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:
-
- Sandia National Lab. (SNL-NM), 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. (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. 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. 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 = {Thu Jun 22 00:00:00 EDT 2017},
month = {Thu Jun 22 00:00:00 EDT 2017}
}
Web of Science
Works referenced in this record:
A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data
journal, January 2007
- Babuška, Ivo; Nobile, Fabio; Tempone, Raúl
- SIAM Journal on Numerical Analysis, Vol. 45, Issue 3
Fekete points and convergence towards equilibrium measures on complex manifolds
journal, January 2011
- Berman, Robert; Boucksom, Sébastien; Nyström, David Witt
- Acta Mathematica, Vol. 207, Issue 1
Bergman kernels for weighted polynomials and weighted equilibrium measures of $\mathbb{C}^{n}$
journal, January 2009
- Berman, Robert J.
- Indiana University Mathematics Journal, Vol. 58, Issue 4
Sparse high order FEM for elliptic sPDEs
journal, March 2009
- Bieri, Marcel; Schwab, Christoph
- Computer Methods in Applied Mechanics and Engineering, Vol. 198, Issue 13-14
Adaptive sparse polynomial chaos expansion based on least angle regression
journal, March 2011
- Blatman, Géraud; Sudret, Bruno
- Journal of Computational Physics, Vol. 230, Issue 6
An Orthogonality Property of the Legendre Polynomials
journal, January 2016
- Bos, L.; Narayan, A.; Levenberg, N.
- Constructive Approximation, Vol. 45, Issue 1
Global sensitivity analysis using sparse grid interpolation and polynomial chaos
journal, November 2012
- Buzzard, Gregery T.
- Reliability Engineering & System Safety, Vol. 107
Decoding by Linear Programming
journal, December 2005
- Candes, E. J.; Tao, T.
- IEEE Transactions on Information Theory, Vol. 51, Issue 12
Near-Optimal Signal Recovery From Random Projections: Universal Encoding Strategies?
journal, January 2006
- Candes, Emmanuel J.; Tao, Terence
- IEEE Transactions on Information Theory, Vol. 52, Issue 12, p. 5406-5425
Stable signal recovery from incomplete and inaccurate measurements
journal, January 2006
- Candès, Emmanuel J.; Romberg, Justin K.; Tao, Terence
- Communications on Pure and Applied Mathematics, Vol. 59, Issue 8, p. 1207-1223
Atomic Decomposition by Basis Pursuit
journal, January 2001
- Chen, Scott Shaobing; Donoho, David L.; Saunders, Michael A.
- SIAM Review, Vol. 43, Issue 1
Adaptive Smolyak Pseudospectral Approximations
journal, January 2013
- Conrad, Patrick R.; Marzouk, Youssef M.
- SIAM Journal on Scientific Computing, Vol. 35, Issue 6
Compressed sensing
journal, April 2006
- Donoho, D. L.
- IEEE Transactions on Information Theory, Vol. 52, Issue 4
Stable recovery of sparse overcomplete representations in the presence of noise
journal, January 2006
- Donoho, D. L.; Elad, M.; Temlyakov, V. N.
- IEEE Transactions on Information Theory, Vol. 52, Issue 1
A non-adapted sparse approximation of PDEs with stochastic inputs
journal, April 2011
- Doostan, Alireza; Owhadi, Houman
- Journal of Computational Physics, Vol. 230, Issue 8
Least angle regression
journal, April 2004
- Tibshirani, Robert; Johnstone, Iain; Hastie, Trevor
- The Annals of Statistics, Vol. 32, Issue 2
On the convergence of generalized polynomial chaos expansions
journal, October 2011
- Ernst, Oliver G.; Mugler, Antje; Starkloff, Hans-Jörg
- ESAIM: Mathematical Modelling and Numerical Analysis, Vol. 46, Issue 2
Stochastic Collocation Methods via $\ell_1$ Minimization Using Randomized Quadratures
journal, January 2017
- Guo, Ling; Narayan, Akil; Zhou, Tao
- SIAM Journal on Scientific Computing, Vol. 39, Issue 1
Compressive sampling of polynomial chaos expansions: Convergence analysis and sampling strategies
journal, January 2015
- Hampton, Jerrad; Doostan, Alireza
- Journal of Computational Physics, Vol. 280
Enhancing adaptive sparse grid approximations and improving refinement strategies using adjoint-based a posteriori error estimates
journal, January 2015
- Jakeman, J. D.; Wildey, T.
