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Sparse Recovery via ℓq-Minimization for Polynomial Chaos Expansions

Part of: Numerical approximation and computational geometry Approximations and expansions Nontrigonometric harmonic analysis

Published online by Cambridge University Press:  12 September 2017

Ling Guo ,
Yongle Liu  and
Liang Yan
Ling Guo*
Affiliation:
Department of Mathematics, Shanghai Normal University, Shanghai, China
Yongle Liu*
Affiliation:
Department of Mathematics, Shanghai Normal University, Shanghai, China
Liang Yan*
Affiliation:
School of Mathematics, Southeast University, Nanjing, 210096, China
*
*Corresponding author. Email addresses:[email protected] (L. Guo), [email protected] (Y. Liu), [email protected] (L. Yan)
*Corresponding author. Email addresses:[email protected] (L. Guo), [email protected] (Y. Liu), [email protected] (L. Yan)
*Corresponding author. Email addresses:[email protected] (L. Guo), [email protected] (Y. Liu), [email protected] (L. Yan)
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Abstract

In this paper we consider the algorithm for recovering sparse orthogonal polynomials using stochastic collocation via ℓq minimization. The main results include: 1) By using the norm inequality between ℓq and ℓ2 and the square root lifting inequality, we present several theoretical estimates regarding the recoverability for both sparse and non-sparse signals via ℓq minimization; 2) We then combine this method with the stochastic collocation to identify the coefficients of sparse orthogonal polynomial expansions, stemming from the field of uncertainty quantification. We obtain recoverability results for both sparse polynomial functions and general non-sparse functions. We also present various numerical experiments to show the performance of the ℓq algorithm. We first present some benchmark tests to demonstrate the ability of ℓq minimization to recover exactly sparse signals, and then consider three classical analytical functions to show the advantage of this method over the standard ℓ1 and reweighted ℓ1 minimization. All the numerical results indicate that the ℓq method performs better than standard ℓ1 and reweighted ℓ1 minimization.

Keywords

Uncertainty quantificationstochastic collocationℓq-minimizationpolynomial chaos expansions

MSC classification

Secondary: 65D05: Interpolation 42C05: Orthogonal functions and polynomials, general theory 41A10: Approximation by polynomials
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 10 , Issue 4 , November 2017 , pp. 775 - 797
Copyright
Copyright © Global-Science Press 2017 

