Liberating the dimension

Kuo, Frances Y.; Sloan, Ian H.; Wasilkowski, Grzegorz W.; Woniakowski, Henryk

doi:10.1016/j.jco.2009.12.003

Cited by 62 publications

(171 citation statements)

References 23 publications

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“…The explicit decomposition formula can be useful for numerical integration and approximation of functions in high dimensions, since there might be a merit in approximating the solution corresponding to the lower order part (see, e.g., [10]). We may also use the explicit decomposition formula in a theoretical analysis where no actual computation is required, noting that it may be easier to observe or identify certain properties of one particular decomposition term when it is expressed explicitly rather than recursively (see, e.g., [6]).…”

Section: Discussionmentioning

confidence: 99%

On decompositions of multivariate functions

et al. 2009

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Abstract. We present formulas that allow us to decompose a function f of d variables into a sum of 2 d terms f u indexed by subsets u of {1, . . . , d}, where each term f u depends only on the variables with indices in u. The decomposition depends on the choice of d commuting projections {P j } d j=1 , where P j (f ) does not depend on the variable x j . We present an explicit formula for f u , which is new even for the anova and anchored decompositions; both are special cases of the general decomposition. We show that the decomposition is minimal in the following sense: if f is expressible as a sum in which there is no term that depends on all of the variables indexed by the subset z, then, for every choice of {P j } d j=1 , the terms f u = 0 for all subsets u containing z. Furthermore, in a reproducing kernel Hilbert space setting, we give sufficient conditions for the terms f u to be mutually orthogonal.

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Section: Discussionmentioning

confidence: 99%

On decompositions of multivariate functions

et al. 2009

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“…Here we follow the infinite dimensional setting in [20], but with the anchor at the center of the unit cube rather than at a corner. For F a function depending on infinitely many variables y = (y 1 , y 2 , .…”

Section: Qmc Integration In the Infinite Dimensional Settingmentioning

confidence: 99%

“…The cost can be dramatically reduced if we use instead multi-level and/or changing dimension algorithms which have been analyzed in a number of recent papers, see e.g. [20,23,15,12,26] for infinite dimensional integration, and e.g. [4,1,2] for applications in PDEs.…”

Section: F Y Kuo Ch Schwab and I H Sloanmentioning

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.

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368

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“…We now describe the fast CBC construction for the generic criterion at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/S1446181112000077 [34] where e The overall CBC construction cost is then O(sN ln N) operations.…”

Section: Fast Cbc Construction For Pod Weightsmentioning

confidence: 99%

Quasi-Monte Carlo methods for high-dimensional integration: the standard (weighted Hilbert space) setting and beyond

Kuo¹,

Schwab²,

Sloan³

2012

ANZIAMJ

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This paper is a contemporary review of quasi-Monte Carlo (QMC) methods, that is, equal-weight rules for the approximate evaluation of high-dimensional integrals over the unit cube [0,1] s . It first introduces the by-now standard setting of weighted Hilbert spaces of functions with square-integrable mixed first derivatives, and then indicates alternative settings, such as non-Hilbert spaces, that can sometimes be more suitable. Original contributions include the extension of the fast component-by-component (CBC) construction of lattice rules that achieve the optimal convergence order (a rate of almost 1/N, where N is the number of points, independently of dimension) to so-called "product and order dependent" (POD) weights, as seen in some recent applications. Although the paper has a strong focus on lattice rules, the function space settings are applicable to all QMC methods. Furthermore, the error analysis and construction of lattice rules can be adapted to polynomial lattice rules from the family of digital nets.

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Liberating the dimension

Cited by 62 publications

References 23 publications

On decompositions of multivariate functions

On decompositions of multivariate functions

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

Quasi-Monte Carlo methods for high-dimensional integration: the standard (weighted Hilbert space) setting and beyond

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