On asymptotics and estimates for the uniform norms of the Lagrange interpolation polynomials corresponding to the Chebyshev nodal points

Dzjadyk, V K; Ivanov, V. V.

doi:10.1007/bf01982005

Cited by 34 publications

(15 citation statements)

References 5 publications

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“…It is known that [13]. Using Lemma 3.2, the best approximation E mi (u) to functions u ∈ C 0 (Γ 1 ; W (D)) that admit an analytic extension as described by Assumption 1.8 is bounded by (3.2).…”

Section: Interpolation Estimates For the Clenshaw-curtis Abscissasmentioning

confidence: 97%

A Sparse Grid Stochastic Collocation Method for Partial Differential Equations with Random Input Data

Nobile¹,

Tempone²,

Webster³

2008

SIAM J. Numer. Anal.

904

952

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This work proposes and analyzes a Smolyak-type sparse grid stochastic collocation method for the approximation of statistical quantities related to the solution of partial differential equations with random coefficients and forcing terms (input data of the model). To compute solution statistics, the sparse grid stochastic collocation method uses approximate solutions, produced here by finite elements, corresponding to a deterministic set of points in the random input space. This naturally requires solving uncoupled deterministic problems as in the Monte Carlo method. If the number of random variables needed to describe the input data is moderately large, full tensor product spaces are computationally expensive to use due to the curse of dimensionality. In this case the sparse grid approach is still expected to be competitive with the classical Monte Carlo method. Therefore, it is of major practical relevance to understand in which situations the sparse grid stochastic collocation method is more efficient than Monte Carlo. This work provides error estimates for the fully discrete solution using L q norms and analyzes the computational efficiency of the proposed method. In particular, it demonstrates algebraic convergence with respect to the total number of collocation points and quantifies the effect of the dimension of the problem (number of input random variables) in the final estimates. The derived estimates are then used to compare the method with Monte Carlo, indicating for which problems the former is more efficient than the latter. Computational evidence complements the present theory and shows the effectiveness of the sparse grid stochastic collocation method compared to full tensor and Monte Carlo approaches.

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Section: Interpolation Estimates For the Clenshaw-curtis Abscissasmentioning

confidence: 97%

A Sparse Grid Stochastic Collocation Method for Partial Differential Equations with Random Input Data

Nobile¹,

Tempone²,

Webster³

2008

SIAM J. Numer. Anal.

904

952

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show abstract

“…For Lagrange interpolants based on Clenshaw-Curtis abscissas (4.4), we have [15] λ ln ≤ 2 π log (m (l n ) − 1) + 1 for l n ≥ 2.…”

Section: Error Estimates For Fixed Lmentioning

confidence: 99%

Accelerating Stochastic Collocation Methods for Partial Differential Equations with Random Input Data

Galindo¹,

Jantsch²,

Webster³

et al. 2016

SIAM/ASA J. Uncertainty Quantification

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Abstract. This work proposes and analyzes a generalized acceleration technique for decreasing the computational complexity of using stochastic collocation (SC) methods to solve partial differential equations (PDEs) with random input data. The SC approaches considered in this effort consist of sequentially constructed multi-dimensional Lagrange interpolant in the random parametric domain, formulated by collocating on a set of points so that the resulting approximation is defined in a hierarchical sequence of polynomial spaces of increasing fidelity. Our acceleration approach exploits the construction of the SC interpolant to accelerate the underlying ensemble of deterministic solutions. Specifically, we predict the solution of the parametrized PDE at each collocation point on the current level of the SC approximation by evaluating each sample with a previously assembled lower fidelity interpolant, and then use such predictions to provide deterministic (linear or nonlinear) iterative solvers with improved initial approximations. As a concrete example, we develop our approach in the context of SC approaches that employ sparse tensor products of globally defined Lagrange polynomials on nested one-dimensional Clenshaw-Curtis abscissas. This work also provides a rigorous computational complexity analysis of the resulting fully discrete sparse grid SC approximation, with and without acceleration, which demonstrates the effectiveness of our proposed methodology in reducing the total number of iterations of a conjugate gradient solution of the finite element systems at each collocation point. Numerical examples include both linear and nonlinear parametrized PDEs, which are used to illustrate the theoretical results and the improved efficiency of this technique compared with several others.

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“…Using analogous estimations and (3.1), Dzjadik and Ivanov [14] got the value of 4" within the error 0.45. They had no knowledge of the paper of L. Brutman [15] written in t,978, where using a quite serious analysis of s x), he proved that…”

Section: Z~k<=n 2~k~_nmentioning

confidence: 97%

On the optimal Lebesgue constants for polynomial interpolation

Vértesi

1986

Acta Math Hung

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On asymptotics and estimates for the uniform norms of the Lagrange interpolation polynomials corresponding to the Chebyshev nodal points

Cited by 34 publications

References 5 publications

A Sparse Grid Stochastic Collocation Method for Partial Differential Equations with Random Input Data

A Sparse Grid Stochastic Collocation Method for Partial Differential Equations with Random Input Data

Accelerating Stochastic Collocation Methods for Partial Differential Equations with Random Input Data

On the optimal Lebesgue constants for polynomial interpolation

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