Tractability of infinite-dimensional integration in the worst case and randomized settings

Plaskota, Leszek; Wasilkowski, Grzegorz W.

doi:10.1016/j.jco.2011.01.006

Cited by 44 publications

(105 citation statements)

References 14 publications

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“…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.

227

368

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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.

227

368

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“…Passing to a subsequence we may assume that f n → f almost surely. (13) Again passing to a subsequence we may assume…”

Section: Proof Of Uniqueness In Lemma 3 To Verify This Fact Letmentioning

confidence: 99%

On equivalence of weighted anchored and ANOVA spaces of functions with mixed smoothness of order one in L1 or L∞
Hefter
¹
,
Ritter
²
,
Wasilkowski
³

2016
Journal of Complexity
Self Cite
22
2
63
0
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“…The model considered here, as in [24], is infinite dimensional. Previous treatments of infinite-dimensional quadrature include [17,25,27] with QMC methods [21], with MC methods, and [30] with Smolyak (or sparse-grid) quadrature. Applications of QMC methods to the "lognormal case" are considered in [18,19].…”

mentioning

confidence: 99%

Multi-level Quasi-Monte Carlo Finite Element Methods for a Class of Elliptic PDEs with Random Coefficients

2015

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This paper is a sequel to our previous work (Kuo et al. in SIAM J Numer Anal, 2012) where quasi-Monte Carlo (QMC) methods (specifically, randomly shifted lattice rules) are applied to finite element (FE) discretizations of elliptic partial differential equations (PDEs) with a random coefficient represented by a countably infinite number of terms. We estimate the expected value of some linear functional of the solution, as an infinite-dimensional integral in the parameter space. Here, the (singlelevel) error analysis of our previous work is generalized to a multi-level scheme, with the number of QMC points depending on the discretization level and with a level-dependent dimension truncation strategy. In some scenarios, it is shown that the overall error (i.e., the root-mean-square error averaged over all shifts) is of order O(h 2 ), where h is the finest FE mesh width, or O(N −1+δ ) for arbitrary δ > 0, where N denotes the maximal number of QMC sampling points in the parameter space. For these scenarios, the total work for all PDE solves in the multi-level QMC FE method is essentially of the order of one single PDE solve at the finest FE discretization level, for spatial dimension d = 2 with linear elements. The analysis exploits regularity of the parametric solution with respect to both the physical variables (the variables in Communicated by Philippe Ciarlet. 123Found Comput Math the physical domain) and the parametric variables (the parameters corresponding to randomness). As in our previous work, families of QMC rules with "POD weights" ("product and order dependent weights") which quantify the relative importance of subsets of the variables are found to be natural for proving convergence rates of QMC errors that are independent of the number of parametric variables.

show abstract

Tractability of infinite-dimensional integration in the worst case and randomized settings

Cited by 44 publications

References 14 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

On equivalence of weighted anchored and ANOVA spaces of functions with mixed smoothness of order one in L1 or L∞
Hefter
¹
,
Ritter
²
,
Wasilkowski
³

2016
Journal of Complexity
Self Cite
22
2
63
0
View full text Add to dashboard Cite

Multi-level Quasi-Monte Carlo Finite Element Methods for a Class of Elliptic PDEs with Random Coefficients

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Tractability of infinite-dimensional integration in the worst case and randomized settings

Cited by 44 publications

References 14 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

On equivalence of weighted anchored and ANOVA spaces of functions with mixed smoothness of order one in L1 or L∞Hefter1, Ritter2, Wasilkowski3 2016Journal of ComplexitySelf Cite222630View full textAdd to dashboardCite

Multi-level Quasi-Monte Carlo Finite Element Methods for a Class of Elliptic PDEs with Random Coefficients

Contact Info

Product

Resources

About

On equivalence of weighted anchored and ANOVA spaces of functions with mixed smoothness of order one in L1 or L∞
Hefter
¹
,
Ritter
²
,
Wasilkowski
³

2016
Journal of Complexity
Self Cite
22
2
63
0
View full text Add to dashboard Cite