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High-dimensional integration: The quasi-Monte Carlo way*†

Published online by Cambridge University Press:  02 April 2013

Josef Dick ,
Frances Y. Kuo  and
Ian H. Sloan
Josef Dick
Affiliation:
University of New South Wales, Sydney, NSW 2052, Australia E-mail: josef.dick@unsw.edu.au
Frances Y. Kuo
Affiliation:
>University of New South Wales, Sydney, NSW 2052, Australia E-mail: f.kuo@unsw.edu.au
Ian H. Sloan
Affiliation:
University of New South Wales, Sydney, NSW 2052, Australia E-mail: i.sloan@unsw.edu.au King Fahd University of Petroleum and Minerals, Dhahran 34463, Saudi Arabia
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Abstract

This paper is a contemporary review of QMC (‘quasi-Monte Carlo’) methods, that is, equal-weight rules for the approximate evaluation of high-dimensional integrals over the unit cube [0,1]s, where s may be large, or even infinite. After a general introduction, the paper surveys recent developments in lattice methods, digital nets, and related themes. Among those recent developments are methods of construction of both lattices and digital nets, to yield QMC rules that have a prescribed rate of convergence for sufficiently smooth functions, and ideally also guaranteed slow growth (or no growth) of the worst-case error as s increases. A crucial role is played by parameters called ‘weights’, since a careful use of the weight parameters is needed to ensure that the worst-case errors in an appropriately weighted function space are bounded, or grow only slowly, as the dimension s increases. Important tools for the analysis are weighted function spaces, reproducing kernel Hilbert spaces, and discrepancy, all of which are discussed with an appropriate level of detail.

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Type
Research Article
Information
Acta Numerica , Volume 22 , May 2013 , pp. 133 - 288
Copyright
Copyright © Cambridge University Press 2013

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Footnotes

*

Colour online for monochrome figures available at journals.cambridge.org/anu.

†

The URLs cited in this work were correct at the time of going to press, but the publisher and the authors make no undertaking that the citations remain live or are accurate or appropriate.

References

REFERENCES

Achtsis, N. and Nuyens, D. (2012), A component-by-component construction for the trigonometric degree. In Monte Carlo and Quasi-Monte Carlo Methods 2010 (Plaskota, L. and Woźniakowski, H., eds), Springer, pp. 235–253.CrossRefGoogle Scholar
Acworth, P., Broadie, M. and Glasserman, P. (1998), A comparison of some Monte Carlo and quasi-Monte Carlo techniques for option pricing. In Monte Carlo and Quasi-Monte Carlo Methods 1996 (Hellekalek, P, Larcher, G, Niederreiter, H and Zinterhof, P, eds), Springer, pp. 1–18.Google Scholar
Adler, R. J. (1981), The Geometry of Random Fields, Wiley.Google Scholar
Aistleitner, C. (2011), ‘Covering numbers, dyadic chaining and discrepancy’, J. Complexity 27, 531–540.CrossRefGoogle Scholar
Aistleitner, C. and Hofer, M. (2012), ‘Probabilistic error bounds for the discrepancy of mixed sequences’, Monte Carlo Methods Appl. 18, 181–200.CrossRefGoogle Scholar
Antonov, I. A. and Saleev, V. M. (1979), ‘An effective method for the computation of λPτ-sequences’, Ž. Vyčisl. Mat. i Mat. Fiz. 19, 243–245, 271.Google Scholar
Aronszajn, N. (1950), ‘Theory of reproducing kernels’, Trans. Amer. Math. Soc. 68, 337–404.CrossRefGoogle Scholar
Atanassov, E. I. (2004 a), Efficient CPU-specific algorithm for generating the generalized Faure sequences. In Large-Scale Scientific Computing, Vol. 2907 of Lecture Notes in Computer Science, Springer, pp. 121–127.Google Scholar
Atanassov, E. I. (2004 b), ‘On the discrepancy of Halton sequences’, Math. Balkanica 18, 15–32.Google Scholar
Bakhvalov, N. S. (1959), ‘On approximate computation of integrals’ (in Russian), Vestnik MGU, Ser. Math. Mech. Astron. Phys. Chem. 4, 3–18.Google Scholar
Baldeaux, J. (2012), Scrambled polynomial lattice rules for infinite-dimensional integration. In Monte Carlo and Quasi-Monte Carlo Methods 2010 (Plaskota, L. and Woźniakowski, H., eds), Springer, pp. 255–263.CrossRefGoogle Scholar
Baldeaux, J. and Dick, J. (2009), ‘QMC rules of arbitrary high order: Reproducing kernel Hilbert space approach’, Constr. Approx. 30, 495–527.CrossRefGoogle Scholar
Baldeaux, J. and Dick, J. (2011), ‘A construction of polynomial lattice rules with small gain coefficients’, Numer. Math. 119, 271–297.CrossRefGoogle Scholar
Baldeaux, J. and Gnewuch, M. (2013), Optimal randomized multilevel algorithms for infinite-dimensional integration on function spaces with ANOVA-type decomposition. Submitted.CrossRefGoogle Scholar
Baldeaux, J., Dick, J., Greslehner, J. and Pillichshammer, F. (2011), ‘Construction algorithms for higher order polynomial lattice rules’, J. Complexity 27, 281–299.CrossRefGoogle Scholar
Baldeaux, J., Dick, J., Leobacher, G, Nuyens, D. and Pillichshammer, F. (2012), ‘Efficient calculation of the worst-case error and (fast) component-by-component construction of higher order polynomial lattice rules’, Numer. Algorithms 59, 403–431.CrossRefGoogle Scholar
Barth, A., Schwab, C. and Zollinger, N. (2011), ‘Multi-level Monte Carlo finite element method for elliptic PDEs with stochastic coefficients’, Numer. Math. 119, 123–161.CrossRefGoogle Scholar
Bratley, P. and Fox, B. L. (1988), ‘Algorithm 659: Implementing Sobol's quasirandom sequence generator’, ACM Trans. Math. Softw. 14, 88–100.CrossRefGoogle Scholar
Bratley, P, Fox, B. L., and Niederreiter, H. (1992), ‘Implementation and tests of low-discrepancy sequences’, ACM Trans. Model. Comput. Simul 2, 195–213.CrossRefGoogle Scholar
Bungartz, H. and Griebel, M. (2004), Sparse grids. In Acta Numerica, Vol. 13, Cambridge University Press, pp. 147–269.Google Scholar
Caflisch, R. E., Morokoff, W. and Owen, A. (1997), ‘Valuation of mortgage backed securities using Brownian bridges to reduce effective dimension’, J. Comput. Finance 1, 27–46.CrossRefGoogle Scholar
