Integration of nonperiodic functions of two variables by Fibonacci lattice rules

Niederreiter, Harald; Sloan, Ian H.

doi:10.1016/0377-0427(92)00004-s

Cited by 23 publications

(18 citation statements)

References 8 publications

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“…In this work, we present a technique that is based on two successive Fibonacci numbers. Similar techniques have been presented before [28][29][30] for other applications. For the numerical integration on a sphere surface, there is a distinct advantage of using an oblique array of sampling points based on a chosen pair of successive Fibonacci numbers [31].…”

Section: Qmc-based Samplingmentioning

confidence: 90%

Quasi Monte Carlo-based isotropic distribution of gradient directions for improved reconstruction quality of 3D EPR imaging

Ahmad

Deng

Vikram

et al. 2007

Journal of Magnetic Resonance

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In continuous wave (CW) electron paramagnetic resonance imaging (EPRI), high quality of reconstructed image along with fast and reliable data acquisition is highly desirable for many biological applications. An accurate representation of uniform distribution of projection data is necessary to ensure high reconstruction quality. The current techniques for data acquisition suffer from nonuniformities or local anisotropies in the distribution of projection data and present a poor approximation of a true uniform and isotropic distribution. In this work, we have implemented a technique based on Quasi-Monte Carlo method to acquire projections with more uniform and isotropic distribution of data over a 3D acquisition space. The proposed technique exhibits improvements in the reconstruction quality in terms of both mean-square-error and visual judgment. The effectiveness of the suggested technique is demonstrated using computer simulations and 3D EPRI experiments. The technique is robust and exhibits consistent performance for different object configurations and orientations.

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Section: Qmc-based Samplingmentioning

confidence: 90%

Quasi Monte Carlo-based isotropic distribution of gradient directions for improved reconstruction quality of 3D EPR imaging

Ahmad

Deng

Vikram

et al. 2007

Journal of Magnetic Resonance

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“…For the vertex modified rules we only need to specify the weights at the vertices of the unit cube, all other remain unchanged from the standard lattice rule and are 1/N . Two particular choices for the weights w(a) have been proposed [7,8,9]. The first one has constant weights w(a) ≡ 1/(2 s N ) which mimics the trapezoidal rule in each one-dimensional projection:…”

Section: Vertex Modified Lattice Rulesmentioning

confidence: 99%

“…Proof. This can be found by direct calculation using (11) in (7) and comparing terms with the worst-case errors in the Korobov space (8) and the multilinear space (10).…”

Section: Decomposing the Error For The Unanchored Sobolev Spacementioning

confidence: 99%

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The Analysis of Vertex Modified Lattice Rules in a Non-periodic Sobolev Space

Nuyens

Cools

2018

Contemporary Computational Mathematics - A Celebration of the 80th Birthday of Ian Sloan

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“…These are quasi-Monte Carlo rules which are tuned to a particular class of functions and region of integration, namely, periodic functions over a hypercube. Their range of applicability may be extended by a variety of techniques (see, for example, [2], [15], [33], [34], [48]). Lattice rules generalize an earlier type of quasi-Monte Carlo method, the so-called 'method of good lattice points', introduced by Korobov [19], [20], Bakhvalov [1] and Hlawka [14].…”

Section: Introductionmentioning

confidence: 99%

An application of Diophantine approximation to the construction of rank-1 lattice quadrature rules

Langtry

1996

Math. Comp.

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Abstract. Lattice quadrature rules were introduced by Frolov (1977), Sloan (1985) and Sloan and Kachoyan (1987). They are quasi-Monte Carlo rules for the approximation of integrals over the unit cube in R s and are generalizations of 'number-theoretic' rules introduced by Korobov (1959) and Hlawka (1962)-themselves generalizations, in a sense, of rectangle rules for approximating one-dimensional integrals, and trapezoidal rules for periodic integrands.Error bounds for rank-1 rules are known for a variety of classes of integrands. For periodic integrands with unit period in each variable, these bounds are conveniently characterized by the figure of merit ρ, which was originally introduced in the context of number-theoretic rules. The problem of finding good rules of order N (that is, having N nodes) then becomes that of finding rules with large values of ρ. This paper presents a new approach, based on the theory of simultaneous Diophantine approximation, which uses a generalized continued fraction algorithm to construct rank-1 rules of high order.

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Integration of nonperiodic functions of two variables by Fibonacci lattice rules

Cited by 23 publications

References 8 publications

Quasi Monte Carlo-based isotropic distribution of gradient directions for improved reconstruction quality of 3D EPR imaging

Quasi Monte Carlo-based isotropic distribution of gradient directions for improved reconstruction quality of 3D EPR imaging

The Analysis of Vertex Modified Lattice Rules in a Non-periodic Sobolev Space

An application of Diophantine approximation to the construction of rank-1 lattice quadrature rules

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