A dynamical approach to accelerating numerical integration with equidistributed points

Jenkinson, Oliver; Pollicott, Mark

doi:10.1134/s0081543807010166

Cited by 6 publications

(5 citation statements)

References 9 publications

(14 reference statements)

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“…In order to implement this line of reasoning we need therefore to construct, for each s ∈ [k, k + 1], a trace-class operator L s on a Hilbert space H such that e P (A 1 ,...,A N ;s) is an eigenvalue of L s and is equal to the spectral radius of L s , such that L s is trace-class, such that the sequence of approximation numbers of L s decays rapidly to zero, and such that the sequence of traces tr L n s is easy to compute. Once such a family of operators has been constructed the result follows by relatively straightforward manipulations which, while they do not correspond precisely to any prior work, share a degree of familial resemblance with calculations occurring in numerous earlier articles such as [3,32,33,34,35,36,37,40,52,53,54,55,56,57].…”

Section: Overview Of the Methods And Statement Of The Main Technical ...mentioning

confidence: 99%

Fast approximation of the affinity dimension for dominated affine iterated function systems

Morris

2022

Ann. Fenn. Math.

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In 1988 Falconer introduced a formula which predicts the value of the Hausdorff dimension of the attractor of an affine iterated function system. The value given by this formula–sometimes referred to as the affinity dimension–is known to agree with the Hausdorff dimension both generically and in an increasing range of explicit cases. It is however a nontrivial problem to estimate the numerical value of the affinity dimension for specific iterated function systems. In this article we substantially extend an earlier result of Pollicott and Vytnova on the computation of the affinity dimension. Pollicott and Vytnova's work applies to planar invertible affine contractions with positive linear parts under several additional conditions which among other things constrain the affinity dimension to be between 0 and 1. We extend this result by passing from planar self-affine sets to self-affine sets in arbitrary dimensions, relaxing the positivity hypothesis to a domination condition, and removing all other constraints including that on the range of values of the affinity dimension. We provide explicit examples of two- and three-dimensional affine iterated function systems for which the affinity dimension can be calculated to more than 30 decimal places.

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Section: Overview Of the Methods And Statement Of The Main Technical ...mentioning

confidence: 99%

Fast approximation of the affinity dimension for dominated affine iterated function systems

Morris

2022

Ann. Fenn. Math.

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show abstract

“…In order to implement this line of reasoning we need therefore to construct, for each s ∈ [k, k + 1], a trace-class operator L s on a Hilbert space H such that e P (A1,...,A N ;s) is an eigenvalue of L s and is equal to the spectral radius of L s , such that L s is trace-class, such that the sequence of approximation numbers of L s decays rapidly to zero, and such that the sequence of traces tr L n s is easy to compute. Once such a family of operators has been constructed the result follows by relatively straightforward manipulations which, while they do not correspond precisely to any prior work, share a degree of familial resemblance with calculations occurring in numerous earlier articles such as [3,32,33,34,35,36,37,40,48,49,50,51,52,53].…”

Section: Overview Of the Methods And Statement Of The Main Technical ...mentioning

confidence: 99%

Fast approximation of the affinity dimension for dominated affine iterated function systems

Morris¹

2018

Preprint

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In 1988 K. Falconer introduced a formula which predicts the value of the Hausdorff dimension of the attractor of an affine iterated function system. The value given by this formula -sometimes referred to as the affinity dimension -is known to agree with the Hausdorff dimension both generically and in an increasing range of explicit cases. It is however a nontrivial problem to estimate the numerical value of the affinity dimension for specific iterated function systems. In this article we substantially extend an earlier result of M. Pollicott and P. Vytnova on the computation of the affinity dimension. Pollicott and Vytnova's work applies to planar invertible affine contractions with positive linear parts under several additional conditions which among other things constrain the affinity dimension to be between 0 and 1. We extend this result by passing from planar self-affine sets to self-affine sets in arbitrary dimensions, relaxing the positivity hypothesis to a domination condition, and removing all other constraints including that on the range of values of the affinity dimension. We provide some explicit examples of two-and threedimensional affine iterated function systems for which the affinity dimension can be calculated to more than 30 decimal places.

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“…In [CJ], Cipriano and Jurga study approximations of integrals with respect to stationary probability measures associated to iterated function systems on the interval, using an idea going back to Jenkinson and Pollicott [JP3]. Their setting is an iterated function system {Φ i } K i=1 consisting of Lipschitz contractions Φ i : [0, 1] → [0, 1].…”

Section: Approximation Of Stationary Probability Measures For Iterate...mentioning

confidence: 99%

Lagrange approximation of transfer operators associated with holomorphic data

Bandtlow¹,

Slipantschuk²

2020

Preprint

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We show that spectral data of transfer operators given by holomorphic data can be approximated using an effective numerical scheme based on Lagrange interpolation. In particular, we show that for one-dimensional systems satisfying certain complex contraction properties, spectral data of the approximants converge exponentially to the spectral data of the transfer operator with the exponential rate determined by the respective complex contraction ratios of the underlying systems. We demonstrate the effectiveness of this scheme by numerically computing eigenvalues of transfer operators arising from interval and circle maps, as well as Lyapunov exponents of (positive) random matrix products and iterated function systems, based on examples taken from the literature.

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A dynamical approach to accelerating numerical integration with equidistributed points

Cited by 6 publications

References 9 publications

Fast approximation of the affinity dimension for dominated affine iterated function systems

Fast approximation of the affinity dimension for dominated affine iterated function systems

Fast approximation of the affinity dimension for dominated affine iterated function systems

Lagrange approximation of transfer operators associated with holomorphic data

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