Analysis of the discrete L2 projection on polynomial spaces with random evaluations

Analysis of the discrete L2 projection on polynomial spaces with random evaluations

G. Migliorati, F. Nobile, E. von Schwerin, and R. Tempone. Analysis of the discrete L2 projection on polynomial spaces with random evaluations. Foundations of Computational Mathematics, June 2014, Volume 14, Issue 3, pp 419-456
G. Migliorati, F. Nobile, E. Von Schwerin, And R. Tempone
Approximation theory, Error analysis, Multivariate polynomial approximation, Nonparametric regression, Noise-free data, Generalized polynomial chaos, Point collocation
2014
We analyze the problem of approximating a multivariate function by discrete least-squares projection on a polynomial space starting from random, noise-free observations. An area of possible application of such technique is uncertainty quan-tification for computational models. We prove an optimal convergence estimate, up to a logarithmic factor, in the univariate case, when the observation points are sampled in a bounded domain from a probability density function bounded away from zero and bounded from above, provided the number of samples scales quadratically with the dimension of the polynomial space. Optimality is meant in the sense that the weighted L2 norm of the error committed by the random discrete projection is bounded with high probability from above by the best L∞ error achievable in the given polynomial space, up to logarithmic factors. Several numerical tests are presented in both the univariate and multivariate cases, confirming our theoretical estimates. The numerical tests also clarify how the convergence rate depends on the number of sampling points, on the polynomial degree, and on the smoothness of the target function.
DOI 10.1007/s10208-013-9186-4