Skip to main content
×
Home
    • Aa
    • Aa

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

  • Josef Dick (a1), Frances Y. Kuo (a2) and Ian H. Sloan (a3) (a4)
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.

Copyright
Footnotes
Hide All
*

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.

Footnotes
Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

N. Achtsis and D. Nuyens (2012), A component-by-component construction for the trigonometric degree. In Monte Carlo and Quasi-Monte Carlo Methods 2010 ( L. Plaskota and H. Woźniakowski , eds), Springer, pp. 235253.

P. Acworth , M. Broadie and P. Glasserman (1998), A comparison of some Monte Carlo and quasi-Monte Carlo techniques for option pricing. In Monte Carlo and Quasi-Monte Carlo Methods 1996 ( P Hellekalek , G Larcher , H Niederreiter and P Zinterhof , eds), Springer, pp. 118.

C. Aistleitner (2011), ‘Covering numbers, dyadic chaining and discrepancy’, J. Complexity 27, 531540.

C. Aistleitner and M. Hofer (2012), ‘Probabilistic error bounds for the discrepancy of mixed sequences’, Monte Carlo Methods Appl. 18, 181200.

N. Aronszajn (1950), ‘Theory of reproducing kernels’, Trans. Amer. Math. Soc. 68, 337404.

J. Baldeaux (2012), Scrambled polynomial lattice rules for infinite-dimensional integration. In Monte Carlo and Quasi-Monte Carlo Methods 2010 ( L. Plaskota and H. Woźniakowski , eds), Springer, pp. 255263.

J. Baldeaux and J. Dick (2009), ‘QMC rules of arbitrary high order: Reproducing kernel Hilbert space approach’, Constr. Approx. 30, 495527.

J. Baldeaux and J. Dick (2011), ‘A construction of polynomial lattice rules with small gain coefficients’, Numer. Math. 119, 271297.

J. Baldeaux , J. Dick , J. Greslehner and F. Pillichshammer (2011), ‘Construction algorithms for higher order polynomial lattice rules’, J. Complexity 27, 281299.

J. Baldeaux , J. Dick , G Leobacher , D. Nuyens and F. Pillichshammer (2012), ‘Efficient calculation of the worst-case error and (fast) component-by-component construction of higher order polynomial lattice rules’, Numer. Algorithms 59, 403431.

A. Barth , C. Schwab and N. Zollinger (2011), ‘Multi-level Monte Carlo finite element method for elliptic PDEs with stochastic coefficients’, Numer. Math. 119, 123161.

P. Bratley and B. L. Fox (1988), ‘Algorithm 659: Implementing Sobol's quasirandom sequence generator’, ACM Trans. Math. Softw. 14, 88100.

P Bratley , B. L. Fox , and H. Niederreiter (1992), ‘Implementation and tests of low-discrepancy sequences’, ACM Trans. Model. Comput. Simul 2, 195213.

H. Bungartz and M. Griebel (2004), Sparse grids. In Acta Numerica, Vol. 13, Cambridge University Press, pp. 147269.

R. E. Caflisch , W. Morokoff and A. Owen (1997), ‘Valuation of mortgage backed securities using Brownian bridges to reduce effective dimension’, J. Comput. Finance 1, 2746.

B. Chazelle (2000), The Discrepancy Method: Randomness and Complexity, Cambridge University Press.

H E. Chrestenson (1955), ‘A class of generalized Walsh functions’, Pacific J. Math. 5, 1731.

K. A. Cliffe , M. B. Giles , R. Scheichl and A. L. Teckentrup (2011), ‘Multilevel Monte Carlo methods and applications to elliptic PDEs with random coefficients’, Comput. Vis. Sci. 14, 315.

R. Cools (1997), Constructing cubature formulae: The science behind the art. In Acta Numerica, Vol. 6, Cambridge University Press, pp. 154.

R. Cools and J. N. Lyness (2001), ‘Three- and four-dimensional if-optimal lattice rules of moderate trigonometric degree’, Math. Comp. 70, 15491567.

R. Cools and D. Nuyens (2008), A Belgian view on lattice rules. In Monte Carlo and Quasi-Monte Carlo Methods 2006 ( A. Keller , S. Heinrich and H. Niederreiter , eds), Springer, pp. 321.

R. Cools , F. Y. Kuo and D. Nuyens (2006), ‘Constructing embedded lattice rules for multivariate integration’, SIAM J. Sci. Comput. 28, 21622188.

R. Cools , F. Y. Kuo and D. Nuyens (2010), ‘Constructing lattice rules based on weighted degree of exactness and worst case error’, Computing 87, 6389.

J. Creutzig , S. Dereich , T. Müller-Gronbach and K. Ritter (2009), ‘Infinite- dimensional quadrature and approximation of functions’, Found. Comp. Math. 9, 391429.

P. J. Davis and P. Rabinowitz (1984), Methods of Numerical Integration, second edition, Academic.

L. Devroye (1986), Nonuniform Random Variate Generation, Springer.

J. Dick (2004), ‘On the convergence rate of the component-by-component construction of good lattice rules’, J. Complexity 20, 493522.

J. Dick (2007 a), ‘Explicit constructions of quasi-Monte Carlo rules for the numerical integration of high-dimensional periodic functions’, SIAM J. Numer. Anal. 45, 21412176.

J. Dick (2007 b), ‘The construction of extensible polynomial lattice rules with small weighted star discrepancy’, Math. Comp. 76, 20772085.

J. Dick (2008), ‘Walsh spaces containing smooth functions and quasi-Monte Carlo rules of arbitrary high order’, SIAM J. Numer. Anal. 46, 15191553.

J. Dick (2009 a), ‘The decay of the Walsh coefficients of smooth functions’, Bull. Aust. Math. Soc. 80, 430453.

J. Dick (2009 b), On Quasi-Monte Carlo rules achieving higher order convergence. In Monte Carlo and Quasi-Monte Carlo Methods 2008 ( P. L'Ecuyer and A. B. Owen , eds), Springer, pp. 7396.

J. Dick (2011 a), ‘Higher order scrambled digital nets achieve the optimal rate of the root mean square error for smooth integrands’, Ann. Statist. 39, 13721398.

J. Dick (2011 b), ‘Quasi-Monte Carlo numerical integration on ℝs: Digital nets and worst-case error’, SIAM J. Numer. Anal. 49, 16611691.

J. Dick (2012), ‘Random weights, robust lattice rules and the geometry of the cbcrc algorithm’, Numer. Math. 122, 443467.

J. Dick and J. Baldeaux (2009), Equidistribution properties of generalized nets and sequences. In Monte Carlo and Quasi-Monte Carlo Methods 2008 ( P. L'Ecuyer and A. B. Owen , eds), Springer, pp. 305322.

J. Dick and P. Kritzer (2010), ‘Duality theory and propagation rules for generalized digital nets’, Math. Comp. 79, 9931017.

J. Dick and F. Y. Kuo (2004 a), ‘Reducing the construction cost of the component-by-component construction of good lattice rules’, Math. Comp. 73, 19671988.

J. Dick and F. Y. Kuo (2004 b), Constructing good lattice rules with millions of points. In Monte Carlo and Quasi-Monte Carlo Methods 2002 ( H. Niederreiter , ed.), Springer, pp. 181197.

J. Dick and F. Pillichshammer (2005), ‘Multivariate integration in weighted Hilbert spaces based on Walsh functions and weighted Sobolev spaces’, J. Complexity 21, 149195.

J. Dick and F. Pillichshammer (2007), ‘strong tractability of multivariate integration of arbitrary high order using digitally shifted polynomial lattice rules’, J. Complexity 23, 436453.

J. Dick and F. Pillichshammer (2010), Digital Nets and Sequences, Cambridge University Press.

J. Dick , P. Kritzer , G. Leobacher and F. Pillichshammer (2007 a), ‘Constructions of general polynomial lattice rules based on the weighted star discrepancy’, Finite Fields Appl. 13, 10451070.

J. Dick , P. Kritzer , F. Pillichshammer and W. C. Schmid (2007 b), ‘On the existence of higher order polynomial lattices based on a generalized figure of merit’, J. Complexity 23, 581593.

J. Dick , F. Y. Kuo , F. Pillichshammer and I. H. Sloan (2005), ‘Construction algorithms for polynomial lattice rules for multivariate integration’, Math. Comp. 74, 18951921.

J. Dick , G. Larcher , F. Pillichshammer and H. Woźniakowski (2011), ‘Exponential convergence and tractability of multivariate integration for Korobov spaces’, Math. Comp. 80, 905930.

J. Dick , F. Pillichshammer , and B. J. Waterhouse (2008), ‘The construction of good extensible rank-1 lattices’, Math. Comp. 77, 23452374.

J. Dick , I. H. Sloan , X. Wang and H. Woźniakowski (2006), ‘Good lattice rules in weighted Korobov spaces with general weights’, Numer. Math. 103, 6397.

B. Doerr , M. Gnewuch , P. Kritzer and F. Pillichshammer (2008), ‘Component-by-component construction of low-discrepancy point sets of small size’, Monte Carlo Methods Appl. 14, 129149.

