Dimension-Adaptive Sparse Grid Quadrature for Integrals with Boundary Singularities

Griebel, Michael; Oettershagen, Jens

doi:10.1007/978-3-319-04537-5_5

Cited by 11 publications

(12 citation statements)

References 39 publications

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“…to be not only finite, but also small, as explained in the introduction, see (4). Due to the tensor product form ( 6), one can show (in a similar way to the derivation of the norm of I d, ) that the following holds.…”

Section: Multivariate Integrationmentioning

confidence: 71%

“…Such integrals in the univariate case are often approximated by Gaussian quadratures that enjoy exponential rate of convergence, see, e.g., [1]. There are also generalized Gaussian rules, see, e.g., [4] and the papers cited there, that achieve exponential rate for integrands with singularities at infinity. We stress that those results are about the asymptotic behavior of the integration error and require analytic integrands.…”

Section: Introductionmentioning

confidence: 99%

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On efficient weighted integration via a change of variables

Kritzer

Pillichshammer

Plaskota

et al. 2018

Preprint

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In this paper, we study the approximation of d-dimensional -weighted integrals over unbounded domains R d + or R d using a special change of variables, so that quasi-Monte Carlo (QMC) or sparse grid rules can be applied to the transformed integrands over the unit cube. We consider a class of integrands with bounded L p norm of mixed partial derivatives of first order, where p ∈ [1, +∞].The main results give sufficient conditions on the change of variables ν which guarantee that the transformed integrand belongs to the standard Sobolev space of functions over the unit cube with mixed smoothness of order one. These conditions depend on and p.The proposed change of variables is in general different than the standard change based on the inverse of the cumulative distribution function. We stress that the standard change of variables leads to integrands over a cube; however, those integrands have singularities which make the application of QMC and sparse grids ineffective. Our conclusions are supported by numerical experiments.

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Section: Multivariate Integrationmentioning

confidence: 71%

Section: Introductionmentioning

confidence: 99%

On efficient weighted integration via a change of variables

Kritzer

Pillichshammer

Plaskota

et al. 2018

Preprint

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show abstract

“…An alternative quadrature approach for integrals with boundary singularities is the Gauss–Laguerre method (Griebel and Oettershagen, 2014). To modify this method for the Riemann–Liouville fractional derivative, we must map the Gauss–Laguerre quadrature domain from [ 0 , ∞ ] to [ 0 , 1 ] .…”

Section: Fractional-order Derivativesmentioning

confidence: 99%

Bayesian inference and uncertainty propagation using efficient fractional-order viscoelastic models for dielectric elastomers

Miles

Pash

Smith

et al. 2020

Journal of Intelligent Material Systems and Structures

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Dielectric elastomers are employed for a wide variety of adaptive structures. Many of these soft elastomers exhibit significant rate-dependencies in their response. Accurately quantifying this viscoelastic behavior is non-trivial and in many cases a nonlinear modeling framework is required. Fractional-order operators have been applied to modeling viscoelastic behavior for many years, and recent research has shown fractional-order methods to be effective for nonlinear frameworks. This implementation can become computationally expensive to achieve an accurate approximation of the fractional-order derivative. Accurate estimation of the elastomer’s viscoelastic behavior to quantify parameter uncertainty motivates the use of Markov Chain Monte Carlo (MCMC) methods. Since MCMC is a sampling based method, requiring many model evaluations, efficient estimation of the fractional derivative operator is crucial. In this paper, we demonstrate the effectiveness of using quadrature techniques to approximate the Riemann–Liouville definition for fractional derivatives in the context of estimating the uncertainty of a nonlinear viscoelastic model. We also demonstrate the use of parameter subset selection techniques to isolate parameters that are identifiable in the sense that they are uniquely determined by measured data. For those identifiable parameters, we employ Bayesian inference to compute posterior distributions for parameters. Finally, we propagate parameter uncertainties through the models to compute prediction intervals for quantities of interest.

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“…Moreover, it is not clear which Gaussian rules should be used when ψ is not a constant function. But, even for ψ ≡ 1, it is likely that the worst case errors (with respect to F r p,ψ ) of Gaussian rules are much larger than O(n −r ), since the Weierstrass theorem holds only for compact D. A very interesting extension of Gaussian rules to functions with singularities has been proposed in [2]. However, the results of [2] are also asymptotic and it is not clear how the proposed rules behave for functions from spaces F r p,ψ .…”

Section: Introductionmentioning

confidence: 99%

“…But, even for ψ ≡ 1, it is likely that the worst case errors (with respect to F r p,ψ ) of Gaussian rules are much larger than O(n −r ), since the Weierstrass theorem holds only for compact D. A very interesting extension of Gaussian rules to functions with singularities has been proposed in [2]. However, the results of [2] are also asymptotic and it is not clear how the proposed rules behave for functions from spaces F r p,ψ . In the present paper, we deal with functions of bounded smoothness (r < +∞) and provide worst-case error bounds that are minimal.…”

Section: Introductionmentioning

confidence: 99%

On alternative quantization for doubly weighted approximation and integration over unbounded domains

Kritzer¹,

Pillichshammer²,

Plaskota³

et al. 2019

Preprint

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It is known that for a -weighted L q -approximation of single variable functions f with the rth derivatives in a ψ-weighted L p space, the minimal error of approximations that use n samples of f is proportional to ω 1/α α L 1 f (r) ψ Lp n −r+(1/p−1/q) + , where ω = /ψ and α = r − 1/p + 1/q. Moreover, the optimal sample points are determined by quantiles of ω 1/α . In this paper, we show how the error of best approximations changes when the sample points are determined by a quantizer κ other than ω. Our results can be applied in situations when an alternative quantizer has to be used because ω is not known exactly or is too complicated to handle computationally. The results for q = 1 are also applicable to -weighted integration over unbounded domains.

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Dimension-Adaptive Sparse Grid Quadrature for Integrals with Boundary Singularities

Cited by 11 publications

References 39 publications

On efficient weighted integration via a change of variables

On efficient weighted integration via a change of variables

Bayesian inference and uncertainty propagation using efficient fractional-order viscoelastic models for dielectric elastomers

On alternative quantization for doubly weighted approximation and integration over unbounded domains

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