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Optimal stochastic quadrature formulas for convex functions
BIT, 1994Optimal stochastic (Monte Carlo) quadrature formulas for defined classes of convex functions are studied. Specifically, non-adaptive Monte Carlo methods are seen to be no better than deterministic methods, but adaptive Monte Carlo methods are shown to exhibit a superior performance.
Novak, E., Petras, K.
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An optimal quadrature formula in the Sobolev space
Uzbek Mathematical Journal, 2021This paper studies the problem of construction of optimal quadrature formulas for approximate calculation of integrals with trigonometric weight in the L(2m)(0, 1) space for any ω = 0, ω ∈ R. Here explicit formulas for the optimal coefficients are obtained. We study the order of convergence of the optimal formulas for the case m = 1, 2.
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The Coefficients of Optimal Quadrature Formulas
20062 (R) are studied by means of a variational method. Here w(x) is a weight function, χΩ(x) is the characteristic function of the interval Ω, and c(β) are the coefficients of the quadrature formula. The results generalize some results by A. Sard, L. F. Meyers, I. J. Schoenberg, S. D. Silliman (1-4), and others derived by the method of splines.
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Optimal quadrature formulas for a sphere
USSR Computational Mathematics and Mathematical Physics, 1977Abstract The problem of finding the statistically optimal quadrature formula for a sphere is posed. A system of linear algebraic equations satisfied by weights of the quadrature formula is written down. Two examples are given. Asymptotic estimates of the relative error are given.
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Optimal quadrature formulae for differentiable functions
Calcolo, 1986The existence of an optimal quadrature formula of the form \(\int^{b}_{a}f| x| dx\approx \sum^{n}_{k=1}\sum^{\nu_ k-1}_{\lambda =0}a_{k\lambda}f^{(\lambda)}(xk)\) with preassigned multiplicities \((\nu_ k)_ 1^ n\) in the classes \(LW_ q^ r:=\{f\in C^{(r-1)}:f^{(r-1)}\)- abs.
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An optimal quadrature formula in K2(P3) space
AIP Conference Proceedings, 2021In this paper in K2(P3) Hilbert space the problem of construction of optimal quadrature formula in the sense of Sard is considered and using S.L. Sobolev’s method a new optimal quadrature formula is obtained. For the optimal coefficients explicit formulas are given.
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Robust Structural Optimization Using Gauss-type Quadrature Formula
Transactions of the Korean Society of Mechanical Engineers A, 2009In robust design, the mean and variance of design performance are frequently used to measure the design performance and its robustness under uncertainties. In this paper, we present the Gauss-type quadrature formula as a rigorous method for mean and variance estimation involving arbitrary input distributions and further extend its use to robust design ...
Sang-Hoon Lee +3 more
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Asymptotic Properties of Optimal Quadrature Formulas
1979In [9], Sard introduced a notion of “best” for quadrature formulas which may be described as follows. Let 0 = t0 < t1 < ... < tN = 1 be fixed points, and consider the formula $${Q_N}(f) \equiv \sum\limits_{i = 0}^N {{c_i}f({t_i}) \simeq \int_0^1 f (\tau )} d\tau \equiv I(f)$$ (1) .
David L. Barrow, Philip W. Smith
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A numerical method to obtain optimal quadrature formulas
International Journal for Numerical Methods in Engineering, 1978AbstractThis paper present a numerical method to obtain optimal quadrature formulas of Gauss type and Radau type in the sense of Sard. Using the relation between optimal quadrature formulas and nonospline functions, the optimal quadrature formula can be obtained by solving a set of no‐linear simultaneous algebraic equations induced from the ...
Wang, A. H., Klein, R. L.
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On optimization of weight quadrature formulas
Ukrainian Mathematical Journal, 1995We obtain asymptotically optimal quadrature formulas on the classH ω [-1, 1] for an arbitrary continuous weight function which is positive on [-1, 1] almost everywhere and for a wide class of moduli of continuity ω(t).
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