A generalized discrepancy and quadrature error bound

By Fred J. Hickernell. 1998

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Citations in the corpus

Convergence guarantees for kernel-based quadrature rules in misspecified settings (2016, methodology): “Specifically, we derive convergence rates that depend on the (unknown) lesser smoothness of the integrand, where the degree of smoothness is expressed via powers of RKHSs or via Sobolev spaces.”
Uniform Design: Theory and Application (2000, methodology): “For the centered L2 discrepancy, Hickernell (1998) gave an analytical ex- pression similar to (4.1) as follows: (cL2(p,))2 = - ; Ix/g - ,512) + $2 2 fi [1+ ; (.7&i - .51 k=l j=l i=l 1 - .5/ - ; /xki - x:ji 11 .”
Efficient Nearly Orthogonal and Space-Filling Latin Hypercubes (2007, unknown): “(3), where designs are normalized to [O, 1] in each dimension], is often used instead (Hickernell 1998; Fang, Ma, and Winker 2000b).”
Space-Filling Fractional Factorial Designs (2014, methodology): “Hickernell (1998) used the tool of reproducing kernel Hilbert spaces to define several discrepancies, such as the centered L2-discrepancy (CD) and the wrap-around L2-discrepancy (WD).”
Gaussian Processes and Kernel Methods: A Review on Connections and Equivalences (2018, methodology): “We then discuss the problem of sampling or numerical integration, for which GPs and kernel methods have also played fundamental roles in the form of integral transforms [O’Hagan, 1991, Hickernell, 1998, Briol et al., 2018, Dick et al., 2013].”
Tractability of Multivariate Problems (2008, methodology): “Essentially the same formulas were presented by Hickernell [119], who considered spaces similar to Fd without assuming the condition f( 1 2 ) = 0 and considered algorithms with aj = n −1.”
Minimax and maximin space-filling designs: some properties and methods for construction (2017, unknown): “Along the same vein, discrepancy (see Section 2.5.1) relies on volumes rather than distances and corresponds to kernels that favour functions that can be modelled well by sums of few factors interactions, see Hickernell (1998a,b).”
Quasi-Regression and the Relative Importance of the ANOVA Components of a Function (2002, unknown): “Using ̂2 r;b , estimate 2 I;d;m = Pr2R(I;d;m) 2 r 2 I without bias as follows: ̂2 I;d;m = X r2R(I;d;m) ̂2 r;b = n n 1 X r2R(I;d;m) ̂2 r 1 n2 n Xk=1S2 I;d;m(uk)f2(uk)!; (10) where S2 I;d;m(u) = X r2R(I;d;m) 2r(u): It is now possible to onstru t an approximation for the lower bound d;m(p) = 1 2 X ”
Comparison of Methods for the Computation of Multivariate t Probabilities (2002, methodology): “Hickernell (2001) has recently shown that this type of periodization combined with the randomization provides 0(p-2+?) asymptotic errors.”
A note on generalized aberration in factorial designs (2001, background): “The following analytical expression of the WD can be easily derived (Hickernell (1998b)).”
An Efficient Algorithm for Constructing Optimal Design of Computer Experiments (2005, methodology): “Hickernell (1998) proposed three formulas of L2 discrepancy, among which the centered L2-discrepancy (CL2) seems the most interesting.”
Digital Nets and Sequences: Discrepancy Theory and Quasi-Monte Carlo Integration (2010, methodology): “In the following we use the approach of Hickernell [99] and Sloan & Woźniakowski [247].”
Exploring multi-dimensional spaces: a Comparison of Latin Hypercube and Quasi Monte Carlo Sampling Techniques (2015, background): “Hickernell [26] pointed out that the discrepancy (7.”