Modern Perspectives in Applied Mathematics: Theory and Numerics of PDEs

Hermite spectral methods and discrete least square projection with random evaluations

Tao Tang

Hong Kong Baptist University

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.