Library Subscription: Guest
Begell Digital Portal Begell Digital Library eBooks Journals References & Proceedings Research Collections
International Journal for Uncertainty Quantification

Impact factor: 0.967

ISSN Print: 2152-5080
ISSN Online: 2152-5099

Open Access

International Journal for Uncertainty Quantification

DOI: 10.1615/Int.J.UncertaintyQuantification.2012003925
pages 279-293

STOCHASTIC COLLOCATION ALGORITHMS USING 𝓁1-MINIMIZATION

Liang Yan
Department of Mathematics, Southeast University, Nanjing, 210096, China
Ling Guo
Department of Mathematics, Purdue University, West Lafayette, Indiana 47907, USA
Dongbin Xiu
Ohio State University

ABSTRACT

The idea of 𝓁1-minimization is the basis of the widely adopted compressive sensing method for function approximation. In this paper, we extend its application to high-dimensional stochastic collocation methods. To facilitate practical implementation, we employ orthogonal polynomials, particularly Legendre polynomials, as basis functions, and focus on the cases where the dimensionality is high such that one can not afford to construct high-degree polynomial approximations. We provide theoretical analysis on the validity of the approach. The analysis also suggests that using the Chebyshev measure to precondition the 𝓁1-minimization, which has been shown to be numerically advantageous in one dimension in the literature, may in fact become less efficient in high dimensions. Numerical tests are provided to examine the performance of the methods and validate the theoretical findings.