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A multivariate fast discrete Walsh transform with an application to function interpolation

Authors: Kwong-Ip Liu, Josef Dick and Fred J. Hickernell
Journal: Math. Comp. 78 (2009), 1573-1591
MSC (2000): Primary 42C10, 41A15
Published electronically: January 9, 2009
MathSciNet review: 2501064
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Abstract | References | Similar Articles | Additional Information

Abstract: For high dimensional problems, such as approximation and integration, one cannot afford to sample on a grid because of the curse of dimensionality. An attractive alternative is to sample on a low discrepancy set, such as an integration lattice or a digital net. This article introduces a multivariate fast discrete Walsh transform for data sampled on a digital net that requires only $ \mathcal{O}(N \log N)$ operations, where $ N$ is the number of data points. This algorithm and its inverse are digital analogs of multivariate fast Fourier transforms.

This fast discrete Walsh transform and its inverse may be used to approximate the Walsh coefficients of a function and then construct a spline interpolant of the function. This interpolant may then be used to estimate the function's effective dimension, an important concept in the theory of numerical multivariate integration. Numerical results for various functions are presented.

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  • 1. N. Aronszajn, Theory of reproducing kernels. Trans. Amer. Math. Soc., 68, 337-404, 1950. MR 0051437 (14:479c)
  • 2. H. Bungartz and M. Griebel, Sparse grids. Acta Numer., 13, 147-269, 2004. MR 2249147 (2007e:65102)
  • 3. R. Caflisch, R. Morokoff and A. Owen, Valuation of mortgage backed securities using Brownian bridges to reduce the effective dimension. J. Comput. Finance, 1, 27-46, 1997.
  • 4. H.E. Chrestenson, A class of generalized Walsh functions. Pacific J. Math., 5, 17-31, 1955. MR 0068659 (16:920c)
  • 5. J.W. Cooley and J.W. Tukey, An algorithm for the machine calculation of complex Fourier series. Math. Comp., 19, 297-301, 1965. MR 0178586 (31:2843)
  • 6. J. Dick and F. Pillichshammer, Multivariate integration in weighted Hilbert spaces based on Walsh functions and weighted Sobolev spaces. J. Complexity, 21, 149-195, 2005. MR 2123222 (2005k:41089)
  • 7. B. Efron and C. Stein, The Jackknife estimate of variance. Ann. Statist., 9, 586-596, 1981. MR 615434 (82k:62074)
  • 8. G. E. Fasshauer, Meshfree Approximation Methods with Matlab, Interdisciplinary Mathematical Sciences Series, vol. 6, World Scientific Publishing Co., Singapore, 2007. MR 2357267 (2008i:65002)
  • 9. H. Faure, Discrépance de suites associées à un système de numération (en dimension $ s$). Acta Arith., 41, 337-351, 1982. MR 677547 (84m:10050)
  • 10. G. Larcher, A class of low-discrepancy point-sets and its application to numerical integration by number-theoretical methods. Grazer Math. Ber., 318, 69-80, 1993. MR 1227403 (94e:11089)
  • 11. G. Larcher and C. Traunfellner, On the numerical integration of Walsh series by number-theoretic methods. Math. Comp., 63, 277-291, 1994. MR 1234426 (94j:65030)
  • 12. D. Li and F.J. Hickernell, Trigonometric spectral collocation methods on lattices. Contemp. Math., 330, 121-132, 2003. MR 2011715 (2004j:65198)
  • 13. H. Niederreiter, Low-discrepancy point sets. Monatsh. Math., 102, 155-167, 1986. MR 861937 (87m:11074)
  • 14. H. Niederreiter, Random Number Generation and Quasi-Monte Carlo Methods. No. 63 in CBMS-NSF Series in Applied Mathematics. SIAM, Philadelphia, 1992. MR 1172997 (93h:65008)
