Abstract:
This talk is concerned with approximating multivariate functions in
unbounded domains by using discrete least-square projection with random
points evaluations. Particular attentions will be given to
functions with random Gaussian parameters. We first demonstrate that the
traditional Hermite polynomials chaos expansion suffers from the
instability in the sense that an unapplicable
number of points, which is relevant to the dimension of the approximation
space, is needed to guarantee the stability in the least squares
framework. We then propose to use the Hermite functions
(rather than polynomials) as bases in the expansion. Approximating a
function by Hermite polynomials or functions was rejected by
Gottlieb-Orszag in their 1977 classical book due
to poor resolution properties.
This difficulty will also be seen in the discrete least-square projection
setting. We will propose several ways to
improve the stability and speed up the rate of convergence. |