Skip to main content Accessibility help

Sparse Recovery via q -Minimization for Polynomial Chaos Expansions

  • Ling Guo (a1), Yongle Liu (a1) and Liang Yan (a2)


In this paper we consider the algorithm for recovering sparse orthogonal polynomials using stochastic collocation via q minimization. The main results include: 1) By using the norm inequality between q and 2 and the square root lifting inequality, we present several theoretical estimates regarding the recoverability for both sparse and non-sparse signals via q 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 q algorithm. We first present some benchmark tests to demonstrate the ability of q minimization to recover exactly sparse signals, and then consider three classical analytical functions to show the advantage of this method over the standard 1 and reweighted 1 minimization. All the numerical results indicate that the q method performs better than standard 1 and reweighted 1 minimization.


Corresponding author

*Corresponding author. Email addresses: (L. Guo), (Y. Liu), (L. Yan)


Hide All
[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. 317355.
[2] Van Den Berg, E. and Friedlander, M., Spgl1: A solver for large-scale sparse reconstruction,, 2007.
[3] Cai, T., Wang, L. and Xu, G., New bounds for restricted isometry constants, IEEE T. Inform. Theory, 56 (2010), pp. 43884394.
[4] Cai, T., Wang, L. and Xu, G., Shifting inequality and recovery of sparse signals, IEEE T. Signal Proces., 58 (2010), pp. 13001308.
[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. 668681.
[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. 589592.
[7] Candès, E. J., Romberg, J. and Tao, T., Stable signal recovery from incomplete and inaccurate measurements, Comm. Pure Appl. Math., 56 (2006), pp. 12071223.
[8] Candès, E. J. and Tao, T., Decoding by linear programming, IEEE T. Inform. Theory, 51 (2005), pp. 42034215.
[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. 22032214.
[10] Donoho, D. L., Compressed sensing, IEEE T. Inform. Theory, 52 (2006), pp. 12891306.
[11] Doostan, A. and Owhadi, H., A non-adapted sparse approximation of pdes with stochastic inputs, J. Comput. Phys, 230 (2011), pp. 30153034.
[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. 317339.
[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. 395407.
[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. 365381.
[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. A333A359.
[16] Hsia, Y. and Sheu, R. L., On RIC Bounds of Compressed Sensing Matrices for Approximating Sparse Solutions Using q Quasi Norms, Mathematics, 2013.
[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. A1114A1144.
[18] Lai, M. J. and Wang, J., An unconstrained ℓq minimization for sparse solution under determined linear systems, SIAM J. Optimization, 21 (2011), pp. 82101.
[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. 171191.
[20] Liu, W. H., Gong, D. and Xu, Z., One-Bit compressed sensing by greedy algorithms, Numer. Math. Theor. Meth. Appl., 9 (2016), pp. 169184.
[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. 919954.
[22] Narayan, A. and Zhou, T., Stochastic collocation methods on unstructured meshes, Commun. Comput. Phys., 18 (2015), pp. 136.
[23] Rauhut, H. and Ward, R., Sparse legendre expansions via ℓ1-minimization, J. Approx. Theory, 164 (2012), pp. 517533.
[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. 465488.
[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. 11541158.
[26] Tang, T. and Zhou, T., Recent developments in high order numerical methods for uncertainty quantification, Science in China: Mathematics, 45(7) (2015), pp. 891928.
[27] Xiu, D., Efficient collocational approach for parametric uncertainty analysis, Commun. Comput. Phys., 2 (2007), pp. 293309.
[28] Xiu, D., Fast numerical methods for stochastic computations: a review, Commun. Comput. Phys., 5 (2009), pp. 242272.
[29] Xiu, D. and Karniadakis, G. E., The wiener-askey polynomial chaos for stochastic differential equations, SIAM J. Sci. Comput., 24 (2002), pp. 619644.
[30] Xu, Z. and Zhou, T., On sparse interpolation and the design of deterministic interpolation points, SIAM J. Sci. Comput., 36 (2014), pp. A1752A1769.
[31] Yan, L., Guo, L. and Xiu, D., Stochastic collocation algorithms using ℓ1 minimization, Int. J. Uncertain Quantification, 2 (2012), pp. 279293.
[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.


MSC classification


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed