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Title: 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 Hermite-specific algorithms.

Authors:
 [1];  [2];  [3]
  1. Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
  2. Univ. of Utah, Salt Lake City, UT (United States)
  3. Chinese Academy of Sciences, Beijing (China)
Publication Date:
Research Org.:
Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
Sponsoring Org.:
USDOE National Nuclear Security Administration (NNSA)
OSTI Identifier:
1375028
Report Number(s):
SAND-2016-1610J
Journal ID: ISSN 1064-8275; 619917
Grant/Contract Number:  
AC04-94AL85000
Resource Type:
Accepted Manuscript
Journal Name:
SIAM Journal on Scientific Computing
Additional Journal Information:
Journal Volume: 39; Journal Issue: 3; Journal ID: ISSN 1064-8275
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. https://doi.org/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. https://doi.org/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. https://doi.org/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 Hermite-specific algorithms.},
doi = {10.1137/16m1063885},
journal = {SIAM Journal on Scientific Computing},
number = 3,
volume = 39,
place = {United States},
year = {2017},
month = {6}
}

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