TY - JOUR

T1 - On Discrete Least-Squares Projection in Unbounded Domain with Random Evaluations and its Application to Parametric Uncertainty Quantification

AU - Tang, Tao

AU - Zhou, Tao

N1 - Funding information:
t Department of Mathematics, The Hong Kong Baptist University, Kowloon Tong, Kowloon, Hong Kong, China ([email protected]). This author’s work was supported by Hong Kong Research Grants Council (RGC), Hong Kong Baptist University, and an NSFC-RGC joint research grant.
^ Institute of Computational Mathematics and Scientific/Engineering Computing, AMSS, The Chinese Academy of Sciences, Beijing, China ([email protected]). This author’s work was supported by the National Natural Science Foundation of China (Grants 91130003 and 11201461).
Publisher copyright:
© 2014, Society for Industrial and Applied Mathematics

PY - 2014/9/25

Y1 - 2014/9/25

N2 - This work is concerned with approximating multivariate functions in an unbounded domain by using a discrete least-squares projection with random point evaluations. Particular attention is given to functions with random Gaussian or gamma parameters. We first demonstrate that the traditional Hermite (Laguerre) polynomials chaos expansion suffers from the instability in the sense that an unfeasible 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/Laguerre functions (rather than polynomials) as bases in the expansion. The corresponding design points are obtained by mapping the uniformly distributed random points in bounded intervals to the unbounded domain, which involved a mapping parameter L. By using the Hermite/Laguerre functions and a proper mapping parameter, the stability can be significantly improved even if the number of design points scales linearly (up to a logarithmic factor) with the dimension of the approximation space. Apart from the stability, another important issue is the rate of convergence. To speed up the convergence, an effective scaling factor is introduced, and a principle for choosing quasi-optimal scaling factor is discussed. Applications to parametric uncertainty quantification are illustrated by considering a random ODE model together with an elliptic problem with lognormal random input.

AB - This work is concerned with approximating multivariate functions in an unbounded domain by using a discrete least-squares projection with random point evaluations. Particular attention is given to functions with random Gaussian or gamma parameters. We first demonstrate that the traditional Hermite (Laguerre) polynomials chaos expansion suffers from the instability in the sense that an unfeasible 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/Laguerre functions (rather than polynomials) as bases in the expansion. The corresponding design points are obtained by mapping the uniformly distributed random points in bounded intervals to the unbounded domain, which involved a mapping parameter L. By using the Hermite/Laguerre functions and a proper mapping parameter, the stability can be significantly improved even if the number of design points scales linearly (up to a logarithmic factor) with the dimension of the approximation space. Apart from the stability, another important issue is the rate of convergence. To speed up the convergence, an effective scaling factor is introduced, and a principle for choosing quasi-optimal scaling factor is discussed. Applications to parametric uncertainty quantification are illustrated by considering a random ODE model together with an elliptic problem with lognormal random input.

KW - Hermite functions

KW - Least-squares projection

KW - Scaling

KW - Stability

KW - Unbounded domain

KW - Uncertainty quantification

UR - http://www.scopus.com/inward/record.url?scp=84911423308&partnerID=8YFLogxK

U2 - 10.1137/140961894

DO - 10.1137/140961894

M3 - Journal article

AN - SCOPUS:84911423308

SN - 1064-8275

VL - 36

SP - A2272-A2295

JO - SIAM Journal on Scientific Computing

JF - SIAM Journal on Scientific Computing

IS - 5

ER -