B. Doerr , M. Gnewuch and M. Wahlström (2009), Implementation of a component-by-component algorithm to generate small low-discrepancy samples. In Monte Carlo and Quasi-Monte Carlo Methods 2008 ( P. L'Ecuyerand A. B. Owen , eds), Springer, pp. 323338.

B. Doerr , M. Gnewuch and M. Wahlstrom (2010), ‘Algorithmic construction of low-discrepancy point sets via dependent randomized rounding’, J. Complexity 26, 490507.

K. T. Fang and Y. Wang (1994), Number-Theoretic Methods in Statistics, Chapman & Hall.

N. J. Fine (1949), ‘On the Walsh functions’, Trans. Amer. Math. Soc. 65, 372414.

B. L. Fox (1986), ‘Algorithm 647: Implementation and relative efficiency of quasir-andom sequence generators, ACM Trans. Math. Softw. 12, 362376.

R. G. Ghanem and P. D. Spanos (1991), Stochastic Finite Elements: A Spectral Approach, Springer.

M. B. Giles (2008), ‘Multilevel Monte Carlo path simulation’, Oper. Res. 56, 607617.

M. Gnewuch (2008), ‘Bracketing numbers for axis-parallel boxes and applications to geometric discrepancy, J. Complexity 24, 154172.

M. Gnewuch (2012 a), ‘Weighted geometric discrepancies and numerical integration on reproducing kernel Hilbert spaces’, J. Complexity 28, 217.

M. Gnewuch (2013), Lower error bounds for randomized multilevel and changing dimension algorithms. In Monte Carlo and Quasi-Monte Carlo Methods 2012 ( J. Dick , F. Y. Kuo , G. W. Peters and I. H. Sloan , eds), Springer, to appear.

M. Gnewuch , A. Srivastav and C. Winzen (2009), ‘Finding optimal volume subin-tervals with/k-points and calculating the star discrepancy are NP-hard problems’, J. Complexity 25, 115127.

M. Gnewuch , M. Wahlstriom and C. Winzen (2012), ‘A new randomized algorithm to approximate the star discrepancy based on threshold accepting, SIAM J. Numer. Anal. 50, 781807.

I. G. Graham , F. Y. Kuo , D. Nuyens , R. Scheichl and I. H. Sloan (2011), ‘Quasi-Monte Carlo methods for elliptic PDEs with random coefficients and applications, J. Comput. Phys. 230, 36683694.

M. Griebel (2006), Sparse grids and related approximation schemes for higher dimensional problems. In Foundations of Computational Mathematics, Santander, 2005, Cambridge University Press, pp. 106161.

M. Griebel , F. Y. Kuo and I. H. Sloan (2013), ‘The smoothing effect of integration in ℝd and the ANOVA decomposition’, Math. Comp. 82, 383400.

J. H. Halton (1960), ‘On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals, Numer. Math. 2, 8490.

J. M. Hammersley and D. C. Handscomb (1964), Monte Carlo Methods, Methuen.

T. Hansen , G. L. Mullen and H. Niederreiter (1993), ‘Good parameters for a class of node sets in quasi-Monte Carlo integration, Math. Comp. 61, 225234.

S. Heinrich (1998), ‘Monte Carlo complexity of global solution of integral equations, J. Complexity 14, 151175.

S. Heinrich and E. Sindambiwe (1999), ‘Monte Carlo complexity of parametric integration, J. Complexity 15, 317341.

S. Heinrich , F. Hickernell and R. X. Yue (2004), ‘Optimal quadrature for Haar wavelet spaces’, Math. Comp. 73, 259277.

S. Heinrich , E. Novak , G. W. Wasilkowski , H. Woźniakowski (2001), ‘The inverse of the star-discrepancy depends linearly on the dimension’, Acta Arith. 96, 279302.

F. J. Hickernell (1996 a), ‘Quadrature error bounds with applications to lattice rules’, SIAM J. Numer. Anal. 33, 19952016.

Erratum: ‘Quadrature error bounds with applications to lattice rules’, SIAM J. Numer. Anal. 34, 853866 (1997).

F. J. Hickernell (1996 b), ‘The mean square discrepancy of randomized nets’, ACM Trans. Modeling Comput. Simul. 6, 274296.

F. J. Hickernell (1998 a), ‘A generalized discrepancy and quadrature error bound’, Math. Comp. 67, 299322.

F. J. Hickernell (1998 b), Lattice rules: How well do they measure up? In Random and Quasi-Random Point Sets ( P. Hellekalek and G. Larcher , eds), Springer, pp. 109166.

F. J. Hickernell (1999), ‘Goodness-of-fit statistics, discrepancies and robust designs’, Statist. Probab. Lett. 44, 7378.

