Tent-transformed lattice rules for integration and approximation of multivariate non-periodic functions

Cools, Ronald; Kuo, Frances Y.; Nuyens, Dirk; Suryanarayana, Gowri

doi:10.1016/j.jco.2016.05.004

Cited by 30 publications

(47 citation statements)

References 32 publications

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“…Whether or not deterministic tent-transformed lattice rules can achieve O(N −2+ε ) convergence in H(K sob 2,γ,s ) has remained unknown for a while after the work of Hickernell [13]. It can be seen from our first main result that the results in [2,8] cannot reach this question. Our second main result gives an affirmative answer to this question.…”

Section: Summary Of Main Findingsmentioning

confidence: 91%

Lattice rules in non-periodic subspaces of Sobolev spaces

2018

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We investigate quasi-Monte Carlo (QMC) integration over the sdimensional unit cube based on rank-1 lattice point sets in weighted nonperiodic Sobolev spaces H(K sob α,γ,s ) and their subspaces of high order smoothness α > 1, where γ denotes a set of the weights. A recent paper by Dick, Nuyens and Pillichshammer has studied QMC integration in half-period cosine spaces with smoothness parameter α >

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Section: Summary Of Main Findingsmentioning

confidence: 91%

Lattice rules in non-periodic subspaces of Sobolev spaces

2018

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“…Let e N,s,γ (z) be the worst-case error in the unanchored weighted Sobolev space H sob s,γ using the QMC rule (1/N ) N −1 k=0 f φ k N z . Then it is known due to [9] and [3] that e N,s,π 2 γ (z) ≤ e…”

Section: Korobov Spaces and Related Sobolev Spacesmentioning

confidence: 99%

“…where π 2 γ = (π 2|u| γ u ) ∅ =u⊆ [s] , and that the CBC construction with quality criterion given by the worst-case error of the Korobov space H(K s,2,γ ) can be used to construct tent-transformed lattice rules which achieve the almost optimal convergence order in the space H sob s,π 2 γ under appropriate conditions on the weights γ (see [3,Corollary 1]). Hence we also have a direct connection between integration in the Korobov space using lattice rules and integration in the unanchored Sobolev space using tent-transformed lattice rules.…”

Section: Korobov Spaces and Related Sobolev Spacesmentioning

confidence: 99%

Constructing lattice points for numerical integration by a reduced fast successive coordinate search algorithm

Ebert

Kritzer

2019

Journal of Computational and Applied Mathematics

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In this paper, we study an efficient algorithm for constructing node sets of high-quality quasi-Monte Carlo integration rules for weighted Korobov, Walsh, and Sobolev spaces. The algorithm presented is a reduced fast successive coordinate search (SCS) algorithm, which is adapted to situations where the weights in the function space show a sufficiently fast decay. The new SCS algorithm is designed to work for the construction of lattice points, and, in a modified version, for polynomial lattice points, and the corresponding integration rules can be used to treat functions in different kinds of function spaces. We show that the integration rules constructed by our algorithms satisfy error bounds of optimal convergence order. Furthermore, we give details on efficient implementation such that we obtain a considerable speed-up of previously known SCS algorithms. This improvement is illustrated by numerical results. The speed-up obtained by our results may be of particular interest in the context of QMC for PDEs with random coefficients, where both the dimension and the required number of points are usually very large. Furthermore, our main theorems yield previously unknown generalizations of earlier results.

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“…• See [32,Theorem 15] To apply this abstract theory to a practical integral over R s , it is important to realize that the choice of φ can be tuned as part of the process of transformation to express the integral in the form (7). (This point will become clearer when we describe the maximum likelihood application in Subsection 3.2.)…”

Section: Setting 2: Qmc Integration Over R Smentioning

confidence: 99%

Hot New Directions for Quasi-Monte Carlo Research in Step with Applications

Kuo

Nuyens

2018

Springer Proceedings in Mathematics &Amp; Statistics

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This article provides an overview of some interfaces between the theory of quasi-Monte Carlo (QMC) methods and applications. We summarize three QMC theoretical settings: first order QMC methods in the unit cube [0, 1] s and in R s , and higher order QMC methods in the unit cube. One important feature is that their error bounds can be independent of the dimension s under appropriate conditions on the function spaces. Another important feature is that good parameters for these QMC methods can be obtained by fast efficient algorithms even when s is large. We outline three different applications and explain how they can tap into the different QMC theory. We also discuss three cost saving strategies that can be combined with QMC in these applications. Many of these recent QMC theory and methods are developed not in isolation, but in close connection with applications. * Frances Y. Kuo ( ):

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Tent-transformed lattice rules for integration and approximation of multivariate non-periodic functions

Cited by 30 publications

References 32 publications

Lattice rules in non-periodic subspaces of Sobolev spaces

Lattice rules in non-periodic subspaces of Sobolev spaces

Constructing lattice points for numerical integration by a reduced fast successive coordinate search algorithm

Hot New Directions for Quasi-Monte Carlo Research in Step with Applications

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