Lattice rule algorithms for multivariate approximation in the average case setting

AbstractWe 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 γd,j for j=1,2,…,d. The weight γd,j moderates the behavior of functions with respect to the jth variable, and small γd,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 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 supd⩾1∑j=1dγd,js<∞ for some positive s<1, and tractability holds iff supd⩾1∑j=1dγd,jt/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

Fast approximation by periodic kernel-based lattice-point interpolation with application in uncertainty quantification

This paper deals with the kernel-based approximation of a multivariate periodic function by interpolation at the points of an integration lattice -- a setting that, as pointed out by Zeng, Leung, Hickernell (MCQMC2004, 2006) and Zeng, Kritzer, Hickernell (Constr. Approx., 2009), allows fast evaluation by fast Fourier transform, so avoiding the need for a linear solver. The main contribution of the paper is the application to the approximation problem for uncertainty quantification of elliptic partial differential equations, with the diffusion coefficient given by a random field that is periodic in the stochastic variables, in the model proposed recently by Kaarnioja, Kuo, Sloan (SIAM J. Numer. Anal., 2020). The paper gives a full error analysis, and full details of the construction of lattices needed to ensure a good (but inevitably not optimal) rate of convergence and an error bound independent of dimension. Numerical experiments support the theory.Comment: 37 pages, 5 figure