F. J. Hickernell (2002), Obtaining O(N−2+ε) convergence for lattice quadrature rules. In Monte Carlo and Quasi-Monte Carlo Methods 2000 ( K.T. Fang , F. J. Hickernell and H. Niederreiter , eds), Springer, pp. 274289.

F. J. Hickernell and H. Niederreiter (2003), ‘The existence of good extensible rank-1 lattices’, J. Complexity 19, 286300.

F. J. Hickernell and X. Wang (2002), ‘The error bounds and tractability of quasi-Monte Carlo algorithms in infinite dimension’, Math. Comp. 71, 16411661.

F. J. Hickernell and H. Woźniakowski (2000), ‘Integration and approximation in arbitrary dimensions’, Adv. Comput. Math. 12, 2558.

F. J. Hickernell and H. Woźniakowski (2001), ‘Tractability of multivariate integration for periodic functions’, J. Complexity 17, 660682.

F. J. Hickernell and R. X. Yue (2000), ‘The mean square discrepancy of scrambled (t, s)-sequences’, SIAM J. Numer. Anal. 38, 10891112.

F. J. Hickernell , H. S. Hong , P. L. Ecuyer , and C. Lemieux (2000), ‘Extensible lattice sequences for quasi-Monte Carlo quadrature’, SIAM J. Sci. Comput. 22, 11171138.

F. J. Hickernell , P. Kritzer , F. Y. Kuo and D. Nuyens (2012), ‘Weighted compound integration rules with higher order convergence for all N’, Numer. Algorithms 59, 161183.

F. J. Hickernell , C. Lemieux and A. B. Owen (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, 131.

F. Hickernell , T. Muiller-Gronbach , B. Niu and K. Ritter (2010), ‘Monte Carlo algorithms for infinite-dimensional integration on ℝN’, J. Complexity 26, 229254.

A. Hinrichs (2004), ‘Covering numbers, Vapnik-Červonenkis classes and bounds for the star-discrepancy’, J. Complexity 20, 477483.

A. Hinrichs , F. Pillichshammer and W. C. Schmid (2008), ‘Tractability properties of the weighted star discrepancy’, J. Complexity 24, 134143.

E. Hlawka (1961), ‘Funktionen von beschränkter Variation in der Theorie der Gleichverteilung’, Ann. Mat. Pura Appl. 54, 325333.

E. Hlawka (1962), ‘Zur angeniaherten Berechnung mehrfacher Integrale’, Monatsh. Math. 66, 140151.

R. Hofer and P. Kritzer (2011), ‘On hybrid sequences built from Niederreiter-Halton sequences and Kronecker sequences’, Bull. Aust. Math. Soc. 84, 238254.

R. Hofer and G. Larcher (2010), ‘On existence and discrepancy of certain digital Niederreiter-Halton sequences’, Acta Arith. 141, 369394.

R. Hofer and G. Larcher (2012), ‘Metrical results on the discrepancy of Halton-Kronecker sequences’, Math. Z. 271, 111.

R. Hofer and H. Niederreiter (2013), ‘A construction of (t, s)-sequences with finite-row generating matrices using global function fields’, Finite Fields Appl. 21, 97110.

R. Hofer , P. Kritzer , G. Larcher and F. Pillichshammer (2009), ‘Distribution properties of generalized van der Corput-Halton sequences and their subsequences’, Int. J. Number Theory 5, 719746.

S. Joe (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 ( H. Niederreiter , ed.), Springer, pp. 293298.

S. Joe (2006), Construction of good rank-1 lattice rules based on the weighted star discrepancy. In Monte Carlo and Quasi-Monte Carlo Methods 2004 ( H. Niederreiter and D. Talay , eds), Springer, pp. 181196.

S. Joe and F. Y. Kuo (2008), ‘Constructing Sobol' sequences with better two-dimensional projection’, SIAM J. Sci. Comput. 30, 26352654.

L. Kämmerer , S. Kunis and D. Potts (2012), ‘Interpolation lattices for hyperbolic cross trigonometric polynomials’, J. Complexity 28, 7692.

A. Keller (2006), Myths of computer graphics. In Monte Carlo and Quasi-Monte Carlo Methods 2004 ( H. Niederreiter and D. Talay , eds), Springer, pp. 217243.

A. Keller (2013), Quasi-Monte Carlo image synthesis in a nutshell. In Monte Carlo and Quasi-Monte Carlo Methods 2012 ( J. Dick , F. Y. Kuo , G. W. Peters and I. H. Sloan , eds), Springer, to appear.

P. Kritzer and F. Pillichshammer (2007), ‘Constructions of general polynomial lattices for multivariate integration’, Bull. Austral. Math. Soc. 76, 93110.

