Abstract
This article considers the error of integrating multivariate Haar wavelet series by quasi-Monte Carlo rules using scrambled digital nets. Both the worst-case and random-case errors are analyzed. It is shown that scrambled net quadrature has optimal order. Moreover, there is a simple formula for the worst-case error.
This work was partially supported by a Hong Kong Research Grants Council grant HKBU/2030/99P, by Hong Kong Baptist University grant FRG/97-98/II-99, by Shanghai NSF Grant 00JC14057, and by a Shanghai Higher Education STF Grant.
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References
H. Faure, Discrépance de suites associées à un systéme de numération (en dimensions), Acta Arith. 41 (1982), 337–351.
S. Heinrich, F. J. Hickernell, and R. X. Yue, Optimal quadrature for Haar wavelet spaces, 2001, submitted for publication to Math. Comp.
P. Hellekalek and G. Larcher (eds.), Random and quasi-random point sets, Lecture Notes in Statistics, vol. 138, Springer-Verlag, New York, 1998.
F. J. Hickernell and H. S. Hong, The asymptotic efficiency of randomized nets for quadrature, Math. Comp. 68 (1999), 767–791.
F. J. Hickernell and H. Woźniakowski, The price of pessimism for multidimensional quadrature, J. Complexity 17 (2001), to appear.
F. J. Hickernell and R. X. Yue, The mean square discrepancy of scrambled (t, s)-sequences, SIAM J. Numer. Anal. 38 (2001), 1089–1112.
H. S. Hong and F. J. Hickernell, Implementing scrambled digital nets, 2001, submitted for publication to ACM TOMS.
G. Larcher, Digital point sets: Analysis and applications, In Hellekalek and Larcher [3], pp. 167–222.
-, On the distribution of digital sequences, Monte Carlo and quasi-Monte Carlo methods 1996 (H. Niederreiter, P. Hellekalek, G. Larcher, and P. Zinterhof, eds.), Lecture Notes in Statistics, vol. 127, Springer-Verlag, New York, 1998, pp. 109–123.
H. Niederreiter, Low discrepancy and low dispersion sequences, J. Number Theory 30 (1988), 51–70.
-, Random number generation and quasi-Monte Carlo methods, CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, 1992.
H. Niederreiter and C. Xing, Quasirandom points and global function fields, Finite Fields and Applications (S. Cohen and H. Niederreiter, eds.), London Math. Society Lecture Note Series, no. 233, Cambridge University Press, 1996, pp. 269–296.
-, Nets, (t, s)-sequences and algebraic geometry, In Hellekalek and Larcher [3], pp. 267–302.
A. B. Owen, Randomly permuted (t, m, s)-nets and (t, s)-sequences, Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing (H. Niederreiter and P. J.-S. Shiue, eds.), Lecture Notes in Statistics, vol. 106, Springer-Verlag, New York, 1995, pp. 299–317.
-, Monte Carlo variance of scrambled net quadrature, SIAM J. Numer. Anal. 34 (1997), 1884–1910.
I. M. Sobol’, Multidimensional quadrature formulas and Haar functions (in Russian), Izdat. “Nauka”, Moscow, 1969.
R. X. Yue and F. J. Hickernell, The discrepancy of digital nets, 2001, submitted to J. Complexity.
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Heinrich, S., Hickernell, F.J., Yue, RX. (2001). Integration of Multivariate Haar Wavelet Series. In: Tang, Y.Y., Yuen, P.C., Li, Ch., Wickerhauser, V. (eds) Wavelet Analysis and Its Applications. WAA 2001. Lecture Notes in Computer Science, vol 2251. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45333-4_14
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DOI: https://doi.org/10.1007/3-540-45333-4_14
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