The construction of good extensible rank-1 lattices

Dick, Josef; Pillichshammer, Friedrich; Waterhouse, Benjamin J.

doi:10.1090/s0025-5718-08-02009-7

Cited by 46 publications

(53 citation statements)

References 33 publications

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“…The original algorithm requires a fixed value of N as input, and changing N means that the lattice rule has to be constructed anew. Modified algorithms for obtaining lattice rules that are extensible in N are given in [6,9].…”

Section: Qmc Fe Methods For Elliptic Pdes With Random Coefficientsmentioning

confidence: 99%

Quasi-Monte Carlo Finite Element Methods for a Class of Elliptic Partial Differential Equations with Random Coefficients

Kuo¹,

Schwab²,

Sloan³

2012

SIAM J. Numer. Anal.

227

368

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Section: Qmc Fe Methods For Elliptic Pdes With Random Coefficientsmentioning

confidence: 99%

Quasi-Monte Carlo Finite Element Methods for a Class of Elliptic Partial Differential Equations with Random Coefficients

Kuo¹,

Schwab²,

Sloan³

2012

SIAM J. Numer. Anal.

227

368

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“…There are many variants of this algorithm, including those which aim for good embedded or extensible lattice rules (e.g. Cools, Kuo & Nuyens, 2006;Dick, Pillichshammer & Waterhouse, 2006). With a clever implementation, these algorithms can produce, in a very short computational time, good lattice rules with thousands of dimensions and millions of points that achieve close to O (N −1 ) convergence.…”

Section: Lattice Rules Random Shifts and Weightsmentioning

confidence: 99%

Quasi-Monte Carlo for Highly Structured Generalised Response Models

Kuo

Dunsmuir

Sloan

et al. 2007

Methodol Comput Appl Probab

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Highly structured generalised response models, such as generalised linear mixed models and generalised linear models for time series regression, have become an indispensable vehicle for data analysis and inference in many areas of application. However, their use in practice is hindered by high-dimensional intractable integrals. Quasi-Monte Carlo (QMC) is a dynamic research area in the general problem of high-dimensional numerical integration, although its potential for statistical applications is yet to be fully explored. We survey recent research in QMC, particularly lattice rules, and report on its application to highly structured generalised response models. New challenges for QMC are identified and new methodologies are developed. QMC methods are seen to provide significant improvements compared with ordinary Monte Carlo methods. ABSTRACTHighly structured generalised response models, such as generalised linear mixed models and generalised linear models for time series regression, have become an indispensable vehicle for data analysis and inference in many areas of application. However, their use in practice is hindered by high-dimensional intractable integrals. Quasi-Monte Carlo (QMC) is a dynamic research area in the general problem of high-dimensional numerical integration, although its potential for statistical applications is yet to be fully explored. We survey recent research in QMC, particularly lattice rules, and report on its application to highly structured generalised response models. New challenges for QMC are identified and new methodologies are developed. QMC methods are seen to provide significant improvements compared with ordinary Monte Carlo methods.

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“…Previously, numerical integration of periodic functions has been analyzed for functions which are α times differentiable in each variable with α < ∞; see for example [6,7,12,15,18]. Our approach for infinitely times differentiable functions is similar to the approach in those papers.…”

Section: Introductionmentioning

confidence: 99%

Exponential convergence and tractability of multivariate integration for Korobov spaces

Dick¹,

Larcher²,

Pillichshammer³

et al. 2011

Math. Comp.

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Abstract. In this paper we study multivariate integration for a weighted Korobov space for which the Fourier coefficients of the functions decay exponentially fast. This implies that the functions of this space are infinitely times differentiable. Weights of the Korobov space monitor the influence of each variable and each group of variables. We show that there are numerical integration rules which achieve an exponential convergence of the worst-case integration error. We also investigate the dependence of the worst-case error on the number of variables s, and show various tractability results under certain conditions on weights of the Korobov space. Tractability means that the dependence on s is never exponential, and sometimes the dependence on s is polynomial or there is no dependence on s at all.

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The construction of good extensible rank-1 lattices

Cited by 46 publications

References 33 publications

Quasi-Monte Carlo Finite Element Methods for a Class of Elliptic Partial Differential Equations with Random Coefficients

Quasi-Monte Carlo Finite Element Methods for a Class of Elliptic Partial Differential Equations with Random Coefficients

Quasi-Monte Carlo for Highly Structured Generalised Response Models

Exponential convergence and tractability of multivariate integration for Korobov spaces

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