Gaussian quadrature

Brass, Helmut; Petras, Knut

doi:10.1090/surv/178/06

Cited by 8 publications

(13 citation statements)

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“…There exists a large number of error estimates involving lower order derivatives of the integrand, available in the work by Brass et al [4,5]. Alternatively, one can estimate the error through a contour integral, which is what we will do in the following section.…”

Section: The Gauss-legendre Quadrature Rulementioning

confidence: 99%

Error estimation for quadrature by expansion in layer potential evaluation

Klinteberg

Tornberg

2016

Adv Comput Math

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In boundary integral methods it is often necessary to evaluate layer potentials on or close to the boundary, where the underlying integral is difficult to evaluate numerically. Quadrature by expansion (QBX) is a new method for dealing with such integrals, and it is based on forming a local expansion of the layer potential close to the boundary. In doing so, one introduces a new quadrature error due to nearly singular integration in the evaluation of expansion coefficients. Using a method based on contour integration and calculus of residues, the quadrature error of nearly singular integrals can be accurately estimated. This makes it possible to derive accurate estimates for the quadrature errors related to QBX, when applied to layer potentials in two and three dimensions. As examples we derive estimates for the Laplace and Helmholtz single layer potentials. These results can be used for parameter selection in practical applications.

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Section: The Gauss-legendre Quadrature Rulementioning

confidence: 99%

Error estimation for quadrature by expansion in layer potential evaluation

Klinteberg

Tornberg

2016

Adv Comput Math

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“…Let d be the number of sample points to be used in the approximation and

s_{h}

the roots of the Hermite polynomial

Q_{d} false(s false) false(h = 1, 2, \dots, d false)

with associated weights

w_{h}

. Then by applying the Gauss–Hermite approximation to the integral in

l_{R} false(θ false)

we have

\begin{matrix} right l_{R} (boldθ) & center \approx & left ln h (\overset{π}{true}) + \sum_{i = 1}^{K} ln [\sum_{h = 1}^{d} w_{h, i} f_{y_{i}} (s_{h, i}, boldθ)] . \end{matrix}

By Theorem 5.1.9 of Brass & Petras (), for each i ,

i = 1, \dots, K

, the Gauss–Hermite approximation converges to the exact integral as

d \to \infty

.…”

Section: The Model and Parameter Estimationmentioning

confidence: 97%

Estimation of a generalized linear mixed model for response‐adaptive designs in multi‐centre clinical trials

Selvaratnam

Oyet

et al. 2017

Can J Statistics

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Response‐adaptive designs are important alternatives to equal allocation in clinical trials because equal treatment allocation has been found to have ethical issues. In this article we discuss the implementation of response‐adaptive designs in multi‐centre clinical trials. We develop a generalized linear mixed model (GLMM) for analyzing data obtained from multi‐centre clinical trials and use the maximum likelihood (ML) approach to estimate the model parameters. We apply influence function techniques to derive the asymptotic properties of our estimators. The advantage of using the influence function approach is that it leads to a closed form expression for the asymptotic covariance of the estimated parameters. To our knowledge such a closed form expression does not currently exist in the literature. The performance of the ML estimator under various response‐adaptive designs is examined through simulation studies. We use our simulation studies to compare the asymptotic covariance matrix, based on the influence function to that based on the inverse of the Hessian matrix obtained from the likelihood function of the observations. The techniques are applied to real data obtained from a multi‐centre clinical trial designed to compare two cream preparations (active drug/control) for treating an infection. The Canadian Journal of Statistics 45: 310–325; 2017 © 2017 Statistical Society of Canada

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“…For ℓ > 1 we have the nodes These rules are called Clenshaw-Curtis (CC) quadrature rules. It is well known that the CC-rules are positive rules, that is a ℓ j > 0 for all j and ℓ, see [4]. Observe that the nodes of the U ℓ are nested, since…”

Section: The Ccs Algorithmmentioning

confidence: 99%

On weak tractability of the Clenshaw–Curtis Smolyak algorithm

Hinrichs

Novak

Ullrich

2014

Journal of Approximation Theory

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We consider the problem of integration of d-variate analytic functions defined on the unit cube with directional derivatives of all orders bounded by 1. We prove that the Clenshaw Curtis Smolyak algorithm leads to weak tractability of the problem. This seems to be the first positive tractability result for the Smolyak algorithm for a normalized and unweighted problem. The space of integrands is not a tensor product space and therefore we have to develop a different proof technique. We use the polynomial exactness of the algorithm as well as an explicit bound on the operator norm of the algorithm.

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Gaussian quadrature

Cited by 8 publications

References 0 publications

Error estimation for quadrature by expansion in layer potential evaluation

Error estimation for quadrature by expansion in layer potential evaluation

Estimation of a generalized linear mixed model for response‐adaptive designs in multi‐centre clinical trials

On weak tractability of the Clenshaw–Curtis Smolyak algorithm

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