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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:[email protected] (L. Guo)
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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 

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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