F. Y. Kuo (2003), ‘Component-by-component constructions achieve the optimal rate of convergence for multivariate integration in weighted Korobov and Sobolev spaces’, J. Complexity 19, 301320.

F. Y. Kuo and S. Joe (2002), ‘Component-by-component construction of good lattice rules with a composite number of points’, J. Complexity 18, 943976.

F. Y. Kuo and S. Joe (2003), ‘Component-by-component construction of good intermediate-rank lattice rules’, SIAM J. Numer. Anal. 41, 14651486.

F. Y. Kuo , W. T. M . Dunsmuir, I. H. Sloan , M. P. Wand , R. S. Womersley (2008 a), ‘Quasi-Monte Carlo for highly structured generalised response models’, Method. Comput. Appl. Probab. 10, 239275.

K. Y. Kuo , C. Schwab and I. H. Sloan (2011), ‘Quasi-Monte Carlo methods for high-dimensional integration: The standard (weighted Hilbert space) setting and beyond’, ANZIAM J. 53, 137.

F. Y. Kuo , C. Schwab and I. H. Sloan (2012), Quasi-Monte Carlo finite element methods for a class of elliptic partial differential equations with random coefficients, SIAM J. Numer. Anal. 50, 33513374.

F. Y. Kuo , I. H. Sloan , G. W. Wasilkowski , and B. J. Waterhouse (2010 a), Randomly shifted lattice rules with the optimal rate of convergence for unbounded integrands’, J. Complexity 26, 135160.

F. Y. Kuo , I. H. Sloan , G. W. Wasilkowski and H. Woźniakowski (2010 b), ‘On decompositions of multivariate functions’, Math. Comp. 79, 953966.

F. Y. Kuo , I. H. Sloan , G. W. Wasilkowski and H. Woźniakowski (2010 c), ‘Liberating the dimension’, J. Complexity 26, 422454.

F. Y. Kuo , I. H. Sloan , and H. Woźniakowski (2006 a), Lattice rules for multivariate approximation in the worst case setting. In Monte Carlo and Quasi-Monte Carlo Methods 2004 ( H. Niederreiter and D. Talay , eds), Springer, pp. 289330.

F. Y. Kuo , I. H. Sloan and H. Woźniakowski (2007), ‘Periodization strategy may fail in high dimensions’, Numer. Algorithms 46, 369391.

F. Y. Kuo , I. H. Sloan , and H. Woźniakowski (2008 b), ‘Lattice rule algorithms for multivariate approximation in the average case setting’, J. Complexity 24, 283323.

F. Y. Kuo , G. W. Wasilkowski , and B. J. Waterhouse (2006 b), ‘Randomly shifted lattice rules for unbounded integrals’, J. Complexity 22, 630651.

F. Y. Kuo , G. W. Wasilkowski , and H. Woźniakowski (2009), ‘Lattice algorithms for multivariate ∞ approximation in the worst case setting’, Constr. Approx. 30, 475493.

G. Larcher and C. Traunfellner (1994), ‘On the numerical integration of Walsh series by number-theoretic methods’, Math. Comp. 63, 277291.

G. Larcher , A. Lauß , H. Niederreiter and W. C. Schmid (1996 a), ‘Optimal polynomials for (t, m, s)-nets and numerical integration of multivariate Walsh series’, SIAM J. Numer. Anal. 33, 22392253.

G. Larcher , W. C. Schmid and R. Wolf (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, 701716.

G. Larcher , W. C. Schmid and R. Wolf (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, 5567.

D. Laurie (1996), ‘Periodizing transformations for numerical integration’, J. Comput. Appl. Math. 66, 337344.

P. L' Ecuyer and C. Lemieux (2000), ‘Variance reduction via lattice rules’, Management Sci. 46, 12141235.

P. L'Ecuyer and D. Munger (2012), On figures of merit for randomly shifted lattice rules. In Monte Carlo and Quasi-Monte Carlo Methods 2010 ( L. Plaskota and H. Woźniakowski , eds), Springer, pp. 133159.

P. L'Ecuyer , D. Munger and B. Tuffin (2010), ‘On the distribution of integration error by randomly-shifted lattice rules’. Electron. J. Stat. 4, 950993.

C. Lemieux (2009), Monte Carlo and Quasi-Monte Carlo Sampling, Springer.

C. Lemieux and P. L' Ecuyer (2001), ‘On selection criteria for lattice rules and other quasi-Monte Carlo point sets’, Math. Comput. Simulation 55, 139148.

D. Li and F. J. Hickernell (2003), Trigonometric spectral collocation methods on lattices. In Recent Advances in Scientific Computing and Partial Differential Equations ( S. Y. Cheng , C.-W. Shu and T. Tang , eds), Vol. 330 of AMS Series in Contemporary Mathematics, AMS, pp. 121132.

