A Generalized Sampling and Preconditioning Scheme for Sparse Approximation of Polynomial Chaos Expansions

Jakeman, John D.; Narayan, Akil; Zhou, Tao

doi:10.1137/16m1063885

Title: A Generalized Sampling and Preconditioning Scheme for Sparse Approximation of Polynomial Chaos Expansions

Journal Article · Thu Jun 22 00:00:00 EDT 2017 · SIAM Journal on Scientific Computing

DOI:https://doi.org/10.1137/16m1063885· OSTI ID:1375028

Jakeman, John D. ^[1]; Narayan, Akil ^[2]; Zhou, Tao ^[3]

Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
Univ. of Utah, Salt Lake City, UT (United States)
Chinese Academy of Sciences, Beijing (China)

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.

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Research Organization:: Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)

Sponsoring Organization:: USDOE National Nuclear Security Administration (NNSA)

Grant/Contract Number:: AC04-94AL85000

OSTI ID:: 1375028

Report Number(s):: SAND-2016-1610J; 619917

Journal Information:: SIAM Journal on Scientific Computing, Vol. 39, Issue 3; ISSN 1064-8275

Publisher:: SIAMCopyright Statement

Country of Publication:: United States

Language:: English

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Cited by: 42 works

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