On tensor product approximation of analytic functions

Griebel, Michael; Oettershagen, Jens

doi:10.1016/j.jat.2016.02.006

Cited by 30 publications

(45 citation statements)

References 33 publications

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“…In addition, we provide an estimate on the quadrature error in terms of the number of multiindices contained in the anisotropic sparse index set. Note that this is very similar to the analysis in [3,13]. From (9), we deduce that the error of the anisotropic sparse grid quadrature can be written as (26) I…”

Section: Error Estimation For the Anisotropic Sparse Grid Quadraturesupporting

confidence: 71%

“…Due to the summability properties of {τ n } n , the constant c(κ) can obviously be bounded independent of the dimension m. It remains to estimate the sum in the above estimate which has been extensively studied in [13]. The following result from [13] is particularly useful for the considered situation.…”

Section: Error Estimation For the Anisotropic Sparse Grid Quadraturementioning

confidence: 99%

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Novel results for the anisotropic sparse grid quadrature

Haji-Ali¹,

Harbrecht²,

Peters³

et al. 2018

Journal of Complexity

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This article is dedicated to the anisotropic sparse grid quadrature for functions which are analytically extendable into an anisotropic tensor product domain. Taking into account this anisotropy, we end up with a dimension independent error versus cost estimate of the proposed quadrature. In addition, we provide a novel and improved estimate for the cardinality of the underlying anisotropic index set. To validate the theoretical findings, we present several examples ranging from simple quadrature problems to diffusion problems on random domains. These examples demonstrate the remarkable convergence behaviour of the anisotropic sparse grid quadrature in applications.2000 Mathematics Subject Classification. 65D30, 65C30, 60H25.

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Section: Error Estimation For the Anisotropic Sparse Grid Quadraturesupporting

confidence: 71%

Section: Error Estimation For the Anisotropic Sparse Grid Quadraturementioning

confidence: 99%

Novel results for the anisotropic sparse grid quadrature

Haji-Ali¹,

Harbrecht²,

Peters³

et al. 2018

Journal of Complexity

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show abstract

“…To construct the approximations U i k [f i ], we employ weighted (L)-Leja sequences (see, e.g., [16,32]). Given the weight function w : X i → R + , weighted (L)-Leja sequences are constructed recursively as follows:…”

Section: 3mentioning

confidence: 99%

Multilevel Adaptive Sparse Leja Approximations for Bayesian Inverse Problems

Farcaș¹,

Latz²,

Ullmann³

et al. 2020

SIAM J. Sci. Comput.

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Deterministic interpolation and quadrature methods are often unsuitable to address Bayesian inverse problems depending on computationally expensive forward mathematical models. While interpolation may give precise posterior approximations, deterministic quadrature is usually unable to efficiently investigate an informative and thus concentrated likelihood. This leads to a large number of required expensive evaluations of the mathematical model. To overcome these challenges, we formulate and test a multilevel adaptive sparse Leja algorithm. At each level, adaptive sparse grid interpolation and quadrature are used to approximate the posterior and perform all quadrature operations, respectively. Specifically, our algorithm uses coarse discretizations of the underlying mathematical model to investigate the parameter space and to identify areas of high posterior probability. Adaptive sparse grid algorithms are then used to place points in these areas, and ignore other areas of small posterior probability. The points are weighted Leja points. As the model discretization is coarse, the construction of the sparse grid is computationally efficient. On this sparse grid, the posterior measure can be approximated accurately with few expensive, fine model discretizations. The efficiency of the algorithm can be enhanced further by exploiting more than two discretization levels. We apply the proposed multilevel adaptive sparse Leja algorithm in numerical experiments involving elliptic inverse problems in 2D and 3D space, in which we compare it with Markov chain Monte Carlo sampling and a standard multilevel approximation.

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“…Recently, the Leja points have shown great promise for use in sparse polynomial approximation methods in high dimensions (Chkifa et al, 2013;Narayan & Jakeman, 2014;Griebel & Oettershagen, 2016). The key property is that, by definition, a set of n Leja points is contained in the set of size n + 1, a property that is not shared by other sets of common interpolation nodes, especially for weighted interpolation on unbounded domains.…”

Section: P Jantsch Et Almentioning

confidence: 99%

On the Lebesgue constant of weighted Leja points for Lagrange interpolation on unbounded domains

Jantsch

Webster

Zhang

2018

IMA Journal of Numerical Analysis

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This work focuses on weighted Lagrange interpolation on an unbounded domain and analyzes the Lebesgue constant for a sequence of weighted Leja points. The standard Leja points are a nested sequence of points defined on a compact subset of the real line and can be extended to unbounded domains with the introduction of a weight function w : R → [0, 1]. Due to a simple recursive formulation in one dimension, such abscissas provide a foundation for high-dimensional approximation methods such as sparse grid collocation, deterministic least squares and compressed sensing. Just as in the unweighted case of interpolation on a compact domain, we use results from potential theory to prove that the Lebesgue constant for the Leja points grows subexponentially with the number of interpolation nodes.

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On tensor product approximation of analytic functions

Cited by 30 publications

References 33 publications

Novel results for the anisotropic sparse grid quadrature

Novel results for the anisotropic sparse grid quadrature

Multilevel Adaptive Sparse Leja Approximations for Bayesian Inverse Problems

On the Lebesgue constant of weighted Leja points for Lagrange interpolation on unbounded domains

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