- Journal of Computational Physics, Vol. 280
Enhancing -minimization estimates of polynomial chaos expansions using basis selection
journal, May 2015
- Jakeman, J. D.; Eldred, M. S.; Sargsyan, K.
- Journal of Computational Physics, Vol. 289
Orthonormal polynomials with generalized Freud-type weights
journal, March 2003
- Kasuga, T.; Sakai, R.
- Journal of Approximation Theory, Vol. 121, Issue 1
Orthogonal polynomials for exponential weights x 2 ρ e - 2 Q ( x ) on [ 0 , d )
journal, June 2005
- Levin, Eli; Lubinsky, Doron
- Journal of Approximation Theory, Vol. 134, Issue 2
Orthogonal polynomials for exponential weights x 2 ρ e - 2 Q ( x ) on [ 0 , d ) , II
journal, March 2006
- Levin, Eli; Lubinsky, Doron
- Journal of Approximation Theory, Vol. 139, Issue 1-2
Where does the sup norm of a weighted polynomial live?: A generalization of incomplete polynomials
journal, December 1985
- Mhaskar, H. N.; Saff, E. B.
- Constructive Approximation, Vol. 1, Issue 1
Approximation of Quantities of Interest in Stochastic PDEs by the Random Discrete $L^2$ Projection on Polynomial Spaces
journal, January 2013
- Migliorati, G.; Nobile, F.; von Schwerin, E.
- SIAM Journal on Scientific Computing, Vol. 35, Issue 3
Adaptive Leja Sparse Grid Constructions for Stochastic Collocation and High-Dimensional Approximation
journal, January 2014
- Narayan, Akil; Jakeman, John D.
- SIAM Journal on Scientific Computing, Vol. 36, Issue 6
A Christoffel function weighted least squares algorithm for collocation approximations
journal, November 2016
- Narayan, Akil; Jakeman, John D.; Zhou, Tao
- Mathematics of Computation, Vol. 86, Issue 306
Stochastic Collocation Methods on Unstructured Grids in High Dimensions via Interpolation
journal, January 2012
- Narayan, Akil; Xiu, Dongbin
- SIAM Journal on Scientific Computing, Vol. 34, Issue 3
Generalized Jacobi Weights, Christoffel Functions, and Jacobi Polynomials
journal, March 1994
- Nevai, Paul; Erdélyi, Tamás; Magnus, Alphonse P.
- SIAM Journal on Mathematical Analysis, Vol. 25, Issue 2
A weighted -minimization approach for sparse polynomial chaos expansions
journal, June 2014
- Peng, Ji; Hampton, Jerrad; Doostan, Alireza
- Journal of Computational Physics, Vol. 267
On Asymptotic Properties of Polynomials Orthogonal on the real axis
journal, February 1984
- Rakhmanov, E. A.
