On the Lebesgue constant of Leja sequences for the unit disk and its applications to multivariate interpolation

Calvi, Jean-Paul; van, Manh

doi:10.1016/j.jat.2011.02.001

Cited by 40 publications

(67 citation statements)

References 10 publications

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“…Let us finally mention that if the boundary of K is sufficiently regular, Fejer points that we mentioned above are known to be close, in a certain sense, to Fekete points for K , see [21] and [22]. Finally, we mention that the Lebesgue constant of Leja sequences for the disk and of their images by conformal mappings are studied in [11].…”

Section: Corollary 2 a Particular Leja Sequence Ismentioning

confidence: 88%

Pseudo Leja sequences

Białas-Cież

Calvi

2010

Annali di Matematica

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We study pseudo Leja sequences attached to a compact set in the complex plane. The requirements are weaker than those of ordinary Leja sequences, but these sequences still provide excellent points for interpolation of analytic functions and their computation is much easier. We also apply them to the construction of excellent sets of nodes for multivariate interpolation of analytic functions on product sets.

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Section: Corollary 2 a Particular Leja Sequence Ismentioning

confidence: 88%

Pseudo Leja sequences

Białas-Cież

Calvi

2010

Annali di Matematica

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“…In this paper, we improve further these bounds and give direct bounds for the norms D k of the difference operators, which are useful in view of the discussion in the previous section. Our techniques of proof share several common points with those developed in [2,3,4], yet it is shorter and exploit to a considerable extent the properties of Leja sequences on the unit disk.…”

Section: Leja Sequences and Their Projections 31 Leja Sequence On Thmentioning

confidence: 99%

“…which corresponds to the Newton form, with 2) with the convention that I −1 = 0. Now, for an arbitrary lower set Λ ⊂ N d , we introduce the grid of points…”

Section: Sparse Polynomial Interpolationmentioning

confidence: 99%

“…Recently, Calvi and Phung [2,3] have shown that the Lebesgue constants of Leja sequences on U the unit disk and theirs real projection on [−1, 1], the so-called -Leja sequences, are moderate and have growth asymptotically bounded in O(k log k) and O(k 3 log k) respectively.…”

Section: Leja Sequences and Their Projections 31 Leja Sequence On Thmentioning

confidence: 99%

“…Leja sequences E = (e j ) j≥0 on U considered in [2,4] have all theirs initial value e 0 ∈ ∂U the unit circle. In view of the definition (3.6), the maximum principle implies e j ∈ ∂U for any j ≥ 1.…”

Section: Leja Sequences and Their Projections 31 Leja Sequence On Thmentioning

confidence: 99%

See 2 more Smart Citations

On the Stability of Polynomial Interpolation Using Hierarchical Sampling

Cohen

Chkifa

2015

Sampling Theory, a Renaissance

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Motivated by the development of non-intrusive methods for high dimensional parametric PDE's, we study the stability of a sparse high dimensional polynomial interpolation procedure introduced in [6]. A key aspect of this procedure is its hierarchical structure: the sampling set is progressively enriched together with the polynomial space. The evaluation points are selected from a grid obtained by tensorization of a univariate sequence. The Lebesgue constant that quantifies the stability of the resulting interpolation operator depends on the choice of this sequence. Here we study the -Leja sequence, obtained by the projection of Leja sequences on the complex unit circle, with initial value 1, onto [−1, 1]. For this sequence, we prove cubic growth in the number of points for the Lebesgue constant of the multivariate interpolation operator, independently of the number of variable and of the shape of the polynomial space.

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An hp‐adaptive multi‐element stochastic collocation method for surrogate modeling with information re‐use

Galetzka

Loukrezis

Georg

et al. 2023

Numerical Meth Engineering

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This article introduces an hp-adaptive multi-element stochastic collocation method, which additionally allows to re-use existing model evaluations during either h-or p-refinement. The collocation method is based on weighted Leja nodes. After h-refinement, local interpolations are stabilized by adding and sorting Leja nodes on each newly created sub-element in a hierarchical manner.For p-refinement, the local polynomial approximations are based on total-degree or dimension-adaptive bases. The method is applied in the context of forward and inverse uncertainty quantification to handle non-smooth or strongly localized response surfaces. The performance of the proposed method is assessed in several test cases, also in comparison to competing methods.

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On the Lebesgue constant of Leja sequences for the unit disk and its applications to multivariate interpolation

Abstract: International audienc

Cited by 40 publications

References 10 publications

Pseudo Leja sequences

Pseudo Leja sequences

On the Stability of Polynomial Interpolation Using Hierarchical Sampling

An hp‐adaptive multi‐element stochastic collocation method for surrogate modeling with information re‐use

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