G. Li , J. Schoendorf , T.-S. Ho and H. Rabitz (2004), ‘Multicut-HDMR with an application to an ionospheric model’, J. Comput. Chem. 25, 11491156.

W. L. Loh (2003), ‘On the asymptotic distribution of scrambled net quadrature’, Ann. Statist. 31, 12821324.

J. Lyness and T. Sørevik (2006), ‘Five-dimensional/k-optimal lattice rules’, Math. Comp. 75, 14671480.

J. Matousěk (1998 a), ‘The exponent of discrepancy is at least 1.0669’, J. Complexity 14, 448453.

J. Matousěk (1998 b), ‘On the L2 discrepancy for anchored boxes’, J. Complexity 14, 527556.

J. Matousěk (1999), Geometric Discrepancy: An Illustrated Guide, Vol. 18 of Algorithms and Combinatorics, Springer.

M. Matsumoto and T. Yoshiki (2013), Existence of higher order convergent quasi-Monte Carlo rules via Walsh figure of merit. In Monte Carlo and Quasi-Monte Carlo Methods 2012 ( J. Dick , F. Y. Kuo , G. W. Peters and I. H. Sloan , eds), Springer, to appear.

M. Matsumoto , M. Saito , K. Matoba (2013), ‘A computable figure of merit for quasi-Monte Carlo point sets’, Math. Comp., toappear.

H. Niederreiter (1978), ‘Quasi-Monte Carlo methods and pseudo-random numbers’, Bull. Amer. Math. Soc. 84, 9571041.

H. Niederreiter (1987), ‘Point sets and sequences with small discrepancy’, Monatsh. Math. 104, 273337.

H. Niederreiter (1988), ‘Low-discrepancy and low-dispersion sequences’, J. Number Theory 30, 5170.

H. Niederreiter (2003), ‘The existence of good extensible polynomial lattice rules’, Monatsh. Math. 139, 295307.

H. Niederreiter (2004), Digital nets and coding theory. In Coding, Cryptography and Combinatorics ( K. Q. Feng , H. Niederreiter and C. P. Xing , eds), Birkhauser, pp. 247257.

H. Niederreiter (2009), ‘On the discrepancy of some hybrid sequences’, Acta Arith. 138, 373398.

H. Niederreiter (2010 b), ‘Further discrepancy bounds and an Erdõs-Turán-Koksma inequality for hybrid sequences’, Monatsh. Math. 161, 193222.

H. Niederreiter (2012), ‘Improved discrepancy bounds for hybrid sequences involving Halton sequences’, Acta Arith. 155, 7184.

H. Niederreiter and F. Pillichshammer (2009), ‘Construction algorithms for good extensible lattice rules’, Constr. Approx. 30, 361393.

H. Niederreiter and C. P. Xing (1996 a), ‘Low-discrepancy sequences and global function fields with many rational places’, Finite Fields Appl. 2, 241273.

B. Niu and F. J. Hickernell (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 ( P. L'Ecuyer and A. B. Owen , eds), Springer, pp. 545560.

B. Niu , F. J. Hickernell , T. Müller-Gronbach and K. Ritter (2011), ‘Deterministic multi-level algorithms for infinite-dimensional integration on ℝ’, J. Complexity 27, 331351.

E. Novak and H. Woázniakowski (2001), ‘Intractability results for integration and discrepancy’, J Complexity, 17, 388441.

E. Novak and H. Woázniakowski (2008), Tractability of Multivariate Problems, Vol. I: Linear Information, EMS.

E. Novak and H. Woázniakowski (2010), Tractability of Multivariate Problems, Vol. II: Standard Information for Functionals, EMS.

E. Novak and H. Woźniakowski (2012), Tractability of Multivariate Problems, Vol. III: Standard Information for Operators, EMS.

D. Nuyens and R. Cools (2006 a), ‘Fast algorithms for component-by-component construction of rank-1 lattice rules in shift-invariant reproducing kernel Hilbert spaces’, Math. Comp. 75, 903920.

D. Nuyens and R. Cools (2006 b), ‘Fast component-by-component construction of rank-1 lattice rules with a non-prime number of points’, J. Complexity 22, 428.

D. Nuyens and R. Cools (2006 c), Fast component-by-component construction, a reprise for different kernels. In Monte Carlo and Quasi-Monte Carlo Methods 2004 ( H. Niederreiter and D. Talay , eds), Springer, pp. 373387.

G. Okten (1996), ‘A probabilistic result on the discrepancy of a hybrid Monte Carlo sequence and applications’, Monte Carlo Methods Appl. 2, 255270.

