of financial derivatives,The Journal of Portfolio Management 22(1995) 113–120

    Article  Google Scholar 

  26. I. H. Sloan and S. Joe, Lattice Methods for Multiple Integration,Clarendon Press, Oxford (1994)

    MATH  Google Scholar 

  27. I. H. Sloan and H. Woźniakowski, When are Quasi-Monte Carlo algorithm efficient for high dimensional integration?,J. Complexity1 (1998) 1–33

    Article  Google Scholar 

  28. S. Tezuka, Uniform random numbers: theory and practice,Kluwer Academic, Dordrecht (1995)

    Book  MATH  Google Scholar 

  29. C. Thomas-Agnan, Computing a Family of Reproducing Kernels for Statistical Applications, Numerical Algorithms 13 (1996) 21–32

    Article  MATH  MathSciNet  Google Scholar 

  30. J. F. Traub, G. W. Wasilkowski, and H. Woźniakowski, Information-Based Complexity, Academic Press, New York (1988)

    MATH  Google Scholar 

  31. J. F. Traub and A. G. Werschulz, Complexity and Information, Cambridge University Press, Cambridge (1998)

    MATH  Google Scholar 

  32. G. Wahba, Spline Models for Observational Data, SIAM-NSF Regional Conference Series in Appi. Math, SIAM 59 Philadelphia (1990)

    Google Scholar 

  33. G. W. Wasilkowski, Integration and approximation of multivariate functions Average case complexity with isotropic Wiener measure, Bull. Amer. Soc. 28 (1993) 308–314; full version J. Approx. Theory 77 (1994) 212–227

    Article  MATH  MathSciNet  Google Scholar 

  34. G. W. Wasilkowski, Integration and approximation of multivariate functions Average case complexity with isotropic Wiener measure, full version J. Approx. Theory 77 (1994) 212 – 227

    Article  MATH  MathSciNet  Google Scholar 

  35. G. W. Wasilkowski, and H. Woźniakowski, Explicit Cost Bounds of Algorithms for Multivariate Tensor Product, ProblemsJ. of Complexity11 (1995) 1 – 56

    Article  MATH  Google Scholar 

  36. G. W. Wasilkowski and H. Woźniakowski, On tractability of path integration,J. of Math. Physics37 (4) (1996) 2071 – 2088

    Article  MATH  Google Scholar 

  37. G. W. Wasilkowski and H. Woźniakowski, The exponent of discrepancy is at most 1.4778…,Math. Computation 66(1997) 1125 – 1132

    Article  MATH  Google Scholar 

  38. H. Woźniakowski, Tractability and strong tractability of linear multivariate problems,J. Complexity10 (1994) 96 – 128

    Article  MATH  MathSciNet  Google Scholar 

  39. H. Wozniakowski, Tractability and strong tractability of multivariate tensor product problems, in Proceedings of 2nd Gauss

    Google Scholar 

  40. Symposium, München, 1993,J. Computing and Information4 (1994) 1 – 19

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Computer Science, Columbia University, New York, NY, 10027, USA

    Henryk Woźniakowski

  2. Institute of Applied Mathematics and Mechanics, University of Warsaw, ul. Banacha 2, 02-097, Warszawa, Poland

    Henryk Woźniakowski

Authors
  1. Henryk Woźniakowski
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Editors and Affiliations

  1. Institute of Discrete Mathematics, Austrian Academy of Sciences, Sonnenfelsgasse 19, 1010, Vienna, Austria

    Harald Niederreiter

  2. Claremont Graduate University, 925 N. Dartmouth Avenue, 91711, Claremont, CA, USA

    Jerome Spanier

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© 2000 Springer-Verlag Berlin Heidelberg

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Woźniakowski, H. (2000). Efficiency of Quasi-Monte Carlo Algorithms for High Dimensional Integrals. In: Niederreiter, H., Spanier, J. (eds) Monte-Carlo and Quasi-Monte Carlo Methods 1998. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-59657-5_7

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  • DOI: https://doi.org/10.1007/978-3-642-59657-5_7

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-66176-4

  • Online ISBN: 978-3-642-59657-5

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