New Bounds on the Lebesgue Constants of Leja Sequences on the Unit Disc and on $$\mathfrak {R}$$-Leja Sequences

Chkifa, Moulay Abdellah

doi:10.1007/978-3-319-22804-4_9

Cited by 3 publications

(6 citation statements)

References 11 publications

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“…Their advantage is that they strongly exploit the binary structure of the Leja sequences. It was proved in [4] that for any k expanded in the form (4), the following inequalities hold :…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

“…In the present paper, we will follow the same approach as in [4]. Our estimate for the "quadratic" Lebesgue constant is given by next proposition.…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

“…(9) e 2j+1 = −e 2j , e 2 2j = e 2 2j+1 = e j , j ≥ 0, it was shown in [4] that : λ 2N,2 (z) = λ N,2 (z 2 ), N ≥ 0, |z| ≤ 1 and as consequences :…”

Section: Proof Of Proposition1mentioning

confidence: 99%

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A sharp bound on the Lebesgue constant for Leja points in the unit disk

Ounaïes

2017

Journal of Approximation Theory

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“…Their advantage is that they strongly exploit the binary structure of the Leja sequences. It was proved in [4] that for any k expanded in the form (4), the following inequalities hold :…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

“…In the present paper, we will follow the same approach as in [4]. Our estimate for the "quadratic" Lebesgue constant is given by next proposition.…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

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A sharp bound on the Lebesgue constant for Leja points in the unit disk

Ounaïes

2017

Journal of Approximation Theory

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“…Their projections on the real axis are called the -Leja points, and coincide with the Gauss–Lobatto points for values of of the form for . The growth of the Lebesgue constant has been studied by Calvi and Phung (2011, 2012), Chkifa (2013) and Chkifa (2015) for these two families of points. In the case of complex Leja points, this constant is defined as in (6.64), but taking the supremum over functions defined everywhere and bounded over the complex unit disc.…”

Section: Sparse Polynomial Interpolationmentioning

confidence: 99%

“…It is proved in Chkifa (2013) that the linear bound (6.83) holds for the complex Leja points. For the -Leja points, quadratic bounds of the type with , are established in Chkifa (2015).…”

Section: Sparse Polynomial Interpolationmentioning

confidence: 99%

Approximation of high-dimensional parametric PDEs

Cohen¹,

DeVore²

2015

Acta Numerica

234

266

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Parametrized families of PDEs arise in various contexts such as inverse problems, control and optimization, risk assessment, and uncertainty quantification. In most of these applications, the number of parameters is large or perhaps even infinite. Thus, the development of numerical methods for these parametric problems is faced with the possible curse of dimensionality. This article is directed at (i) identifying and understanding which properties of parametric equations allow one to avoid this curse and (ii) developing and analyzing effective numerical methodd which fully exploit these properties and, in turn, are immune to the growth in dimensionality.The first part of this article studies the smoothness and approximability of the solution map, that is, the map a → u(a) where a is the parameter value and u(a) is the corresponding solution to the PDE. It is shown that for many relevant parametric PDEs, the parametric smoothness of this map is typically holomorphic and also highly anisotropic in that the relevant parameters are of widely varying importance in describing the solution. These two properties are then exploited to establish convergence rates of n-term approximations to the solution map for which each term is separable in the parametric and physical variables. These results reveal that, at least on a theoretical level, the solution map can be well approximated by discretizations of moderate complexity, thereby showing how the curse of dimensionality is broken. This theoretical analysis is carried out through concepts of approximation theory such as best n-term approximation, sparsity, and n-widths. These notions determine a priori the best possible performance of numerical methods and thus serve as a benchmark for concrete algorithms.The second part of this article turns to the development of numerical algorithms based on the theoretically established sparse separable approximations. The numerical methods studied fall into two general categories. The first uses polynomial expansions in terms of the parameters to approximate the solution map. The second one searches for suitable low dimensional spaces for simultaneously approximating all members of the parametric family. The numerical implementation of these approaches is carried out through adaptive and greedy algorithms. An a priori analysis of the performance of these algorithms establishes how well they meet the theoretical benchmarks.

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Newton interpolation using $$\Re $$-Leja sequences

Chkifa

2022

Bit Numer Math

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New Bounds on the Lebesgue Constants of Leja Sequences on the Unit Disc and on $$\mathfrak {R}$$-Leja Sequences

Cited by 3 publications

References 11 publications

A sharp bound on the Lebesgue constant for Leja points in the unit disk

A sharp bound on the Lebesgue constant for Leja points in the unit disk

Approximation of high-dimensional parametric PDEs

Newton interpolation using $$\Re $$-Leja sequences

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