- Mathematics of the USSR-Sbornik, Vol. 47, Issue 1
Sparse Legendre expansions via <mml:math altimg="si12.gif" display="inline" overflow="scroll" xmlns:xocs="http://www.elsevier.com/xml/xocs/dtd" xmlns:xs="http://www.w3.org/2001/XMLSchema" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns="http://www.elsevier.com/xml/ja/dtd" xmlns:ja="http://www.elsevier.com/xml/ja/dtd" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:tb="http://www.elsevier.com/xml/common/table/dtd" xmlns:sb="http://www.elsevier.com/xml/common/struct-bib/dtd" xmlns:ce="http://www.elsevier.com/xml/common/dtd" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:cals="http://www.elsevier.com/xml/common/cals/dtd"><mml:msub><mml:mrow><mml:mi>ℓ</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:math>-minimization
journal, May 2012
- Rauhut, Holger; Ward, Rachel
- Journal of Approximation Theory, Vol. 164, Issue 5
Subsampled Gauss Quadrature Nodes for Estimating Polynomial Chaos Expansions
journal, January 2014
- Tang, Gary; Iaccarino, Gianluca
- SIAM/ASA Journal on Uncertainty Quantification, Vol. 2, Issue 1
On Discrete Least-Squares Projection in Unbounded Domain with Random Evaluations and its Application to Parametric Uncertainty Quantification
journal, January 2014
- Tang, Tao; Zhou, Tao
- SIAM Journal on Scientific Computing, Vol. 36, Issue 5
High-Order Collocation Methods for Differential Equations with Random Inputs
journal, January 2005
- Xiu, Dongbin; Hesthaven, Jan S.
- SIAM Journal on Scientific Computing, Vol. 27, Issue 3
The Wiener--Askey Polynomial Chaos for Stochastic Differential Equations
journal, January 2002
- Xiu, Dongbin; Karniadakis, George Em
- SIAM Journal on Scientific Computing, Vol. 24, Issue 2
STOCHASTIC COLLOCATION ALGORITHMS USING l1-MINIMIZATION
journal, January 2012
- Yan, Liang; Guo, Ling; Xiu, Dongbin
- International Journal for Uncertainty Quantification, Vol. 2, Issue 3
Reweighted minimization method for stochastic elliptic differential equations
journal, September 2013
- Yang, Xiu; Karniadakis, George Em
- Journal of Computational Physics, Vol. 248
Generalized Jacobi weights, Christoffel functions, and zeros of orthogonal polynomials
journal, May 1992
- Erdélyi, Tamás; Nevai, Paul
- Journal of Approximation Theory, Vol. 69, Issue 2
Coherence motivated sampling and convergence analysis of least squares polynomial Chaos regression
journal, June 2015
- Hampton, Jerrad; Doostan, Alireza
- Computer Methods in Applied Mechanics and Engineering, Vol. 290
Adaptive sparse polynomial chaos expansion based on least angle regression
journal, March 2011
- Blatman, Géraud; Sudret, Bruno
- Journal of Computational Physics, Vol. 230, Issue 6
Compressive sampling of polynomial chaos expansions: Convergence analysis and sampling strategies
journal, January 2015
- Hampton, Jerrad; Doostan, Alireza
- Journal of Computational Physics, Vol. 280
High-Order Collocation Methods for Differential Equations with Random Inputs
journal, January 2005
- Xiu, Dongbin; Hesthaven, Jan S.
- SIAM Journal on Scientific Computing, Vol. 27, Issue 3
Enhancing adaptive sparse grid approximations and improving refinement strategies using adjoint-based a posteriori error estimates
text, January 2014
- Jakeman, John D.; Wildey, Timothy
- arXiv
Works referencing / citing this record:
Concurrent surrogate model selection (COSMOS): optimizing model type, kernel function, and hyper-parameters
journal, September 2017
- Mehmani, Ali; Chowdhury, Souma; Meinrenken, Christoph
- Structural and Multidisciplinary Optimization, Vol. 57, Issue 3
A mixed ℓ 1 regularization approach for sparse simultaneous approximation of parameterized PDEs
journal, November 2019
- Dexter, Nick; Tran, Hoang; Webster, Clayton
- ESAIM: Mathematical Modelling and Numerical Analysis, Vol. 53, Issue 6
Compressed sensing with sparse corruptions: Fault-tolerant sparse collocation approximations
text, January 2017
- Adcock, Ben; Bao, Anyi; Jakeman, John D.
- arXiv
Compressed sensing approaches for polynomial approximation of high-dimensional functions
preprint, January 2017
- Adcock, Ben; Brugiapaglia, Simone; Webster, Clayton G.
- arXiv
Computation of Induced Orthogonal Polynomial Distributions
preprint, January 2017
- Narayan, Akil
- arXiv