Charrier, J., Scheichl, R. and Teckentrup, A. L. (2011), Finite element error analysis of elliptic PDEs with random coefficients and its application to multilevel Monte Carlo methods. Preprint 2/11, Bath Institute For Complex Systems.Google Scholar
Chazelle, B. (2000), The Discrepancy Method: Randomness and Complexity, Cambridge University Press.CrossRefGoogle Scholar
Chrestenson, H E. (1955), ‘A class of generalized Walsh functions’, Pacific J. Math. 5, 17–31.CrossRefGoogle Scholar
Chung, K. L. (1974), A Course in Probability Theory, second edition, Vol. 21 of Probability and Mathematical Statistics, Academic Press.Google Scholar
Cliffe, K. A., Giles, M. B., Scheichl, R. and Teckentrup, A. L. (2011), ‘Multilevel Monte Carlo methods and applications to elliptic PDEs with random coefficients’, Comput. Vis. Sci. 14, 3–15.CrossRefGoogle Scholar
Cools, R. (1997), Constructing cubature formulae: The science behind the art. In Acta Numerica, Vol. 6, Cambridge University Press, pp. 1–54.Google Scholar
Cools, R. and Lyness, J. N. (2001), ‘Three- and four-dimensional if-optimal lattice rules of moderate trigonometric degree’, Math. Comp. 70, 1549–1567.CrossRefGoogle Scholar
Cools, R. and Nuyens, D. (2008), A Belgian view on lattice rules. In Monte Carlo and Quasi-Monte Carlo Methods 2006 (Keller, A., Heinrich, S. and Niederreiter, H., eds), Springer, pp. 3–21.CrossRefGoogle Scholar
Cools, R., Kuo, F. Y. and Nuyens, D. (2006), ‘Constructing embedded lattice rules for multivariate integration’, SIAM J. Sci. Comput. 28, 2162–2188.CrossRefGoogle Scholar
Cools, R., Kuo, F. Y. and Nuyens, D. (2010), ‘Constructing lattice rules based on weighted degree of exactness and worst case error’, Computing 87, 63–89.CrossRefGoogle Scholar
Creutzig, J., Dereich, S., Müller-Gronbach, T. and Ritter, K. (2009), ‘Infinite- dimensional quadrature and approximation of functions’, Found. Comp. Math. 9, 391–429.CrossRefGoogle Scholar
Davis, P. J. and Rabinowitz, P. (1984), Methods of Numerical Integration, second edition, Academic.Google Scholar
Devroye, L. (1986), Nonuniform Random Variate Generation, Springer.CrossRefGoogle Scholar
Dick, J. (2004), ‘On the convergence rate of the component-by-component construction of good lattice rules’, J. Complexity 20, 493–522.CrossRefGoogle Scholar
Dick, J. (2007 a), ‘Explicit constructions of quasi-Monte Carlo rules for the numerical integration of high-dimensional periodic functions’, SIAM J. Numer. Anal. 45, 2141–2176.CrossRefGoogle Scholar
Dick, J. (2007 b), ‘The construction of extensible polynomial lattice rules with small weighted star discrepancy’, Math. Comp. 76, 2077–2085.CrossRefGoogle Scholar
Dick, J. (2008), ‘Walsh spaces containing smooth functions and quasi-Monte Carlo rules of arbitrary high order’, SIAM J. Numer. Anal. 46, 1519–1553.CrossRefGoogle Scholar
Dick, J. (2009 a), ‘The decay of the Walsh coefficients of smooth functions’, Bull. Aust. Math. Soc. 80, 430–453.CrossRefGoogle Scholar
Dick, J. (2009 b), On Quasi-Monte Carlo rules achieving higher order convergence. In Monte Carlo and Quasi-Monte Carlo Methods 2008 (L'Ecuyer, P. and Owen, A. B., eds), Springer, pp. 73–96.CrossRefGoogle Scholar
Dick, J. (2011 a), ‘Higher order scrambled digital nets achieve the optimal rate of the root mean square error for smooth integrands’, Ann. Statist. 39, 1372–1398.CrossRefGoogle Scholar
Dick, J. (2011 b), ‘Quasi-Monte Carlo numerical integration on ℝs: Digital nets and worst-case error’, SIAM J. Numer. Anal. 49, 1661–1691.CrossRefGoogle Scholar
Dick, J. (2012), ‘Random weights, robust lattice rules and the geometry of the cbcrc algorithm’, Numer. Math. 122, 443–467.CrossRefGoogle Scholar
Dick, J. and Baldeaux, J. (2009), Equidistribution properties of generalized nets and sequences. In Monte Carlo and Quasi-Monte Carlo Methods 2008 (L'Ecuyer, P. and Owen, A. B., eds), Springer, pp. 305–322.CrossRefGoogle Scholar
Dick, J. and Gnewuch, M. (2013), Infinite-dimensional integration in weighted Hilbert spaces: Anchored decompositions, optimal deterministic algorithms, and higher order convergence. Submitted.CrossRefGoogle Scholar
Dick, J. and Kritzer, P. (2010), ‘Duality theory and propagation rules for generalized digital nets’, Math. Comp. 79, 993–1017.CrossRefGoogle Scholar
Dick, J. and Kuo, F. Y. (2004 a), ‘Reducing the construction cost of the component-by-component construction of good lattice rules’, Math. Comp. 73, 1967–1988.CrossRefGoogle Scholar
Dick, J. and Kuo, F. Y. (2004 b), Constructing good lattice rules with millions of points. In Monte Carlo and Quasi-Monte Carlo Methods 2002 (Niederreiter, H., ed.), Springer, pp. 181–197.CrossRefGoogle Scholar
Dick, J. and Pillichshammer, F. (2005), ‘Multivariate integration in weighted Hilbert spaces based on Walsh functions and weighted Sobolev spaces’, J. Complexity 21, 149–195.CrossRefGoogle Scholar
Dick, J. and Pillichshammer, F. (2007), ‘strong tractability of multivariate integration of arbitrary high order using digitally shifted polynomial lattice rules’, J. Complexity 23, 436–453.CrossRefGoogle Scholar
Dick, J. and Pillichshammer, F. (2010), Digital Nets and Sequences, Cambridge University Press.CrossRefGoogle Scholar
Dick, J., Kritzer, P., Leobacher, G. and Pillichshammer, F. (2007 a), ‘Constructions of general polynomial lattice rules based on the weighted star discrepancy’, Finite Fields Appl. 13, 1045–1070.CrossRefGoogle Scholar
Dick, J., Kritzer, P., Pillichshammer, F. and Schmid, W. C. (2007 b), ‘On the existence of higher order polynomial lattices based on a generalized figure of merit’, J. Complexity 23, 581–593.CrossRefGoogle Scholar
Dick, J., Kuo, F. Y., Pillichshammer, F. and Sloan, I. H. (2005), ‘Construction algorithms for polynomial lattice rules for multivariate integration’, Math. Comp. 74, 1895–1921.CrossRefGoogle Scholar
Dick, J., Larcher, G., Pillichshammer, F. and Woźniakowski, H. (2011), ‘Exponential convergence and tractability of multivariate integration for Korobov spaces’, Math. Comp. 80, 905–930.CrossRefGoogle Scholar
Dick, J., Nuyens, D. and Pillichshammer, F. (2013), Lattice rules for nonperiodic smooth integrands. Submitted.CrossRefGoogle Scholar
Dick, J., Pillichshammer, F., and Waterhouse, B. J. (2008), ‘The construction of good extensible rank-1 lattices’, Math. Comp. 77, 2345–2374.CrossRefGoogle Scholar
Dick, J., Sloan, I. H., Wang, X. and Woźniakowski, H. (2006), ‘Good lattice rules in weighted Korobov spaces with general weights’, Numer. Math. 103, 63–97.CrossRefGoogle Scholar
Digital Library of Mathematical Functions, National Institute of Standards and Technology.Google Scholar
Doerr, B., Gnewuch, M., Kritzer, P. and Pillichshammer, F. (2008), ‘Component-by-component construction of low-discrepancy point sets of small size’, Monte Carlo Methods Appl. 14, 129–149.CrossRefGoogle Scholar
Doerr, B., Gnewuch, M. and Wahlström, M. (2009), Implementation of a component-by-component algorithm to generate small low-discrepancy samples. In Monte Carlo and Quasi-Monte Carlo Methods 2008 (L'Ecuyerand, P.Owen, A. B., eds), Springer, pp. 323–338.CrossRefGoogle Scholar
Doerr, B., Gnewuch, M. and Wahlstrom, M. (2010), ‘Algorithmic construction of low-discrepancy point sets via dependent randomized rounding’, J. Complexity 26, 490–507.CrossRefGoogle Scholar
Drmota, M. and Tichy, R. F. (1997), Sequences, Discrepancies and Applications, Vol. 1651 of Lecture Notes in Mathematics, Springer.Google Scholar
Fang, K. T. and Wang, Y. (1994), Number-Theoretic Methods in Statistics, Chapman & Hall.CrossRefGoogle Scholar
Faure, H. (1982), ‘Discrepance de suites associées à unsysteme de numeration (en dimensions)’, Acta Arith. 41, 337–351.CrossRefGoogle Scholar
Fine, N. J. (1949), ‘On the Walsh functions’, Trans. Amer. Math. Soc. 65, 372–414.CrossRefGoogle Scholar
Fox, B. L. (1986), ‘Algorithm 647: Implementation and relative efficiency of quasir-andom sequence generators, ACM Trans. Math. Softw. 12, 362–376.CrossRefGoogle Scholar
Ghanem, R. G. and Spanos, P. D. (1991), Stochastic Finite Elements: A Spectral Approach, Springer.CrossRefGoogle Scholar
Giles, M. B. (2008), ‘Multilevel Monte Carlo path simulation’, Oper. Res. 56, 607–617.CrossRefGoogle Scholar
Giles, M., Kuo, F. Y., Sloan, I. H. and Waterhouse, B. J. (2008), ‘Quasi-Monte Carlo for finance applications’, ANZIAM J. 50 (CTAC2008), C308–C323.CrossRefGoogle Scholar
Gnewuch, M. (2008), ‘Bracketing numbers for axis-parallel boxes and applications to geometric discrepancy, J. Complexity 24, 154–172.CrossRefGoogle Scholar
Gnewuch, M. (2009), ‘On probabilistic results for the discrepancy of a hybrid-Monte Carlo sequence, J. Complexity 23, 312–317.CrossRefGoogle Scholar
Gnewuch, M. (2012 a), ‘Weighted geometric discrepancies and numerical integration on reproducing kernel Hilbert spaces’, J. Complexity 28, 2–17.CrossRefGoogle Scholar
Gnewuch, M. (2012 b), ‘Infinite-dimensional integration on weighted Hilbert spaces’, Math. Comp. 280, 2175–2205.CrossRefGoogle Scholar
Gnewuch, M. (2013), Lower error bounds for randomized multilevel and changing dimension algorithms. In Monte Carlo and Quasi-Monte Carlo Methods 2012 (Dick, J., Kuo, F. Y., Peters, G. W. and Sloan, I. H., eds), Springer, to appear.Google Scholar
Gnewuch, M. and Rosca, A. V. (2009), ‘On G-discrepancy and mixed Monte Carlo and quasi-Monte Carlo sequences’, Acta Univ. Apulensis Math. Inform. 18, 97–110.Google Scholar
Gnewuch, M., Mayer, S. and Ritter, K. (2013), On weighted Hilbert spaces and integration of functions of infinitely many variables. Submitted.CrossRefGoogle Scholar
Gnewuch, M., Srivastav, A. and Winzen, C. (2009), ‘Finding optimal volume subin-tervals with/k-points and calculating the star discrepancy are NP-hard problems’, J. Complexity 25, 115–127.CrossRefGoogle Scholar
Gnewuch, M., Wahlstriom, M. and Winzen, C. (2012), ‘A new randomized algorithm to approximate the star discrepancy based on threshold accepting, SIAM J. Numer. Anal. 50, 781–807.CrossRefGoogle Scholar
Goda, T. and Dick, J. (2013), Construction of interlaced scrambled polynomial lattice rules of arbitrary high order. Submitted.CrossRefGoogle Scholar
Graham, I. G., Kuo, F. Y., Nichols, J., Scheichl, R., Schwab, C. and Sloan, I. H. (2013), Quasi-Monte Carlo finite element methods for elliptic PDEs with log-normal random coefficients. In preparation.CrossRefGoogle Scholar
Graham, I. G., Kuo, F. Y., Nuyens, D., Scheichl, R. and Sloan, I. H. (2011), ‘Quasi-Monte Carlo methods for elliptic PDEs with random coefficients and applications, J. Comput. Phys. 230, 3668–3694.CrossRefGoogle Scholar
Griebel, M. (2006), Sparse grids and related approximation schemes for higher dimensional problems. In Foundations of Computational Mathematics, Santander, 2005, Cambridge University Press, pp. 106–161.CrossRefGoogle Scholar
Griebel, M., Kuo, F. Y. and Sloan, I. H. (2013), ‘The smoothing effect of integration in ℝd and the ANOVA decomposition’, Math. Comp. 82, 383–400.CrossRefGoogle Scholar
Halton, J. H. (1960), ‘On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals, Numer. Math. 2, 84–90.CrossRefGoogle Scholar
Hammersley, J. M. and Handscomb, D. C. (1964), Monte Carlo Methods, Methuen.CrossRefGoogle Scholar
Hansen, T., Mullen, G. L. and Niederreiter, H. (1993), ‘Good parameters for a class of node sets in quasi-Monte Carlo integration, Math. Comp. 61, 225–234.CrossRefGoogle Scholar
Heinrich, S. (1998), ‘Monte Carlo complexity of global solution of integral equations, J. Complexity 14, 151–175.CrossRefGoogle Scholar
Heinrich, S. and Sindambiwe, E. (1999), ‘Monte Carlo complexity of parametric integration, J. Complexity 15, 317–341.CrossRefGoogle Scholar
Heinrich, S., Hickernell, F. and Yue, R. X. (2004), ‘Optimal quadrature for Haar wavelet spaces’, Math. Comp. 73, 259–277.CrossRefGoogle Scholar
Heinrich, S., Novak, E., Wasilkowski, G. W., Woźniakowski, H. (2001), ‘The inverse of the star-discrepancy depends linearly on the dimension’, Acta Arith. 96, 279–302.CrossRefGoogle Scholar