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References

[1] Babuska, I., Nobile, F. and Tempone, R., A stochastic collocation method for elliptic partial differential equations with random input data, SIAM Review, 52(2) (2010), pp. 317–355.CrossRefGoogle Scholar
[2] Van Den Berg, E. and Friedlander, M., Spgl1: A solver for large-scale sparse reconstruction, http://www.cs.ubc.ca/labs/scl/spgl, 2007.Google Scholar
[3] Cai, T., Wang, L. and Xu, G., New bounds for restricted isometry constants, IEEE T. Inform. Theory, 56 (2010), pp. 4388–4394.CrossRefGoogle Scholar
[4] Cai, T., Wang, L. and Xu, G., Shifting inequality and recovery of sparse signals, IEEE T. Signal Proces., 58 (2010), pp. 1300–1308.CrossRefGoogle Scholar
[5] Candès, E., Rudelson, M., Tao, T. and Vershynin, R., Error correction via linear programming, in Proceeding of the 46th Annual IEEE Symposium on Foundations of Computer Science, 2005, pp. 668–681.CrossRefGoogle Scholar
[6] Candès, E. J., The restricted isometry property and its implications for compressed sensing, C. R. Acad. Sci. Paris Sér. I Math., 346 (2008), pp. 589–592.CrossRefGoogle Scholar
[7] Candès, E. J., Romberg, J. and Tao, T., Stable signal recovery from incomplete and inaccurate measurements, Comm. Pure Appl. Math., 56 (2006), pp. 1207–1223.CrossRefGoogle Scholar
[8] Candès, E. J. and Tao, T., Decoding by linear programming, IEEE T. Inform. Theory, 51 (2005), pp. 4203–4215.Google Scholar
[9] Davies, M. E. and Gribonval, R., Restricted isometry constants where ℓp sparse recovery can fail for 0 < p ≤ 1, IEEE T. Inform. Theory, 55 (2010), pp. 2203–2214.CrossRefGoogle Scholar
[10] Donoho, D. L., Compressed sensing, IEEE T. Inform. Theory, 52 (2006), pp. 1289–1306.CrossRefGoogle Scholar
[11] Doostan, A. and Owhadi, H., A non-adapted sparse approximation of pdes with stochastic inputs, J. Comput. Phys, 230 (2011), pp. 3015–3034.CrossRefGoogle Scholar
[12] Ernst, O. G., Mugler, A., Starkloff, H. J. and Ullmann, E., On the convergence of generalized polynomial chaos expansions, ESAIM: Mathematical Modelling and Numerical Analysis, 46(2) (2012), pp. 317–339.CrossRefGoogle Scholar
[13] Foucart, S. and Lai, M. J., Sparsest solutions of undetermined linear systems via ℓq-minimization for 0 < q ≤ 1, Appl. Comput. Harmon. Anal., 26 (2009), pp. 395–407.CrossRefGoogle Scholar
[14] Gao, Z. and Zhou, T., On the choice of design points for least square polynomial approximations with application to uncertainty quantification, Commun. Comput. Phys., 16 (2014), pp. 365–381.CrossRefGoogle Scholar
[15] Guo, L., Narayan, A., Zhou, T. and Chen, Y., Stochastic collocation methods via ℓ1 minimization using randomized quadratures, SIAM J. Sci. Comput., 39-1 (2017), pp. A333–A359.CrossRefGoogle Scholar
[16] Hsia, Y. and Sheu, R. L., On RIC Bounds of Compressed Sensing Matrices for Approximating Sparse Solutions Using ℓq Quasi Norms, Mathematics, 2013.Google Scholar
[17] Jakeman, J., Narayan, A. and Zhou, T., A generalized sampling and preconditioning scheme for sparse approximation of polynomial chaos expansions, SIAM J. Sci. Comput., 39-3 (2017), pp. A1114–A1144.Google Scholar
[18] Lai, M. J. and Wang, J., An unconstrained ℓq minimization for sparse solution under determined linear systems, SIAM J. Optimization, 21 (2011), pp. 82–101.CrossRefGoogle Scholar
[19] Liu, Y. L. and Guo, L., Stochastic collocation via l1-minimisation on low discrepancy point sets with application to uncertainty quantification, East Asian J. Appl. Math., 6 (2016), pp. 171–191.Google Scholar
[20] Liu, W. H., Gong, D. and Xu, Z., One-Bit compressed sensing by greedy algorithms, Numer. Math. Theor. Meth. Appl., 9 (2016), pp. 169–184.CrossRefGoogle Scholar
[21] Mathelin, L. and Gallivan, K. A., A compressed sensing approach for partial differential equa- tions with random input data, Commun. Comput. Phys., 12 (2012), pp. 919–954.CrossRefGoogle Scholar
[22] Narayan, A. and Zhou, T., Stochastic collocation methods on unstructured meshes, Commun. Comput. Phys., 18 (2015), pp. 1–36.CrossRefGoogle Scholar
[23] Rauhut, H. and Ward, R., Sparse legendre expansions via ℓ1-minimization, J. Approx. Theory, 164 (2012), pp. 517–533.Google Scholar
[24] Shu, R. W., Jin, J. W. and Jin, S., A stochastic Galerkin method for the Boltzmann equation with multi-dimensional random inputs using sparse Wavelet bases, Numer. Math. Theor. Meth. Appl., 10(2) (2017), pp. 465–488.CrossRefGoogle Scholar
[25] Song, C. B. and Xia, S. T., Sparse signal recovery by ℓq minimization under restricted isometry property, IEEE Signal Proc. Let., 21(9) (2014), pp. 1154–1158.CrossRefGoogle Scholar
[26] Tang, T. and Zhou, T., Recent developments in high order numerical methods for uncertainty quantification, Science in China: Mathematics, 45(7) (2015), pp. 891–928.Google Scholar
[27] Xiu, D., Efficient collocational approach for parametric uncertainty analysis, Commun. Comput. Phys., 2 (2007), pp. 293–309.Google Scholar
[28] Xiu, D., Fast numerical methods for stochastic computations: a review, Commun. Comput. Phys., 5 (2009), pp. 242–272.Google Scholar
[29] Xiu, D. and Karniadakis, G. E., The wiener-askey polynomial chaos for stochastic differential equations, SIAM J. Sci. Comput., 24 (2002), pp. 619–644.CrossRefGoogle Scholar
[30] Xu, Z. and Zhou, T., On sparse interpolation and the design of deterministic interpolation points, SIAM J. Sci. Comput., 36 (2014), pp. A1752–A1769.CrossRefGoogle Scholar
[31] Yan, L., Guo, L. and Xiu, D., Stochastic collocation algorithms using ℓ1 minimization, Int. J. Uncertain Quantification, 2 (2012), pp. 279–293.CrossRefGoogle Scholar
[32] Yan, L., Shin, Y. and Xiu, D., Sparse approximation using ℓ1–ℓ2 minimization and its applications to stochastic collocation, SIAM J. Sci. Comput., 39(1) (2017), pp. A229–A254.CrossRefGoogle Scholar