Lattice rule algorithms for multivariate approximation in the average case setting

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 > 1 and weights d,j for j = 1, 2, . . . , d. The weight d,j moderates the behavior of functions with respect to the jth variable, and small 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 by using a polynomi...

• Type: Journal
• Year: 2008
• Where: JC
• Authors: Frances Y. Kuo, Ian H. Sloan, Henryk Wozniakowski
• Keywords: Function Values, JC 2008, Multivariate Approximation, Strong Tractability