Lattice rule algorithms for multivariate approximation in the average case setting

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 α > 1 and weights γ d , j for j = 1 , 2 , … , d . The weight γ d , j moderates the behavior of functions with respect to the j th variable, and small γ d , j means that functions depend weakly on the j th 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 ε by using a polynomial number of function values in ε - 1 and d in the case of tractability, and only polynomial in ε - 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 γ d , j s < ∞ for some positive s < 1 , and tractability holds iff sup d ⩾ 1 ∑ j = 1 d γ d , j t / log ( d + 1 ) < ∞ for some positive t < 1 . . The previous results for the class of function values have been non-constructive. We provide a construction in this paper and prove tractability and strong tractability error bounds for lattice rule algorithms. This paper can be viewed as a continuation of our previous paper where we studied multivariate approximation for weighted Korobov spaces in the worst case setting. Many technical results from that paper are also useful for the average case setting. The exponents of ε - 1 and d corresponding to our error bounds are not sharp. However, for α close to 1 and for slow decaying weights, we obtain almost the minimal exponent of ε - 1 . We also compare the results from the worst case and the average case settings in weighted Korobov spaces.