Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-10T16:32:54.508Z Has data issue: false hasContentIssue false
  • English
  • Français

Stochastic Collocation via l1-Minimisation on Low Discrepancy Point Sets with Application to Uncertainty Quantification

Part of: Approximations and expansions Partial differential equations, boundary value problems Stochastic analysis

Published online by Cambridge University Press:  12 May 2016

Yongle Liu  and
Ling Guo
Yongle Liu
Affiliation:
Department of Mathematics, Shanghai Normal University, Shanghai, China
Ling Guo*
Affiliation:
Department of Mathematics and E-Institute of Shanghai Universities and Scientific Computing, Shanghai Normal University, Shanghai, China
*
*Corresponding author. Email address:lguo@shnu.edu.cn (L. Guo)
Get access

Abstract

Various numerical methods have been developed in order to solve complex systems with uncertainties, and the stochastic collocation method using l1-minimisation on low discrepancy point sets is investigated here. Halton and Sobol' sequences are considered, and low discrepancy point sets and random points are compared. The tests discussed involve a given target function in polynomial form, high-dimensional functions and a random ODE model. Our numerical results show that the low discrepancy point sets perform as well or better than random sampling for stochastic collocation via l1-minimisation.

Keywords

Stochastic collocationQuasi-Monte Carlo sequencelow discrepancy point setsLegendre polynomialsl1-minimisation

