On discrete least-squares projection in unbounded domain with random evaluations and its application to parametric uncertainty quantification

Tao TANG*, Tao Zhou

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

21 Citations (Scopus)

Abstract

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.

Original languageEnglish
Pages (from-to)A2272-A2295
JournalSIAM Journal of Scientific Computing
Volume36
Issue number5
DOIs
Publication statusPublished - 2014

Scopus Subject Areas

  • Computational Mathematics
  • Applied Mathematics

User-Defined Keywords

  • Hermite functions
  • Least-squares projection
  • Scaling
  • Stability
  • Unbounded domain
  • Uncertainty quantification

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