G. Okten , B. Tuffin and V. Burago (2006), ‘A central limit theorem and improved error bounds for a hybrid-Monte Carlo sequence with applications in computational finance’, J. Complexity 22, 435458.

A. B. Owen (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. 299317.

A. B. Owen (1997 b), ‘Monte Carlo variance of scrambled net quadrature’, SIAM J. Numer. Anal. 34, 18841910.

A. B. Owen (1998), ‘scrambling Sobol' and Niederreiter-Xing points’, J. Complexity 14, 466489.

A. B. Owen (2006), Quasi-Monte Carlo for integrands with point singularities at unknown locations. In Monte Carlo and Quasi-Monte Carlo Methods 2004 ( H. Niederreiter and D. Talay , eds), Springer, pp. 403417.

S. H. Paskov and J. Traub (1995), ‘Faster evaluation of financial derivatives’, J. Portfolio Management 22, 113120.

F. Pillichshammer (2002), ‘Bounds for the quality parameter of digital shift nets over Z2’, Finite Fields Appl. 8, 444454.

F. Pillichshammer and G. Pirsic (2009), Discrepancy of hyperplane nets and cyclic nets. In Monte Carlo and Quasi-Monte Carlo Methods 2008 ( P. L' Ecuyer and A. B. Owen , eds), Springer, pp. 573587.

G. Pirsic (2002), A software implementation of Niederreiter-Xing sequences. In Monte Carlo and Quasi-Monte Carlo Methods 2000 ( K. T. Fang , F. J. Hickernell and H. Niederreiter , eds), Springer, pp. 434445.

G. Pirsic and F. Pillichshammer (2011), ‘Extensible hyperplane nets’, Finite Fields Appl. 17, 407423.

G. Pirsic and W. C. Schmid (2001), ‘Calculation of the quality parameter of digital nets and application to their construction’, J. Complexity 17, 827839.

G. Pirsic , J. Dick and F. Pillichshammer (2006), ‘Cyclic digital nets, hyperplane nets, and multivariate integration in Sobolev spaces’, SIAM J. Numer. Anal. 44, 385411.

L. Plaskota and G. Wasilkowski (2011), ‘Tractability of infinite-dimensional integration in the worst case and randomized setting’, J. Complexity 27, 505518.

L. Plaskota , G. Wasilkowski and H. Woázniakowski (2000), ‘A new algorithm and worst case complexity for Feynman-Kac path integration’, J. Comput. Phys. 164, 335353.

C.-H. Rhee and P. W. Glynn (2012), A new approach to unbiased estimation for SDE's. In Proc. 2012 Winter Simulation Conference ( C. Laroque , J. Him-melspach , R. Pasupathy , O. Rose and A. M. Uhrmacher , eds).

W. C. Schmid (1996), Shift-nets: A new class of binary digital (t, m, s)-nets. In Monte Carlo and Quasi-Monte Carlo Methods 1996 ( H. Niederreiter , P. Hellekalek , G. Larcher and P. Zinterhof , eds), Vol. 127 of Lecture Notes in Statistics, Springer, pp. 369381.

W. C. Schmid (2000), Improvements and extensions of the ‘salzburg Tables’ by using irreducible polynomials. In Monte Carlo and Quasi-Monte Carlo Methods 1998 ( H. Niederreiter and J. Spanier , eds), Springer, pp. 436447.

C. Schwab and C. J. Gittelson (2011), Sparse tensor discretizations of high-dimensional parametric and stochastic PDEs. In Acta Numerica, Vol. 20, Cambridge University Press, pp. 291467.

A. Sidi (1993), A new variable transformation for numerical integration. In Numerical Integration IV: Oberwolfach, 1992 ( H. Brass and G. Hämmerlin , eds), Birkhäuser, pp. 359373.

V. Sinescu and S. Joe (2007), ‘Good lattice rules based on the general weighted star discrepancy’, Math. Comp. 76, 9891004.

V. Sinescu and S. Joe (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 ( A. Keller , S. Heinrich and H. Niederreiter , eds), Springer, pp. 645658.

V. Sinescu and P. L.|Ecuyer (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, 449465.

I. H. Sloan (2007), ‘Finite order integration weights can be dangerous’, Comput. Meth. Appl. Math. 7, 239254.

I. H. Sloan and A, V. Reztsov (2002), ‘Component-by-component construction of good lattice rules’, Math. Comp. 71, 263273.

I. H. Sloan and H. Woźniakowski (1998), ‘When are quasi-Monte Carlo algorithms efficient for high-dimensional integrals?’, J. Complexity 14, 133.

I. H. Sloan and H. Woźniakowski (2001), ‘Tractability of multivariate integration for weighted Korobov classes’, J. Complexity 17, 697721.