MSC classification

Secondary: 41A10: Approximation by polynomials 60H35: Computational methods for stochastic equations 65N35: Spectral, collocation and related methods
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 6 , Issue 2 , May 2016 , pp. 171 - 191
Copyright
Copyright © Global-Science Press 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Bourgain, J., Dilworth, S., Ford, K., Konyagin, S. and D. Kutzarova, , Explicit constructions of RIP matrices and related problems, Duke Math. J. 159, 145–185 (2011).Google Scholar
[2]Candès, E., Romberg, J. and Tao, T., Stable signal recovery from incomplete and inaccurate measurements, Comm. Pure Appl. Math. 59, 1207–1223 (2006).Google Scholar
[3]Candès, E. and Tao, T., Near-optimal signal recovery from random approjection: Universal encoding strategies, IEEE Trans. Inform. Theory 52, 1207–1223 (2006).Google Scholar
[4]Choi, S. K., Grandhi, R. V., Canfield, R. A. and Pettit, C. L., Polynomial chaos expansion with latin hypercube sampling for estimating response variability, AIAA J. 42, 1191–1198 (2004).CrossRefGoogle Scholar
[5]Cohen, A., DeVore, R. and Schwab, C., Convergence rates of best N-term Galerkin approximations for a class of elliptic sPDEs, Found. Comp. Math. 10, 615–646 (2010).Google Scholar
[6]DeVore, R. A., Deterministic constructions of compressed sensing matrices, J. Complexity 23, 918–925 (2007).Google Scholar
[7]Dick, J., Kuo, F. Y. and Sloan, I. H., High-dimensional integration: The Quasi-Monte Carlo way, Acta Numerica 22, 133–288 (2013).Google Scholar
[8]Dick, J. and Pillichssmmer, F., Digital Nets and Sequences: Discrepancy Theory and Quasi-Monte Carlo Integration, Cambridge University Press (2010).Google Scholar
[9]Doostan, A. and Owhadi, H., A non-adapted sparse approximation of PDEs with stochastic inputs, J. Comput. Phys. 230, 3015–3034 (2011).Google Scholar
[10]Drmota, M. and Tichy, R. F., Sequences, Discrepancies and Applications, Springer (1997).Google Scholar
[11]Fishman, G., Monte Carlo: Concepts, Algorithms and Applications, Springer (1996).Google Scholar
[12]Gao, Z. and Zhou, T., Choice of nodal sets for least square polynomial chaos method with application to uncertainty quantification, Commun. Comp. Phys. 16, 365–381 (2014).Google Scholar
[13]Gardner, T. S., Cantor, C. R. and Collins, J. J., Construction of a genetic toggle switch in Escherichia coli, Nature 403, 339–342 (2000).Google Scholar
[14]Ghanem, R. and Spanos, P., Stochastic Finite Elements: A Spectral Approach. Springer (1991).Google Scholar
[15]Hampton, J. and Doostan, A., Compressive sampling of polynomial chaos expansions: Convergence analysis and sampling strategies, J. Comp. Phys. 280, 363–386 (2015).Google Scholar
[16]Iacò, M. R., Low discrepancy sequences: Theory and applications, arXiv:1502.04897 (2015).Google Scholar
[17]Iwen, M. A., Simple deterministically constructiible RIP matirces with sublinear fourier sampling requirements, 43rd Annual Conf. Information Sciences & Systems (CISS), Baltimore, pp. 870–875 (2009).Google Scholar
[18]Iwen, M. A., Combinatorial sublinear-time fourier algorithms, Found. Comp. Math. 10, 303–338 (2010).Google Scholar
[19]Mathelin, L. and Gallivan, K. A., A compressed sensing approach for partial differential equations with random input data, J. Comp. Phys. 12, 919–954 (2012).Google Scholar
[20]Migliorati, G., Nobile, F., Schwerin, E. and Tempone, R., Analysis of the discrete l2 projection on polynomial spaces with random evaluations, Found. Comp. Math. 14, 419–456 (2014).Google Scholar
[21]Narayan, A. and Zhou, T., Stochastic collocation on unstructured multivariate meshes, Commun. Comp. Phys. 18, 1–36 (2015).Google Scholar
[22]Nobile, F., Tempone, R. and Webster, C., A sparse grid stochastic collocation method for partial differential equations with random input data, SIAM J. Num. Anal. 46, 2309–2345 (2008).Google Scholar
[23]Rauhut, H., Compressive sensing and structured random matrices, Theoretical Foundations and Numerical Methods for Sparse Recovery, Fornasier, M. (Ed.), Berlin, New York (De Gruyter), pp. 1–92 (2010).Google Scholar
[24]Rauhut, H. and Ward, R., Sparse Legendre expansions via (l1-minimisation, J. Approximation Theory 164, 517–533 (2012).Google Scholar
[25]Sobol, I. M.', On the distribution of points in a cube and the approximate evaluation of integrals, USSR Comp. Math. & Math. Phys. 7, 86–112 (1967).Google Scholar
[26]Tang, T. and Zhou, T., Convergence analysis for stochastic collocation methods to scalar hyperbolic equations with a random wave speed, Commun. Comput. Phys. 8, 226–248 (2010).Google Scholar
[27]Tang, T. and Zhou, T., Recent developments in high order numerical methods for uncertainty quantification, Scientia Sinica Mathematica 45, 891–928 (2015).Google Scholar
[28]Todor, R. A. and Schwab, C., Convergence rates for sparse chaos approximations of elliptic problems with stochastic coefficients, SIAM J. Num. Anal. 27, 232–261 (2007).Google Scholar
[29]van den Berg, E. and Friedlander, M. P., SPGL1: A solver for large-scale sparse reconstruction, http://www.cs.ubc.ca/labs/scl/spgl1 (2007).Google Scholar
[30]van den Berg, E. and Friedlander, M. P., Probing the Pareto frontier for basis pursuit solutions, SIAM J. Sci. Comp. 31, 890–912 (2008).Google Scholar
[31]Xiu, D., Numerical Methods for Stochastic Computations: A Spectral Method Approach, Princeton University Press (2010).Google Scholar
[32]Xiu, D. and Karniadakis, G. E., The Wiener-Askey polynomial chaos for stochastic differential equations, SIAM J. Sci. Comp. 24, 619–644 (2002).Google Scholar
[33]Xu, Z., Deterministic sampling of sparse trigonometric polynomials, J. Complexity 27, 133–140 (2011).Google Scholar
[34]Xu, Z. and Zhou, T., On sparse interpolation and the design of deterministic interpolation points, SIAM J. Sci. Comp. 36, A1752-A1769 (2014).CrossRefGoogle Scholar
[35]Yan, L., Guo, L. and Xiu, D., Stochastic collocation algorithm using l1-minimisation, Int. J. Uncertainty Quantification 2, 279–293 (2012).Google Scholar
[36]Yang, J. and Zhang, Y., Alternating direction algorithms for l1-problems in compressive sensing, SIAM J. Sci. Comp. 33, 250–278 (2011).Google Scholar
[37]Yin, W., Osher, S., Goldfarb, D. and Darbon, J., Sensing iterative algorithms for (l1-minimisation with applications to compressed sensing, SIAM J. Imaging Sciences 1, 143–168 (2008).Google Scholar