Hickernell, F. J. (1996 a), ‘Quadrature error bounds with applications to lattice rules’, SIAM J. Numer. Anal. 33, 1995–2016.CrossRefGoogle Scholar
Erratum: ‘Quadrature error bounds with applications to lattice rules’, SIAM J. Numer. Anal. 34, 853–866 (1997).CrossRefGoogle Scholar
Hickernell, F. J. (1996 b), ‘The mean square discrepancy of randomized nets’, ACM Trans. Modeling Comput. Simul. 6, 274–296.CrossRefGoogle Scholar
Hickernell, F. J. (1998 a), ‘A generalized discrepancy and quadrature error bound’, Math. Comp. 67, 299–322.CrossRefGoogle Scholar
Hickernell, F. J. (1998 b), Lattice rules: How well do they measure up? In Random and Quasi-Random Point Sets (Hellekalek, P. and Larcher, G., eds), Springer, pp. 109–166.CrossRefGoogle Scholar
Hickernell, F. J. (1999), ‘Goodness-of-fit statistics, discrepancies and robust designs’, Statist. Probab. Lett. 44, 73–78.CrossRefGoogle Scholar
Hickernell, F. J. (2002), Obtaining O(N−2+ε) convergence for lattice quadrature rules. In Monte Carlo and Quasi-Monte Carlo Methods 2000 (Fang, K.T., Hickernell, F. J. and Niederreiter, H., eds), Springer, pp. 274–289.CrossRefGoogle Scholar
Hickernell, F. J. and Hong, H. S. (2002), Quasi-Monte Carlo methods and their randomisations. In Applied Probability: AMS/IP Studies in Advanced Mathematics, Vol. 26 (Chan, R., Kwok, Y.-K., Yao, D. and Zhang, Q., eds), AMS, pp. 59–77.Google Scholar
Hickernell, F. J. and Niederreiter, H. (2003), ‘The existence of good extensible rank-1 lattices’, J. Complexity 19, 286–300.CrossRefGoogle Scholar
Hickernell, F. J. and Wang, X. (2002), ‘The error bounds and tractability of quasi-Monte Carlo algorithms in infinite dimension’, Math. Comp. 71, 1641–1661.CrossRefGoogle Scholar
Hickernell, F. J. and Woźniakowski, H. (2000), ‘Integration and approximation in arbitrary dimensions’, Adv. Comput. Math. 12, 25–58.CrossRefGoogle Scholar
Hickernell, F. J. and Woźniakowski, H. (2001), ‘Tractability of multivariate integration for periodic functions’, J. Complexity 17, 660–682.CrossRefGoogle Scholar
Hickernell, F. J. and Yue, R. X. (2000), ‘The mean square discrepancy of scrambled (t, s)-sequences’, SIAM J. Numer. Anal. 38, 1089–1112.CrossRefGoogle Scholar
Hickernell, F. J., Hong, H. S., Ecuyer, P. L., and Lemieux, C. (2000), ‘Extensible lattice sequences for quasi-Monte Carlo quadrature’, SIAM J. Sci. Comput. 22, 1117–1138.CrossRefGoogle Scholar
Hickernell, F. J., Kritzer, P., Kuo, F. Y. and Nuyens, D. (2012), ‘Weighted compound integration rules with higher order convergence for all N’, Numer. Algorithms 59, 161–183.CrossRefGoogle Scholar
Hickernell, F. J., Lemieux, C. and Owen, A. B. (2005), ‘Control variates for quasi-Monte Carlo, with comments by Pierre L Ecuyer and Xiao-Li Meng and a rejoinder by the authors. Statist. Sci. 20, 1–31.Google Scholar
Hickernell, F., Muiller-Gronbach, T., Niu, B. and Ritter, K. (2010), ‘Monte Carlo algorithms for infinite-dimensional integration on ℝN’, J. Complexity 26, 229–254.CrossRefGoogle Scholar
Hinrichs, A. (2004), ‘Covering numbers, Vapnik-Červonenkis classes and bounds for the star-discrepancy’, J. Complexity 20, 477–483.CrossRefGoogle Scholar
Hinrichs, A., Pillichshammer, F. and Schmid, W. C. (2008), ‘Tractability properties of the weighted star discrepancy’, J. Complexity 24, 134–143.CrossRefGoogle Scholar
Hlawka, E. (1961), ‘Funktionen von beschränkter Variation in der Theorie der Gleichverteilung’, Ann. Mat. Pura Appl. 54, 325–333.CrossRefGoogle Scholar
Hlawka, E. (1962), ‘Zur angeniaherten Berechnung mehrfacher Integrale’, Monatsh. Math. 66, 140–151.CrossRefGoogle Scholar
Hofer, R. and Kritzer, P. (2011), ‘On hybrid sequences built from Niederreiter-Halton sequences and Kronecker sequences’, Bull. Aust. Math. Soc. 84, 238–254.CrossRefGoogle Scholar
Hofer, R. and Larcher, G. (2010), ‘On existence and discrepancy of certain digital Niederreiter-Halton sequences’, Acta Arith. 141, 369–394.CrossRefGoogle Scholar
Hofer, R. and Larcher, G. (2012), ‘Metrical results on the discrepancy of Halton-Kronecker sequences’, Math. Z. 271, 1–11.CrossRefGoogle Scholar
Hofer, R. and Niederreiter, H. (2013), ‘A construction of (t, s)-sequences with finite-row generating matrices using global function fields’, Finite Fields Appl. 21, 97–110.CrossRefGoogle Scholar
Hofer, R. and Pirsic, G. (2011), ‘An explicit construction of finite-row digital (0,s)-sequences’, Unif. Distrib. Theory 6, 13–30.Google Scholar
Hofer, R., Kritzer, P., Larcher, G. and Pillichshammer, F. (2009), ‘Distribution properties of generalized van der Corput-Halton sequences and their subsequences’, Int. J. Number Theory 5, 719–746.CrossRefGoogle Scholar
Härmann, W., Leydold, J. and Derflinger, G. (2010), Automatic Nonuniform Random Variate Generation, Springer.Google Scholar
Joe, S. (2004), Component by component construction of rank-1 lattice rules having O(n−1(ln(n))d) star discrepancy. In Monte Carlo and Quasi-Monte Carlo Methods 2002 (Niederreiter, H., ed.), Springer, pp. 293–298.CrossRefGoogle Scholar
Joe, S. (2006), Construction of good rank-1 lattice rules based on the weighted star discrepancy. In Monte Carlo and Quasi-Monte Carlo Methods 2004 (Niederreiter, H. and Talay, D., eds), Springer, pp. 181–196.CrossRefGoogle Scholar
Joe, S. and Kuo, F. Y. (2008), ‘Constructing Sobol' sequences with better two-dimensional projection’, SIAM J. Sci. Comput. 30, 2635–2654.CrossRefGoogle Scholar
Kämmerer, L., Kunis, S. and Potts, D. (2012), ‘Interpolation lattices for hyperbolic cross trigonometric polynomials’, J. Complexity 28, 76–92.CrossRefGoogle Scholar
Keller, A. (2006), Myths of computer graphics. In Monte Carlo and Quasi-Monte Carlo Methods 2004 (Niederreiter, H. and Talay, D., eds), Springer, pp. 217–243.CrossRefGoogle Scholar
Keller, A. (2013), Quasi-Monte Carlo image synthesis in a nutshell. In Monte Carlo and Quasi-Monte Carlo Methods 2012 (Dick, J., Kuo, F. Y., Peters, G. W. and Sloan, I. H., eds), Springer, to appear.Google Scholar
