Journal ArticleDOI
Quadrature Error Bounds with Applications to Lattice Rules
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TLDR
In this paper, the authors derived error bounds and worst-case integrands for a large family of quadrature rules for the case of lattice rules with periodicity.Abstract:
Reproducing kernel Hilbert spaces are used to derive error bounds and worst-case integrands for a large family of quadrature rules. In the case of lattice rules applied to periodic integrands these error bounds resemble those previously derived in the literature. However, the theory developed here does not require periodicity and is not restricted to lattice rules. An analysis of variance (ANOVA) decomposition is employed in defining the inner product. It is shown that imbedded rules are superior when integrating functions with large high-order ANOVA effects.read more
Citations
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MonographDOI
Digital Nets and Sequences: Discrepancy Theory and Quasi-Monte Carlo Integration
TL;DR: This comprehensive treatment of contemporary quasi-Monte Carlo methods, digital nets and sequences, and discrepancy theory starts from scratch with detailed explanations of the basic concepts and then advances to current methods used in research.
Journal ArticleDOI
A generalized discrepancy and quadrature error bound
TL;DR: An error bound for multidimensional quadrature is derived that includes the Koksma-Hlawka inequality as a special case and includes as special cases the L p -star discrepancy and P α that arises in the study of lattice rules.
Book
Monte Carlo and Quasi-Monte Carlo Sampling
TL;DR: The Monte Carlo method has been used in many applications, e.g., for algebra, beyond numerical integration, this article, and for error and variance analysis for Halton sequences.
Journal ArticleDOI
High-dimensional integration: The quasi-Monte Carlo way
TL;DR: A survey of recent developments in lattice methods, digital nets, and related themes can be found in this paper, where the authors present a contemporary review of QMC (quasi-Monte Carlo) methods, that is, equalweight rules for the approximate evaluation of high-dimensional integrals over the unit cube [0, 1] s, w heres may be large, or even infinite.
Acta Numerica: High dimensional integration - the Quasi-Monte Carlo way
TL;DR: 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.
References
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Book
Spline models for observational data
TL;DR: In this paper, a theory and practice for the estimation of functions from noisy data on functionals is developed, where convergence properties, data based smoothing parameter selection, confidence intervals, and numerical methods are established which are appropriate to a number of problems within this framework.
Journal ArticleDOI
Theory of Reproducing Kernels.
TL;DR: In this paper, a short historical introduction is given to indicate the different manners in which these kernels have been used by various investigators and discuss the more important trends of the application of these kernels without attempting, however, a complete bibliography of the subject matter.
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Random number generation and quasi-Monte Carlo methods
TL;DR: This chapter discusses Monte Carlo methods and Quasi-Monte Carlo methods for optimization, which are used for numerical integration, and their applications in random numbers and pseudorandom numbers.
Journal ArticleDOI
The Jackknife Estimate of Variance
Bradley Efron,Charles Stein +1 more
TL;DR: In this paper, it was shown that the natural jackknife variance estimate tends always to be biased upwards, a theorem to this effect being proved for the natural Jackknife estimate of $\operatorname{Var} S(X_1, X_2, \cdots, X_{n-1})$ based on the symmetric function of i.i.d. random variables.
Book
Lattice Methods for Multiple Integration
TL;DR: In this paper, the use of lattice methods for the approximate integration of smooth periodic functions over the unit cube in any number of dimensions is discussed, and the authors show that the lattice method can be used to approximate any periodic function over a unit cube.