On Sparse Interpolation and the Design of Deterministic Interpolation Points

Xu, Zhiqiang; Zhou, Tao

doi:10.1137/13094596x

Cited by 40 publications

(32 citation statements)

References 32 publications

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“…I.e., ρ is the uniform measure on Γ = [−1, 1] d , the PCE basis functions ψ j are tensor-product Legendre polynomials, and Ξ is constructed via iid sampling from the Chebyshev (arcsine) measure. The use of Chebyshev sampling when approximating with a Legendre polynomial basis (where available data is only function values) has been widely investigated [37,49,24], and can produce better results (compared to uniform sampling) when large-degree approximations are required. Here we shall show how inclusion of gradient information can be accomplished in a systematic way.…”

Section: A Gradient Enhanced Compressed Sensing Approachmentioning

confidence: 99%

A gradient enhanced ℓ1-minimization for sparse approximation of polynomial chaos expansions

Guo

Narayan

Zhou

2018

Journal of Computational Physics

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We investigate a gradient-enhanced 1 -minimization for constructing sparse polynomial chaos expansions. In addition to function evaluations, measurements of the function gradient is also included to accelerate the identification of expansion coefficients. By designing appropriate preconditioners to the measurement matrix, we show gradient-enhanced 1 minimization leads to stable and accurate coefficient recovery. The framework for designing preconditioners is quite general and it applies to recover of functions whose domain is bounded or unbounded. Comparisons between the gradient enhanced approach and the standard 1 -minimization are also presented and numerical examples suggest that the inclusion of derivative information can guarantee sparse recovery at a reduced computational cost.

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Section: A Gradient Enhanced Compressed Sensing Approachmentioning

confidence: 99%

A gradient enhanced ℓ1-minimization for sparse approximation of polynomial chaos expansions

Guo

Narayan

Zhou

2018

Journal of Computational Physics

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“…The focus of this paper is on determinstic sampling schemes; such methods have also been investigated [21,37,7]. We remark again that polynomials grids constructed via Fekete or Leja methods are closely connected to our procedure [5,4,1].…”

Section: Introductionmentioning

confidence: 99%

Weighted Approximate Fekete Points: Sampling for Least-Squares Polynomial Approximation

Guo¹,

Narayan²,

Yan³

et al. 2018

SIAM J. Sci. Comput.

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We propose and analyze a weighted greedy scheme for computing deterministic sample configurations in multidimensional space for performing least-squares polynomial approximations on L 2 spaces weighted by a probability density function. Our procedure is a particular weighted version of the approximate Fekete points method, with the weight function chosen as the (inverse) Christoffel function. Our procedure has theoretical advantages: when linear systems with optimal condition number exist, the procedure finds them. In the one-dimensional setting with any density function, our greedy procedure almost always generates optimally-conditioned linear systems. Our method also has practical advantages: our procedure is impartial to compactness of the domain of approximation, and uses only pivoted linear algebraic routines. We show through numerous examples that our sampling design outperforms competing randomized and deterministic designs when the domain is both low and high dimensional.Accurately estimating the coefficients f j of the expansion is important since these coefficients can be easily manipulated to infer revealing properties, such as statistical moments or parametric sensitivities. Many numerical techniques on how to obtain the polynomial coefficients in UQ problems have been developed in recent years. While early development often 1 The initial sample for the greedy method can be any point on R not coinciding with N − 1 isolated points; see Thereom 3.2 for details.

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“…One of the popular stochastic methodologies is generalized polynomial chaos (gPC) [3], which is an extension of the standard polynomial chaos method [4]. In addition, advanced gPC algorithms for high-dimensional stochastic problems have been developed [5][6][7][8][9][10][11][12][13][14][15].…”

Section: Introductionmentioning

confidence: 99%

An Adaptive WENO Collocation Method for Differential Equations with Random Coefficients

et al. 2016

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Abstract:The stochastic collocation method for solving differential equations with random inputs has gained lots of popularity in many applications, since such a scheme exhibits exponential convergence with smooth solutions in the random space. However, in some circumstance the solutions do not fulfill the smoothness requirement; thus a direct application of the method will cause poor performance and slow convergence rate due to the well known Gibbs phenomenon. To address the issue, we propose an adaptive high-order multi-element stochastic collocation scheme by incorporating a WENO (Weighted Essentially non-oscillatory) interpolation procedure and an adaptive mesh refinement (AMR) strategy. The proposed multi-element stochastic collocation scheme requires only repetitive runs of an existing deterministic solver at each interpolation point, similar to the Monte Carlo method. Furthermore, the scheme takes advantage of robustness and the high-order nature of the WENO interpolation procedure, and efficacy and efficiency of the AMR strategy. When the proposed scheme is applied to stochastic problems with non-smooth solutions, the Gibbs phenomenon is mitigated by the WENO methodology in the random space, and the errors around discontinuities in the stochastic space are significantly reduced by the AMR strategy. The numerical experiments for some benchmark stochastic problems, such as the Kraichnan-Orszag problem and Burgers' equation with random initial conditions, demonstrate the reliability, efficiency and efficacy of the proposed scheme.

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On Sparse Interpolation and the Design of Deterministic Interpolation Points

Cited by 40 publications

References 32 publications

A gradient enhanced ℓ1-minimization for sparse approximation of polynomial chaos expansions

A gradient enhanced ℓ1-minimization for sparse approximation of polynomial chaos expansions

Weighted Approximate Fekete Points: Sampling for Least-Squares Polynomial Approximation

An Adaptive WENO Collocation Method for Differential Equations with Random Coefficients

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