Koksma, J. F. (1942/1943), ‘Een algemeene stelling uit de theorie der gelijkmatige verdeeling modulo 1’, Mathematica B (Zutphen) 11, 7–11.Google Scholar
Korobov, N. M. (1959), ‘The approximate computation of multiple integrals’ (in Russian), Dokl. Akad. Nauk SSSR 124, 1207–1210.Google Scholar
Korobov, N. M. (1963), Number-Theoretic Methods in Approximate Analysis, Fiz-matgiz.Google Scholar
Korobov, N. M. (1982), ‘On the calculation of optimal coefficients’ (in Russian), Dokl. Akad. Nauk SSSR 267, 289–292.Google Scholar
Kritzer, P. and Pillichshammer, F. (2007), ‘Constructions of general polynomial lattices for multivariate integration’, Bull. Austral. Math. Soc. 76, 93–110.CrossRefGoogle Scholar
Kuipers, L. and Niederreiter, H. (1974), Uniform Distribution ofSequences, Wiley.Google Scholar
Kuo, F. Y. (2003), ‘Component-by-component constructions achieve the optimal rate of convergence for multivariate integration in weighted Korobov and Sobolev spaces’, J. Complexity 19, 301–320.CrossRefGoogle Scholar
Kuo, F. Y. and Joe, S. (2002), ‘Component-by-component construction of good lattice rules with a composite number of points’, J. Complexity 18, 943–976.CrossRefGoogle Scholar
Kuo, F. Y. and Joe, S. (2003), ‘Component-by-component construction of good intermediate-rank lattice rules’, SIAM J. Numer. Anal. 41, 1465–1486.CrossRefGoogle Scholar
Kuo, F. Y. and Sloan, I. H. (2005), ‘Lifting the curse of dimensionality’, Not. Amer. Math. Soc. 52, 1320–1328.Google Scholar
Kuo, F. Y., M, W. T.. Dunsmuir, Sloan, I. H., Wand, M. P., Womersley, R. S. (2008 a), ‘Quasi-Monte Carlo for highly structured generalised response models’, Method. Comput. Appl. Probab. 10, 239–275.CrossRefGoogle Scholar
Kuo, K. Y., Schwab, C. and Sloan, I. H. (2011), ‘Quasi-Monte Carlo methods for high-dimensional integration: The standard (weighted Hilbert space) setting and beyond’, ANZIAM J. 53, 1–37.CrossRefGoogle Scholar
Kuo, F. Y., Schwab, C. and Sloan, I. H. (2012), Quasi-Monte Carlo finite element methods for a class of elliptic partial differential equations with random coefficients, SIAM J. Numer. Anal. 50, 3351–3374.CrossRefGoogle Scholar
Kuo, F. Y., Schwab, C. and Sloan, I. H. (2013), Multi-level quasi-Monte Carlo finite element methods for a class of elliptic partial differential equations with random coefficients. Submitted.CrossRefGoogle Scholar
Kuo, F. Y., Sloan, I. H., Wasilkowski, G. W., and Waterhouse, B. J. (2010 a), Randomly shifted lattice rules with the optimal rate of convergence for unbounded integrands’, J. Complexity 26, 135–160.CrossRefGoogle Scholar
Kuo, F. Y., Sloan, I. H., Wasilkowski, G. W. and Woźniakowski, H. (2010 b), ‘On decompositions of multivariate functions’, Math. Comp. 79, 953–966.CrossRefGoogle Scholar
Kuo, F. Y., Sloan, I. H., Wasilkowski, G. W. and Woźniakowski, H. (2010 c), ‘Liberating the dimension’, J. Complexity 26, 422–454.CrossRefGoogle Scholar
Kuo, F. Y., Sloan, I. H., and Woźniakowski, H. (2006 a), Lattice rules for multivariate approximation in the worst case setting. In Monte Carlo and Quasi-Monte Carlo Methods 2004 (Niederreiter, H. and Talay, D., eds), Springer, pp. 289–330.CrossRefGoogle Scholar
Kuo, F. Y., Sloan, I. H. and Woźniakowski, H. (2007), ‘Periodization strategy may fail in high dimensions’, Numer. Algorithms 46, 369–391.CrossRefGoogle Scholar
Kuo, F. Y., Sloan, I. H., and Woźniakowski, H. (2008 b), ‘Lattice rule algorithms for multivariate approximation in the average case setting’, J. Complexity 24, 283–323.CrossRefGoogle Scholar
Kuo, F. Y., Wasilkowski, G. W., and Waterhouse, B. J. (2006 b), ‘Randomly shifted lattice rules for unbounded integrals’, J. Complexity 22, 630–651.CrossRefGoogle Scholar
Kuo, F. Y., Wasilkowski, G. W., and Woźniakowski, H. (2009), ‘Lattice algorithms for multivariate ∞ approximation in the worst case setting’, Constr. Approx. 30, 475–493.CrossRefGoogle Scholar
Larcher, G. and Traunfellner, C. (1994), ‘On the numerical integration of Walsh series by number-theoretic methods’, Math. Comp. 63, 277–291.CrossRefGoogle Scholar
Larcher, G., Lauß, A., Niederreiter, H. and Schmid, W. C. (1996 a), ‘Optimal polynomials for (t, m, s)-nets and numerical integration of multivariate Walsh series’, SIAM J. Numer. Anal. 33, 2239–2253.CrossRefGoogle Scholar
Larcher, G., Schmid, W. C. and Wolf, R. (1994), ‘Representation of functions as Walsh series to different bases and an application to the numerical integration of high-dimensional Walsh series’, Math. Comp. 63, 701–716.CrossRefGoogle Scholar
Larcher, G., Schmid, W. C. and Wolf, R. (1996 b), ‘Quasi-Monte Carlo methods for the numerical integration of multivariate Walsh series: Monte Carlo and quasi-Monte Carlo methods’, Math. Comput. Modelling 23, 55–67.CrossRefGoogle Scholar
Laurie, D. (1996), ‘Periodizing transformations for numerical integration’, J. Comput. Appl. Math. 66, 337–344.CrossRefGoogle Scholar
L'Ecuyer, P. (2004), Quasi-Monte Carlo methods in finance. In Proc. 2004 Winter Simulation Conference (Ingalls, R. G., Rossetti, M. D., Smith, J. S. and Peters, B. A., eds), IEEE Computer Society Press, pp. 1645–1655.Google Scholar
Ecuyer, P. L' and Lemieux, C. (2000), ‘Variance reduction via lattice rules’, Management Sci. 46, 1214–1235.CrossRefGoogle Scholar
L'Ecuyer, P. and Munger, D. (2012), On figures of merit for randomly shifted lattice rules. In Monte Carlo and Quasi-Monte Carlo Methods 2010 (Plaskota, L. and Woźniakowski, H., eds), Springer, pp. 133–159.CrossRefGoogle Scholar
L'Ecuyer, P., Munger, D. and Tuffin, B. (2010), ‘On the distribution of integration error by randomly-shifted lattice rules’. Electron. J. Stat. 4, 950–993.Google Scholar
Lemieux, C. (2009), Monte Carlo and Quasi-Monte Carlo Sampling, Springer.Google Scholar
Lemieux, C. and Ecuyer, P. L' (2001), ‘On selection criteria for lattice rules and other quasi-Monte Carlo point sets’, Math. Comput. Simulation 55, 139–148.CrossRefGoogle Scholar