  • 15. H. Niederreiter, Constructions of $ (t,m,s)$-nets and $ (t,s)$-sequences. Finite Fields Appl., 11, 578-600, 2005. MR 2158777 (2006c:11090)
  • 16. H. Niederreiter and G. Pirsic, Duality for digital nets and its applications. Acta Arith., 97, 173-182, 2001. MR 1824983 (2001m:11130)
  • 17. D. Potts, G. Steidl, and M. Tasche, Fast Fourier transforms for nonequispaced data: A tutorial. Modern Sampling Theory: Mathematics and Applications (eds. J.J. Benedetto and P. Ferreira), 249-274, Birkhäuser Boston, 2001. MR 1865690
  • 18. C.M. Rader, Discrete Fourier transforms when the number of data samples is prime. Proc. IEEE, 56, 1107-1108, 1968.
  • 19. I.H. Sloan and S. Joe, Lattice methods for multiple integration. Oxford University Press, Oxford, 1994. MR 1442955 (98a:65026)
  • 20. I.M. Sobol', On the distribution of points in a cube and the approximate evaluation of integrals. U.S.S.R. Comput. Maths. Math. Phys., 7, 86-112, 1967. MR 0219238 (36:2321)
  • 21. I.M. Sobol', Multidimensional quadrature formulas and Haar functions. Nauka, Moscow, 1969. (In Russian) MR 0422968 (54:10952)
  • 22. I.M. Sobol', Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates. Math. Comput. Simulation, 55, 271-280, 2001. MR 1823119
  • 23. G. Wahba, Spline models for observational data. SIAM, Philadelphia, 1990. MR 1045442 (91g:62028)
  • 24. J.L. Walsh, A closed set of normal orthogonal functions. Amer. J. Math., 55, 5-24, 1923. MR 1506485
  • 25. X. Wang and K.T. Fang, The effective dimension and quasi-Monte Carlo integration. J. Complexity, 19, 101-124, 2003. MR 1966664 (2003m:62016)
  • 26. X. Wang and F.J. Hickernell, Randomized Halton sequences. Math. Comput. Modelling, 32, 887-899, 2000. MR 1792105 (2001i:65010)
  • 27. X. Wang and I.H. Sloan, Why are high-dimensional finance problems often of low effective dimension? SIAM J. Sci. Comput., 27, 159-183, 2005. MR 2201179 (2006j:91171)
  • 28. X. Wang and I.H. Sloan, Efficient weighted lattice rules with applications to finance. SIAM J. Sci. Comput., 28, 728-750, 2006. MR 2231728 (2007c:65008)
  • 29. X. Zeng, K.T. Leung and F.J. Hickernell, Error Analysis of Splines for Periodic Problems Using Lattice Design. MCQMC2004 (eds. Niederreiter and Talay), 501-514, 2006. MR 2208728 (2006k:41018)
  • 30. P. Zinterhof, Über die schnelle Lösung von hochdimensionalen Fredholm-Gleichungen vom Faltungstyp mit zahlentheoretischen Methoden (On the fast solution of higher-dimensional Fredholm equations of convolution type by means of number-theoretic methods), Österreich. Akad. Wiss. Math.-Natur. Kl. Sitzungsber. II 196, no. 4-7, 159-169 (1987). MR 963215 (90c:65164)

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Additional Information

Kwong-Ip Liu
Affiliation: Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong, People’s Republic of China

Josef Dick
Affiliation: School of Mathematics and Statistics, University of New South Wales, Sydney 2052, Australia

Fred J. Hickernell
Affiliation: Department of Applied Mathematics, Illinois Institute of Technology, Room E1-208, 10 W. 32nd Street, Chicago, Illinois 60616

Received by editor(s): January 11, 2008
Received by editor(s) in revised form: July 29, 2008
Published electronically: January 9, 2009
Additional Notes: This research was supported in part by the Hong Kong Research Grants Council grant HKBU/2009/04P, the HKSAR Research Grants Council Project No. HKBU200605 and the United States National Science Foundation grant NSF-DMS-0713848.
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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