@Article{NMTMA-10-775,
author = {},
title = {Sparse Recovery via ℓ$_q$-Minimization for Polynomial Chaos Expansions},
journal = {Numerical Mathematics: Theory, Methods and Applications},
year = {2017},
volume = {10},
number = {4},
pages = {775--797},
abstract = {<p style="text-align: justify;">In this paper we consider the algorithm for recovering sparse orthogonal
polynomials using stochastic collocation via ℓ<sub>q</sub> minimization. The main results include:
1) By using the norm inequality between ℓ<sub>q</sub> and ℓ<sub>2</sub> and the square root lifting inequality, we present several theoretical estimates regarding the recoverability for both sparse
and non-sparse signals via ℓ<sub>q</sub> minimization; 2) We then combine this method with the
stochastic collocation to identify the coefficients of sparse orthogonal polynomial expansions, stemming from the field of uncertainty quantification. We obtain recoverability
results for both sparse polynomial functions and general non-sparse functions. We also
present various numerical experiments to show the performance of the ℓ<sub>q</sub> algorithm.
We first present some benchmark tests to demonstrate the ability of ℓ<sub>q</sub> minimization
to recover exactly sparse signals, and then consider three classical analytical functions
to show the advantage of this method over the standard ℓ<sub>1</sub> and reweighted ℓ<sub>1</sub> minimization. All the numerical results indicate that the ℓ<sub>q</sub> method performs better than
standard ℓ<sub>1</sub> and reweighted ℓ<sub>1</sub> minimization.</p>},
issn = {2079-7338},
doi = {https://doi.org/10.4208/nmtma.2017.0001},
url = {http://global-sci.org/intro/article_detail/nmtma/10456.html}
}