Li, D. and Hickernell, F. J. (2003), Trigonometric spectral collocation methods on lattices. In Recent Advances in Scientific Computing and Partial Differential Equations (Cheng, S. Y., Shu, C.-W. and Tang, T., eds), Vol. 330 of AMS Series in Contemporary Mathematics, AMS, pp. 121–132.CrossRefGoogle Scholar
Li, G., Schoendorf, J., Ho, T.-S. and Rabitz, H. (2004), ‘Multicut-HDMR with an application to an ionospheric model’, J. Comput. Chem. 25, 1149–1156.CrossRefGoogle Scholar
Loh, W. L. (2003), ‘On the asymptotic distribution of scrambled net quadrature’, Ann. Statist. 31, 1282–1324.CrossRefGoogle Scholar
Lyness, J. and Sørevik, T. (2006), ‘Five-dimensional/k-optimal lattice rules’, Math. Comp. 75, 1467–1480.CrossRefGoogle Scholar
Maize, E. (1980), Contributions to the theory of error reduction in quasi-Monte Carlo methods. PhD thesis, Claremont Graduate School.Google Scholar
Matousěk, J. (1998 a), ‘The exponent of discrepancy is at least 1.0669’, J. Complexity 14, 448–453.CrossRefGoogle Scholar
Matousěk, J. (1998 b), ‘On the L2 discrepancy for anchored boxes’, J. Complexity 14, 527–556.CrossRefGoogle Scholar
Matousěk, J. (1999), Geometric Discrepancy: An Illustrated Guide, Vol. 18 of Algorithms and Combinatorics, Springer.CrossRefGoogle Scholar
Matsumoto, M. and Yoshiki, T. (2013), Existence of higher order convergent quasi-Monte Carlo rules via Walsh figure of merit. In Monte Carlo and Quasi-Monte Carlo Methods 2012 (Dick, J., Kuo, F. Y., Peters, G. W. and Sloan, I. H., eds), Springer, to appear.Google Scholar
Matsumoto, M., Saito, M., Matoba, K. (2013), ‘A computable figure of merit for quasi-Monte Carlo point sets’, Math. Comp., toappear.CrossRefGoogle Scholar
Niederreiter, H. (1978), ‘Quasi-Monte Carlo methods and pseudo-random numbers’, Bull. Amer. Math. Soc. 84, 957–1041.CrossRefGoogle Scholar
Niederreiter, H. (1987), ‘Point sets and sequences with small discrepancy’, Monatsh. Math. 104, 273–337.CrossRefGoogle Scholar
Niederreiter, H. (1988), ‘Low-discrepancy and low-dispersion sequences’, J. Number Theory 30, 51–70.CrossRefGoogle Scholar
Niederreiter, H. (1992 a), Random Number Generation and Quasi-Monte Carlo Methods, SIAM.Google Scholar
Niederreiter, H. (1992 b), ‘Low-discrepancy point sets obtained by digital constructions over finite fields’, Czechoslovak Math. J. 42, 143–166.CrossRefGoogle Scholar
Niederreiter, H. (2003), ‘The existence of good extensible polynomial lattice rules’, Monatsh. Math. 139, 295–307.CrossRefGoogle Scholar
Niederreiter, H. (2004), Digital nets and coding theory. In Coding, Cryptography and Combinatorics (Feng, K. Q., Niederreiter, H. and Xing, C. P., eds), Birkhauser, pp. 247–257.CrossRefGoogle Scholar
Niederreiter, H. (2009), ‘On the discrepancy of some hybrid sequences’, Acta Arith. 138, 373–398.CrossRefGoogle Scholar
Niederreiter, H. (2010 a), ‘A discrepancy bound for hybrid sequences involving digital explicit inversive pseudorandom numbers’, Unif. Distrib. Theory 5, 53–63.Google Scholar
Niederreiter, H. (2010 b), ‘Further discrepancy bounds and an Erdõs-Turán-Koksma inequality for hybrid sequences’, Monatsh. Math. 161, 193–222.CrossRefGoogle Scholar
Niederreiter, H. (2012), ‘Improved discrepancy bounds for hybrid sequences involving Halton sequences’, Acta Arith. 155, 71–84.CrossRefGoogle Scholar
Niederreiter, H. and Pillichshammer, F. (2009), ‘Construction algorithms for good extensible lattice rules’, Constr. Approx. 30, 361–393.CrossRefGoogle Scholar
Niederreiter, H. and Winterhof, A. (2011), ‘Discrepancy bounds for hybrid sequences involving digital explicit inversive pseudorandom numbers’, Unif. Distrib. Theory 6, 33–56.Google Scholar
Niederreiter, H. and Xing, C. P. (1995), ‘Low-discrepancy sequences obtained from algebraic function fields over finite fields’, Acta Arith. 72, 281–298.CrossRefGoogle Scholar
Niederreiter, H. and Xing, C. P. (1996 a), ‘Low-discrepancy sequences and global function fields with many rational places’, Finite Fields Appl. 2, 241–273.CrossRefGoogle Scholar
Niederreiter, H. and Xing, C. P. (1996 b), Quasirandom points and global function fields. In Finite Fields and Applications Vol. 233 of London Mathematical Society Lecture Note Series, Cambridge University Press, pp. 269–296.Google Scholar
Niu, B. and Hickernell, F. J. (2009), Monte Carlo simulation of stochastic integrals when the cost of function evaluation is dimension dependent. In Monte Carlo and Quasi-Monte Carlo Methods 2008 (L'Ecuyer, P. and Owen, A. B., eds), Springer, pp. 545–560.CrossRefGoogle Scholar
Niu, B., Hickernell, F. J., Müller-Gronbach, T. and Ritter, K. (2011), ‘Deterministic multi-level algorithms for infinite-dimensional integration on ℝℕ’, J. Complexity 27, 331–351.CrossRefGoogle Scholar
Novak, E. and Woázniakowski, H. (2001), ‘Intractability results for integration and discrepancy’, J Complexity, 17, 388–441.CrossRefGoogle Scholar
Novak, E. and Woázniakowski, H. (2008), Tractability of Multivariate Problems, Vol. I: Linear Information, EMS.CrossRefGoogle Scholar
Novak, E. and Woźniakowski, H. (2009), L2 discrepancy and multivariate integration. In Analytic Number Theory: Essays in Honour of Klaus Roth (Chen, W.W.L., Gowers, W. T., Halberstam, H., Schmidt, W. M. and Vaughan, R. C., eds), Cambridge University Press, pp. 359–388.Google Scholar
Novak, E. and Woázniakowski, H. (2010), Tractability of Multivariate Problems, Vol. II: Standard Information for Functionals, EMS.CrossRefGoogle Scholar
Novak, E. and Woźniakowski, H. (2012), Tractability of Multivariate Problems, Vol. III: Standard Information for Operators, EMS.CrossRefGoogle Scholar
Nuyens, D. and Cools, R. (2006 a), ‘Fast algorithms for component-by-component construction of rank-1 lattice rules in shift-invariant reproducing kernel Hilbert spaces’, Math. Comp. 75, 903–920.CrossRefGoogle Scholar
Nuyens, D. and Cools, R. (2006 b), ‘Fast component-by-component construction of rank-1 lattice rules with a non-prime number of points’, J. Complexity 22, 4–28.CrossRefGoogle Scholar
