Fractal measures and polynomial sampling: I.F.S.–Gaussian integration

Mantica, Giorgio

doi:10.1007/s11075-007-9111-5

Cited by 9 publications

(12 citation statements)

References 14 publications

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“…There is a large literature on this type of approximation or “inverse” problem, probably beginning with Elton‐Yan . We refer the reader to , , and the references therein for more information.…”

Section: Introductionmentioning

confidence: 99%

First and second moments for self-similar couplings and Wasserstein distances

Fraser

2015

Math. Nachr.

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We study aspects of the Wasserstein distance in the context of self-similar measures. Computing this distance between two measures involves minimising certain moment integrals over the space of couplings, which are measures on the product space with the original measures as prescribed marginals. We focus our attention on self-similar measures associated to equicontractive iterated function systems satisfying the open set condition and consisting of two maps on the unit interval. We are particularly interested in understanding the restricted family of self-similar couplings and our main achievement is the explicit computation of the 1st and 2nd moment integrals for such couplings. We show that this family is enough to yield an explicit formula for the 1st Wasserstein distance and provide non-trivial upper and lower bounds for the 2nd Wasserstein distance.Mathematics Subject Classification 2010: Primary: 28A80, 28A33, 60B05. Secondary: 28A78.

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Section: Introductionmentioning

confidence: 99%

First and second moments for self-similar couplings and Wasserstein distances

Fraser

2015

Math. Nachr.

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“…Therefore, σ is the distribution of the position of the prey, that, along with the value of δ yields the distribution μ of the predator's positions. The measure μ can be equivalently defined by (2): its approximation properties and the related inverse problems have been discussed in [3,8,9,16].…”

Section: Infinite Affine Iterated Function Systemsmentioning

confidence: 99%

Dynamical systems and numerical analysis: the study of measures generated by uncountable I.F.S.

Mantica

2010

Numer Algor

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“…Following [15][16][17], we will call δ-homogeneous linear iterated function system (δ-HLIFS) balanced measure the unique measure µ such that:…”

Section: Balanced Measures and Linear Refinable Functionalsmentioning

confidence: 99%

“…the Lebesgue measure or to some of its weighted variants. Among the possible generalizations of the problem, the case of singular measures naturally appears, for instance, when dealing with fractal properties of some physical phenomenon, see [3,17].…”

Section: Introductionmentioning

confidence: 99%

An evaluation of Clenshaw–Curtis quadrature rule for integration w.r.t. singular measures

Calabrò

Esposito

2009

Journal of Computational and Applied Mathematics

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MSC: 28A25 65D30 65D32Keywords: Quadrature methods Fractal measures Gauss quadrature Clenshaw-Curtis formulae a b s t r a c t This work is devoted to the study of quadrature rules for integration with respect to (w.r.t.) general probability measures with known moments. Automatic calculation of the Clenshaw-Curtis rules is considered and analyzed. It is shown that it is possible to construct these rules in a stable manner for quadrature w.r.t. balanced measures. In order to make a comparison Gauss rules and their stable implementation for integration w.r.t. balanced measures are recalled. Convergence rates are tested in the case of binomial measures.

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Fractal measures and polynomial sampling: I.F.S.–Gaussian integration

Abstract: Measures generated by Iterated Function Systems can be used in place of atomic measures in Gaussian integration. A stable algorithm for the numerical solution of the related approximation problem -an inverse problem in fractal construction -is proposed.

Cited by 9 publications

References 14 publications

First and second moments for self-similar couplings and Wasserstein distances

First and second moments for self-similar couplings and Wasserstein distances

Dynamical systems and numerical analysis: the study of measures generated by uncountable I.F.S.

An evaluation of Clenshaw–Curtis quadrature rule for integration w.r.t. singular measures

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