I. H. Sloan and H. Woźniakowski (2002), ‘Tractability of integration in non-periodic and periodic weighted tensor product Hilbert spaces’, J. Complexity 18, 479499.

I. H. Sloan , F. Y. Kuo and S. Joe (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, 16091640.

I. H. Sloan , F. Y. Kuo and S. Joe (2002 b) ‘Constructing randomly shifted lattice rules in weighted Sobolev spaces’, SIAM J. Numer. Anal. 40, 16501665.

I. H. Sloan , X. Wang and H. Woźniakowski (2004), ‘Finite-order weights imply tractability of multivariate integration’, J. Complexity 20, 4674.

S. Tezuka (2013), ‘On the discrepancy of generalized Niederreiter sequences’, J. Complexity, toappear.

S. Tezuka and H. Faure (2003), ‘I-binomial scrambling of digital nets and sequences’, J. Complexity 19, 744757.

C. Thomas-Agnan (1996), ‘Computing a family of reproducing kernels for statistical applications’, Numer. Algorithms 13, 2132.

J. L. Walsh (1923), ‘A closed set of normal orthogonal functions’, Amer. J. Math. 45, 524.

X. Wang (2002), ‘A constructive approach to strong tractability using quasi-Monte Carlo algorithms’, J. Complexity 18, 683701.

X. Wang (2003), ‘strong tractability of multivariate integration using quasi-Monte Carlo algorithms’, Math. Comp. 72, 823838.

X. Wang and K.-T. Fang (2003), ‘Effective dimension and quasi-Monte Carlo integration’, J. Complexity 19, 101124.

X. Wang and I. H. Sloan (2005), ‘Why are high-dimensional finance problems often of low effective dimension?’, SIAM J. Sci. Comput. 27, 159183.

X. Wang and I. H. Sloan (2006), ‘Efficient weighted lattice rules with applications to finance’, SIAM J. Sci. Comput. 28, 728750.

X. Wang and I. H. Sloan (2007), ‘Brownian bridge and principal component analysis: Towards removing the curse of dimensionality’, IMA J. Numer. Anal. 27, 631654.

X. Wang and I. H. Sloan (2011), ‘Quasi-Monte Carlo methods in financial engineering: An equivalence principle and dimension reduction’, Oper. Res. 59, 8095.

G. W. Wasilkowski and H. Woźniakowski (1995), ‘Explicit cost bounds of algorithms for multivariate tensor product problems’, J. Complexity 11, 156.

G. W. Wasilkowski and H. Woźniakowski (1996), ‘On tractability of path integration’, J. Math.Phys. 37, 20712088.

G. W. Wasilkowski and H. Woźniakowski (1999), ‘Weighted tensor product algorithms for linear multivariate problems’, J. Complexity 15, 402447.

G. W. Wasilkowski and H. Woźniakowski (2004), ‘Finite-order weights imply tractability of linear multivariate problems’, J. Approx. Theory 130, 5777.

G. W. Wasilkowski and H. Woźniakowski (2010), ‘On the exponent of discrepancy’, Math. Comp. 79, 983992.

A. Werschulz and H. Woźniakowski (2009), ‘Tractability of multivariate approximation over a weighted unanchored Sobolev space’, Constr. Approx. 30, 395421.

H. Weyl (1916), ‘Über die Gleichverteilung von Zahlen mod. Eins’, Math. Ann. 77, 313352.

R. X. Yue and F. J. Hickernell (2001), ‘Integration and approximation based on scramble sampling in arbitrary dimensions’, J. Complexity 17, 881897.

R. X. Yue and F. J. Hickernell (2002), ‘The discrepancy and gain coefficients of scrambled digital nets’, J. Complexity 18, 135151.

R. X. Yue and F. J. Hickernell (2005), ‘strong tractability of integration using scrambled Niederreiter points‘, Math. Comp. 74, 18711893.

R. X. Yue and F. J. Hickernell (2006), ‘strong tractability of quasi-Monte Carlo quadrature using nets for certain Banach spaces’, SIAM J. Numer. Anal. 44, 25592583.

X. Y. Zeng , K. T. Leung , and F. J. Hickernell (2006), Error analysis of splines for periodic problems using lattice designs. In Monte Carlo and Quasi-Monte Carlo Methods 2004 ( H. Niederreiter and D. Talay , eds), Springer, pp. 501514.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Acta Numerica
  • ISSN: 0962-4929
  • EISSN: 1474-0508
  • URL: /core/journals/acta-numerica
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 205 *
Loading metrics...

Abstract views

Total abstract views: 625 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 26th September 2017. This data will be updated every 24 hours.