A Generalized Sampling and Preconditioning Scheme for Sparse Approximation of Polynomial Chaos Expansions
Abstract
We propose an algorithm for recovering sparse orthogonal polynomial expansions via collocation. A standard sampling approach for recovering sparse polynomials uses Monte Carlo sampling, from the density of orthogonality, which results in poor function recovery when the polynomial degree is high. Our proposed approach aims to mitigate this limitation by sampling with respect to the weighted equilibrium measure of the parametric domain and subsequently solves a preconditioned $$\ell^1$$minimization problem, where the weights of the diagonal preconditioning matrix are given by evaluations of the Christoffel function. Our algorithm can be applied to a wide class of orthogonal polynomial families on bounded and unbounded domains, including all classical families. We present theoretical analysis to motivate the algorithm and numerical results that show our method is superior to standard Monte Carlo methods in many situations of interest. In conclusion, numerical examples are also provided to demonstrate that our proposed algorithm leads to comparable or improved accuracy even when compared with Legendre and Hermitespecific algorithms.
 Authors:

 Sandia National Lab. (SNLNM), Albuquerque, NM (United States)
 Univ. of Utah, Salt Lake City, UT (United States)
 Chinese Academy of Sciences, Beijing (China)
 Publication Date:
 Research Org.:
 Sandia National Lab. (SNLNM), Albuquerque, NM (United States)
 Sponsoring Org.:
 USDOE National Nuclear Security Administration (NNSA)
 OSTI Identifier:
 1375028
 Report Number(s):
 SAND20161610J
Journal ID: ISSN 10648275; 619917
 Grant/Contract Number:
 AC0494AL85000
 Resource Type:
 Accepted Manuscript
 Journal Name:
 SIAM Journal on Scientific Computing
 Additional Journal Information:
 Journal Volume: 39; Journal Issue: 3; Journal ID: ISSN 10648275
 Publisher:
 SIAM
 Country of Publication:
 United States
 Language:
 English
 Subject:
 97 MATHEMATICS AND COMPUTING; uncertainty quantification; polynomial chaos; compressed sensing
Citation Formats
Jakeman, John D., Narayan, Akil, and Zhou, Tao. A Generalized Sampling and Preconditioning Scheme for Sparse Approximation of Polynomial Chaos Expansions. United States: N. p., 2017.
Web. doi:10.1137/16m1063885.
Jakeman, John D., Narayan, Akil, & Zhou, Tao. A Generalized Sampling and Preconditioning Scheme for Sparse Approximation of Polynomial Chaos Expansions. United States. doi:10.1137/16m1063885.
Jakeman, John D., Narayan, Akil, and Zhou, Tao. Thu .
"A Generalized Sampling and Preconditioning Scheme for Sparse Approximation of Polynomial Chaos Expansions". United States. doi:10.1137/16m1063885. https://www.osti.gov/servlets/purl/1375028.
@article{osti_1375028,
title = {A Generalized Sampling and Preconditioning Scheme for Sparse Approximation of Polynomial Chaos Expansions},
author = {Jakeman, John D. and Narayan, Akil and Zhou, Tao},
abstractNote = {We propose an algorithm for recovering sparse orthogonal polynomial expansions via collocation. A standard sampling approach for recovering sparse polynomials uses Monte Carlo sampling, from the density of orthogonality, which results in poor function recovery when the polynomial degree is high. Our proposed approach aims to mitigate this limitation by sampling with respect to the weighted equilibrium measure of the parametric domain and subsequently solves a preconditioned $\ell^1$minimization problem, where the weights of the diagonal preconditioning matrix are given by evaluations of the Christoffel function. Our algorithm can be applied to a wide class of orthogonal polynomial families on bounded and unbounded domains, including all classical families. We present theoretical analysis to motivate the algorithm and numerical results that show our method is superior to standard Monte Carlo methods in many situations of interest. In conclusion, numerical examples are also provided to demonstrate that our proposed algorithm leads to comparable or improved accuracy even when compared with Legendre and Hermitespecific algorithms.},
doi = {10.1137/16m1063885},
journal = {SIAM Journal on Scientific Computing},
number = 3,
volume = 39,
place = {United States},
year = {2017},
month = {6}
}
Web of Science