Lattice rule algorithms for multivariate approximation in the average case setting

Authors:
Frances Y. Kuo;Ian H. Sloan;Henryk Woźniakowski
Affiliations:
School of Mathematics and Statistics, University of New South Wales, Sydney NSW 2052, Australia;School of Mathematics and Statistics, University of New South Wales, Sydney NSW 2052, Australia;Department of Computer Science, Columbia University, New York, NY 10027, USA and Institute of Applied Mathematics and Mechanics, University of Warsaw, ul. Banacha 2, 02-097 Warszawa, Poland
Venue:
Journal of Complexity
Year:
2008

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Abstract

We study multivariate approximation for continuous functions in the average case setting. The space of d variate continuous functions is equipped with the zero mean Gaussian measure whose covariance function is the reproducing kernel of a weighted Korobov space with the smoothness parameter @a1 and weights @c"d","j for j=1,2,...,d. The weight @c"d","j moderates the behavior of functions with respect to the jth variable, and small @c"d","j means that functions depend weakly on the jth variable. We study lattice rule algorithms which approximate the Fourier coefficients of a function based on function values at lattice sample points. The generating vector for these lattice points is constructed by the component-by-component algorithm, and it is tailored for the approximation problem. Our main interest is when d is large, and we study tractability and strong tractability of multivariate approximation. That is, we want to reduce the initial average case error by a factor @e by using a polynomial number of function values in @e^-^1 and d in the case of tractability, and only polynomial in @e^-^1 in the case of strong tractability. Necessary and sufficient conditions on tractability and strong tractability are obtained by applying known general tractability results for the class of arbitrary linear functionals and for the class of function values. Strong tractability holds for the two classes in the average case setting iff sup"d"="1@?"j"="1^d@c"d","j^s="1@?"j"="1^d@c"d","j^t/log(d+1)