Nuyens, D. and Cools, R. (2006 c), Fast component-by-component construction, a reprise for different kernels. In Monte Carlo and Quasi-Monte Carlo Methods 2004 (Niederreiter, H. and Talay, D., eds), Springer, pp. 373–387.CrossRefGoogle Scholar
Okten, G. (1996), ‘A probabilistic result on the discrepancy of a hybrid Monte Carlo sequence and applications’, Monte Carlo Methods Appl. 2, 255–270.CrossRefGoogle Scholar
Okten, G., Tuffin, B. and Burago, V. (2006), ‘A central limit theorem and improved error bounds for a hybrid-Monte Carlo sequence with applications in computational finance’, J. Complexity 22, 435–458.CrossRefGoogle Scholar
Owen, A. B. (1995), Randomly permuted (t, m, s)-nets and (t, s)-sequences. In Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing (Las Vegas, NV, 1994), Vol. 106 of Lecture Notes in Statistics, Springer, pp. 299–317.CrossRefGoogle Scholar
Owen, A. B. (1997 a), ‘scrambled net variance for integrals of smooth functions’, Ann. Statist. 25, 1541–1562.CrossRefGoogle Scholar
Owen, A. B. (1997 b), ‘Monte Carlo variance of scrambled net quadrature’, SIAM J. Numer. Anal. 34, 1884–1910.CrossRefGoogle Scholar
Owen, A. B. (1998), ‘scrambling Sobol' and Niederreiter-Xing points’, J. Complexity 14, 466–489.CrossRefGoogle Scholar
Owen, A. B. (2006), Quasi-Monte Carlo for integrands with point singularities at unknown locations. In Monte Carlo and Quasi-Monte Carlo Methods 2004 (Niederreiter, H. and Talay, D., eds), Springer, pp. 403–417.CrossRefGoogle Scholar
Paskov, S. H. and Traub, J. (1995), ‘Faster evaluation of financial derivatives’, J. Portfolio Management 22, 113–120.CrossRefGoogle Scholar
Pillichshammer, F. (2002), ‘Bounds for the quality parameter of digital shift nets over Z2’, Finite Fields Appl. 8, 444–454.Google Scholar
Pillichshammer, F. and Pirsic, G. (2009), Discrepancy of hyperplane nets and cyclic nets. In Monte Carlo and Quasi-Monte Carlo Methods 2008 (Ecuyer, P. L' and Owen, A. B., eds), Springer, pp. 573–587.CrossRefGoogle Scholar
Pirsic, G. (2002), A software implementation of Niederreiter-Xing sequences. In Monte Carlo and Quasi-Monte Carlo Methods 2000 (Fang, K. T., Hickernell, F. J. and Niederreiter, H., eds), Springer, pp. 434–445.CrossRefGoogle Scholar
Pirsic, G. and Pillichshammer, F. (2011), ‘Extensible hyperplane nets’, Finite Fields Appl. 17, 407–423.CrossRefGoogle Scholar
Pirsic, G. and Schmid, W. C. (2001), ‘Calculation of the quality parameter of digital nets and application to their construction’, J. Complexity 17, 827–839.CrossRefGoogle Scholar
Pirsic, G., Dick, J. and Pillichshammer, F. (2006), ‘Cyclic digital nets, hyperplane nets, and multivariate integration in Sobolev spaces’, SIAM J. Numer. Anal. 44, 385–411.CrossRefGoogle Scholar
Plaskota, L. and Wasilkowski, G. (2011), ‘Tractability of infinite-dimensional integration in the worst case and randomized setting’, J. Complexity 27, 505–518.CrossRefGoogle Scholar
Plaskota, L., Wasilkowski, G. and Woázniakowski, H. (2000), ‘A new algorithm and worst case complexity for Feynman-Kac path integration’, J. Comput. Phys. 164, 335–353.CrossRefGoogle Scholar
Rader, C. M. (1968), ‘Discrete Fourier transforms when the number of data samples is prime’, Proc. IEEE 5, 1107–1108.CrossRefGoogle Scholar
Rhee, C.-H. and Glynn, P. W. (2012), A new approach to unbiased estimation for SDE's. In Proc. 2012 Winter Simulation Conference (Laroque, C., Him-melspach, J., Pasupathy, R., Rose, O. and Uhrmacher, A. M., eds).Google Scholar
Schmid, W. C. (1996), Shift-nets: A new class of binary digital (t, m, s)-nets. In Monte Carlo and Quasi-Monte Carlo Methods 1996 (Niederreiter, H., Hellekalek, P., Larcher, G. and Zinterhof, P., eds), Vol. 127 of Lecture Notes in Statistics, Springer, pp. 369–381.CrossRefGoogle Scholar
Schmid, W. C. (2000), Improvements and extensions of the ‘salzburg Tables’ by using irreducible polynomials. In Monte Carlo and Quasi-Monte Carlo Methods 1998 (Niederreiter, H. and Spanier, J., eds), Springer, pp. 436–447.CrossRefGoogle Scholar
Schwab, C. and Gittelson, C. J. (2011), Sparse tensor discretizations of high-dimensional parametric and stochastic PDEs. In Acta Numerica, Vol. 20, Cambridge University Press, pp. 291–467.Google Scholar
Sidi, A. (1993), A new variable transformation for numerical integration. In Numerical Integration IV: Oberwolfach, 1992 (Brass, H. and Hämmerlin, G., eds), Birkhäuser, pp. 359–373.CrossRefGoogle Scholar
Sinescu, V. and Joe, S. (2007), ‘Good lattice rules based on the general weighted star discrepancy’, Math. Comp. 76, 989–1004.CrossRefGoogle Scholar
Sinescu, V. and Joe, S. (2008), Good lattice rules with a composite number of points based on the product weighted star discrepancy. In Monte Carlo and Quasi-Monte Carlo Methods 2006 (Keller, A., Heinrich, S. and Niederreiter, H., eds), Springer, pp. 645–658.CrossRefGoogle Scholar
Sinescu, V. and L.|Ecuyer, P. (2011), ‘Existence and construction of shifted lattice rules with an arbitrary number of points and bounded weighted star discrepancy for general decreasing weights’, J. Complexity 27, 449–465.CrossRefGoogle Scholar
Sloan, I. H. (2007), ‘Finite order integration weights can be dangerous’, Comput. Meth. Appl. Math. 7, 239–254.CrossRefGoogle Scholar
Sloan, I. H. and Joe, S. (1994), Lattice Methods for Multiple Integration, Oxford University Press.CrossRefGoogle Scholar
Sloan, I. H. and ReztsovA, V. A, V. (2002), ‘Component-by-component construction of good lattice rules’, Math. Comp. 71, 263–273.CrossRefGoogle Scholar
Sloan, I. H. and Woźniakowski, H. (1998), ‘When are quasi-Monte Carlo algorithms efficient for high-dimensional integrals?’, J. Complexity 14, 1–33.CrossRefGoogle Scholar
Sloan, I. H. and Woźniakowski, H. (2001), ‘Tractability of multivariate integration for weighted Korobov classes’, J. Complexity 17, 697–721.CrossRefGoogle Scholar
Sloan, I. H. and Woźniakowski, H. (2002), ‘Tractability of integration in non-periodic and periodic weighted tensor product Hilbert spaces’, J. Complexity 18, 479–499.CrossRefGoogle Scholar
