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Title: Compressive Sampling in Multivariate Polynomial Approximation with Corrupted Simulation Samples.


Abstract not provided.

 [1];  [1]; ;  [2]
  1. (Simon Fraser University)
  2. (University of Utah)
Publication Date:
Research Org.:
Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
Sponsoring Org.:
USDOE National Nuclear Security Administration (NNSA)
OSTI Identifier:
Report Number(s):
DOE Contract Number:
Resource Type:
Resource Relation:
Conference: Proposed for presentation at the SIAM Conference on Computational Science and Engineering held February 27 - March 3, 2017 in Atlanta, GA.
Country of Publication:
United States

Citation Formats

Anyi Bao, Ben Adcock, Jakeman, John Davis, and Akil Narayan. Compressive Sampling in Multivariate Polynomial Approximation with Corrupted Simulation Samples.. United States: N. p., 2017. Web.
Anyi Bao, Ben Adcock, Jakeman, John Davis, & Akil Narayan. Compressive Sampling in Multivariate Polynomial Approximation with Corrupted Simulation Samples.. United States.
Anyi Bao, Ben Adcock, Jakeman, John Davis, and Akil Narayan. Wed . "Compressive Sampling in Multivariate Polynomial Approximation with Corrupted Simulation Samples.". United States. doi:.
title = {Compressive Sampling in Multivariate Polynomial Approximation with Corrupted Simulation Samples.},
author = {Anyi Bao and Ben Adcock and Jakeman, John Davis and Akil Narayan},
abstractNote = {Abstract not provided.},
doi = {},
journal = {},
number = ,
volume = ,
place = {United States},
year = {Wed Feb 01 00:00:00 EST 2017},
month = {Wed Feb 01 00:00:00 EST 2017}

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  • Sampling orthogonal polynomial bases via Monte Carlo is of interest for uncertainty quantification of models with random inputs, using Polynomial Chaos (PC) expansions. It is known that bounding a probabilistic parameter, referred to as coherence, yields a bound on the number of samples necessary to identify coefficients in a sparse PC expansion via solution to an ℓ{sub 1}-minimization problem. Utilizing results for orthogonal polynomials, we bound the coherence parameter for polynomials of Hermite and Legendre type under their respective natural sampling distribution. In both polynomial bases we identify an importance sampling distribution which yields a bound with weaker dependence onmore » the order of the approximation. For more general orthonormal bases, we propose the coherence-optimal sampling: a Markov Chain Monte Carlo sampling, which directly uses the basis functions under consideration to achieve a statistical optimality among all sampling schemes with identical support. We demonstrate these different sampling strategies numerically in both high-order and high-dimensional, manufactured PC expansions. In addition, the quality of each sampling method is compared in the identification of solutions to two differential equations, one with a high-dimensional random input and the other with a high-order PC expansion. In both cases, the coherence-optimal sampling scheme leads to similar or considerably improved accuracy.« less
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  • Compressive sensing is a powerful technique for recovering sparse solutions of underdetermined linear systems, which is often encountered in uncertainty quanti cation analysis of expensive and high-dimensional physical models. We perform numerical investigations employing several com- pressive sensing solvers that target the unconstrained LASSO formulation, with a focus on linear systems that arise in the construction of polynomial chaos expansions. With core solvers of l1 ls, SpaRSA, CGIST, FPC AS, and ADMM, we develop techniques to mitigate over tting through an automated selection of regularization constant based on cross-validation, and a heuristic strategy to guide the stop-sampling decision. Practical recommendationsmore » on parameter settings for these tech- niques are provided and discussed. The overall method is applied to a series of numerical examples of increasing complexity, including large eddy simulations of supersonic turbulent jet-in-cross flow involving a 24-dimensional input. Through empirical phase-transition diagrams and convergence plots, we illustrate sparse recovery performance under structures induced by polynomial chaos, accuracy and computational tradeoffs between polynomial bases of different degrees, and practi- cability of conducting compressive sensing for a realistic, high-dimensional physical application. Across test cases studied in this paper, we find ADMM to have demonstrated empirical advantages through consistent lower errors and faster computational times.« less
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  • 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 preconditionedmore » $$\ell^1$$-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.« less
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