Sloan, I. H., Kuo, F. Y. and Joe, S. (2002 a) ‘On the step-by-step construction of quasi-Monte Carlo integration rules that achieve strong tractability error bounds in weighted Sobolev spaces’, Math. Comp. 71, 1609–1640.CrossRefGoogle Scholar
Sloan, I. H., Kuo, F. Y. and Joe, S. (2002 b) ‘Constructing randomly shifted lattice rules in weighted Sobolev spaces’, SIAM J. Numer. Anal. 40, 1650–1665.CrossRefGoogle Scholar
Sloan, I. H., Wang, X. and Woźniakowski, H. (2004), ‘Finite-order weights imply tractability of multivariate integration’, J. Complexity 20, 46–74.CrossRefGoogle Scholar
Sobol', I. M. (1967), ‘Distribution of points in a cube and approximate evaluation of integrals’ (in Russian), Ž. Vyčisl. Mat. i Mat. Fiz. 7, 784–802.Google Scholar
Sobol', I. M. (1969), Multidimensional Quadrature Formulas and Haar Functions (in Russian), Nauka.Google Scholar
Smolyak, S. (1963), ‘Quadrature and interpolation formulas for tensor products of certain classes of functions’, Soviet Math. Dokl. 4, 240–243.Google Scholar
Russian original in Dokl. Akad. Nauk SSSR 148 (1963), 1042–1045.Google Scholar
Stroud, A. H. (1971), Approximate Calculation ofMultiple Integrals, Prentice-Hall.Google Scholar
Teckentrup, A. L., Scheichl, R., Giles, M. B. and Ullmann, E. (2012), Further analysis of multilevel Monte Carlo methods for elliptic PDEs with random coefficient. Technical Report, University of Bath.Google Scholar
Tezuka, S. (2013), ‘On the discrepancy of generalized Niederreiter sequences’, J. Complexity, toappear.CrossRefGoogle Scholar
Tezuka, S. and Faure, H. (2003), ‘I-binomial scrambling of digital nets and sequences’, J. Complexity 19, 744–757.CrossRefGoogle Scholar
Thomas-Agnan, C. (1996), ‘Computing a family of reproducing kernels for statistical applications’, Numer. Algorithms 13, 21–32.CrossRefGoogle Scholar
Traub, J. F., Wasilkowski, G. W. and Woźniakowski, H. (2008), Information-Based Complexity, Academic Press.Google Scholar
van der Vart, A. W. and Wellner, J. A. (2009), Weak Convergence and Empirical Processes, Springer Series in Statistics, Springer.Google Scholar
Wahba, G. (1990), Spline Models for Observational Data, Vol.59 of CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM.Google Scholar
Walsh, J. L. (1923), ‘A closed set of normal orthogonal functions’, Amer. J. Math. 45, 5–24.CrossRefGoogle Scholar
Wang, X. (2002), ‘A constructive approach to strong tractability using quasi-Monte Carlo algorithms’, J. Complexity 18, 683–701.CrossRefGoogle Scholar
Wang, X. (2003), ‘strong tractability of multivariate integration using quasi-Monte Carlo algorithms’, Math. Comp. 72, 823–838.CrossRefGoogle Scholar
Wang, X. and Fang, K.-T. (2003), ‘Effective dimension and quasi-Monte Carlo integration’, J. Complexity 19, 101–124.CrossRefGoogle Scholar
Wang, X. and Sloan, I. H. (2005), ‘Why are high-dimensional finance problems often of low effective dimension?’, SIAM J. Sci. Comput. 27, 159–183.CrossRefGoogle Scholar
Wang, X. and Sloan, I. H. (2006), ‘Efficient weighted lattice rules with applications to finance’, SIAM J. Sci. Comput. 28, 728–750.CrossRefGoogle Scholar
Wang, X. and Sloan, I. H. (2007), ‘Brownian bridge and principal component analysis: Towards removing the curse of dimensionality’, IMA J. Numer. Anal. 27, 631–654.CrossRefGoogle Scholar
Wang, X. and Sloan, I. H. (2011), ‘Quasi-Monte Carlo methods in financial engineering: An equivalence principle and dimension reduction’, Oper. Res. 59, 80–95.CrossRefGoogle Scholar
Wasilkowski, G. W. and Woźniakowski, H. (1995), ‘Explicit cost bounds of algorithms for multivariate tensor product problems’, J. Complexity 11, 1–56.CrossRefGoogle Scholar
Wasilkowski, G. W. and Woźniakowski, H. (1996), ‘On tractability of path integration’, J. Math.Phys. 37, 2071–2088.CrossRefGoogle Scholar
Wasilkowski, G. W. and Woźniakowski, H. (1999), ‘Weighted tensor product algorithms for linear multivariate problems’, J. Complexity 15, 402–447.CrossRefGoogle Scholar
Wasilkowski, G. W. and Woźniakowski, H. (2004), ‘Finite-order weights imply tractability of linear multivariate problems’, J. Approx. Theory 130, 57–77.CrossRefGoogle Scholar
Wasilkowski, G. W. and Woźniakowski, H. (2010), ‘On the exponent of discrepancy’, Math. Comp. 79, 983–992.CrossRefGoogle Scholar
Werschulz, A. and Woźniakowski, H. (2009), ‘Tractability of multivariate approximation over a weighted unanchored Sobolev space’, Constr. Approx. 30, 395–421.CrossRefGoogle Scholar
Weyl, H. (1916), ‘Über die Gleichverteilung von Zahlen mod. Eins’, Math. Ann. 77, 313–352.CrossRefGoogle Scholar
Woźniakowski, H. (2013), Monte Carlo integration. In Encyclopedia of Numerical Analysis.Google Scholar
Xing, C. P. and Niederreiter, H. (1995), ‘A construction of low-discrepancy sequences using global function fields’, Acta Arith. 73, 87–102.CrossRefGoogle Scholar
Yue, R. X. and Hickernell, F. J. (2001), ‘Integration and approximation based on scramble sampling in arbitrary dimensions’, J. Complexity 17, 881–897.CrossRefGoogle Scholar
Yue, R. X. and Hickernell, F. J. (2002), ‘The discrepancy and gain coefficients of scrambled digital nets’, J. Complexity 18, 135–151.CrossRefGoogle Scholar
Yue, R. X. and Hickernell, F. J. (2005), ‘strong tractability of integration using scrambled Niederreiter points‘, Math. Comp. 74, 1871–1893.CrossRefGoogle Scholar
Yue, R. X. and Hickernell, F. J. (2006), ‘strong tractability of quasi-Monte Carlo quadrature using nets for certain Banach spaces’, SIAM J. Numer. Anal. 44, 2559–2583.CrossRefGoogle Scholar
Zeng, X. Y., Leung, K. T., and Hickernell, F. J. (2006), Error analysis of splines for periodic problems using lattice designs. In Monte Carlo and Quasi-Monte Carlo Methods 2004 (Niederreiter, H. and Talay, D., eds), Springer, pp. 501–514.CrossRefGoogle Scholar
Zenger, C. (1991), Sparse grids. In Parallel Algorithms for Partial Differential Equations (Hackbusch, W., ed.), Vol. 31 of Notes on Numerical Fluid Mechanics, Vieweg.Google Scholar