Hostname: page-component-745bb68f8f-b95js Total loading time: 0 Render date: 2025-01-25T20:32:14.576Z Has data issue: false hasContentIssue false
  • English
  • Français

A Compressed Sensing Approach for Partial Differential Equations with Random Input Data

Published online by Cambridge University Press:  20 August 2015

L. Mathelin  and
K. A. Gallivan
L. Mathelin*
Affiliation:
LIMSI-CNRS, BP 133, 91403 Orsay, France
K. A. Gallivan*
Affiliation:
Mathematics Dpt., 208 Love Building, 1017 Academic Way, Florida State University, Tallahassee FL 32306-4510, USA
*
Corresponding author.Email:mathelin@limsi.fr
Email:gallivan@math.fsu.edu
Get access

Abstract

In this paper, a novel approach for quantifying the parametric uncertainty associated with a stochastic problem output is presented. As with Monte-Carlo and stochastic collocation methods, only point-wise evaluations of the stochastic output response surface are required allowing the use of legacy deterministic codes and precluding the need for any dedicated stochastic code to solve the uncertain problem of interest. The new approach differs from these standard methods in that it is based on ideas directly linked to the recently developed compressed sensing theory. The technique allows the retrieval of the modes that contribute most significantly to the approximation of the solution using a minimal amount of information. The generation of this information, via many solver calls, is almost always the bottle-neck of an uncertainty quantification procedure. If the stochastic model output has a reasonably compressible representation in the retained approximation basis, the proposed method makes the best use of the available information and retrieves the dominant modes. Uncertainty quantification of the solution of both a 2-D and 8-D stochastic Shallow Water problem is used to demonstrate the significant performance improvement of the new method, requiring up to several orders of magnitude fewer solver calls than the usual sparse grid-based Polynomial Chaos (Smolyak scheme) to achieve comparable approximation accuracy.

Keywords

60-0835J3535Q30Uncertainty quantificationcompressed sensingcollocation techniquestochastic spectral decompositionSmolyak sparse approximationstochastic collocation
Type
Research Article
Information
Communications in Computational Physics , Volume 12 , Issue 4 , October 2012 , pp. 919 - 954
Copyright
Copyright © Global Science Press Limited 2012

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]Abramowitz, M., and Stegun, I. Handbook of mathematical functions. Dover, 1970.Google Scholar
[2]Babusška, I., Nobile, F., and Tempone, R. A stochastic collocation method for elliptic partial differentialequations with random input data. SIAMJ. Numer. Anal. 45, 3 (2007), 1005–1034.Google Scholar
[3]Becker, S., Bobin, J., and Candès, , Nesta, E.: a fast and accurate first-order method for sparse recovery. Tech. rep., Caltech Institute of Technology, 2009.Google Scholar
[4]Blanchard, J., Cartis, C., and Tanner, J. Compressed sensing: how sharp is the restricted isometry property ? SIAM review (2010). to appear.Google Scholar
[5]Cai, T., Wang, L., and Xu, G.New bounds for restricted isometry constants. Tech. rep., Massachusetts Institute of Technology, 2009.Google Scholar
[6]Cai, T., Xu, G., and Zhang, J. On recovery of sparse signals via ℓ1 minimization. IEEE Trans. Inf. Theory 55 (2009), 3388–3397.Google Scholar
[7]Cameron, R., and Martin, W.The orthogonal development of non-linear fonctionals in series of fourier-hermite fonctionals. Ann. Math. 48, 2 (1947), 385–392.Google Scholar
[8]Candès, E., and Plan, Y.Near-ideal model selection by ℓ1 minimization. Annals of Statistics 37 (2007), 2145–2177.Google Scholar
[9]Candès, E., and Romberg, J. Quantitative robust uncertainty principles and optimally sparse decompositions. Found. Comput. Math. 6, 2 (2006), 227–254.Google Scholar
[10]Candès, E., Romberg, J., and Tao, T. Stable signal recovery from incomplete measurements. Comm. Pure Appl. Math. 59 (2005), 1207–1223.Google Scholar
[11]Candeès, E., and Tao, T.Near-optimal signal recovery from random projections: universal encoding strategies. IEEE Trans. Inform. Theory 52 (2004), 5406–5425.Google Scholar
[12]Candeès, E., Wakin, M. and Boyd, S.Enhancing sparsity by reweighted ℓ1 minimization. J. Fourier Anal. Appl. 17 (2007), 877–905.Google Scholar
[13]Chen, S., Donoho, D., and Saunders, M.Atomic decomposition by basis pursuit. SIAM J. Sci. Comput. 20 (1999), 33–61.Google Scholar
[14]Deb, M., Babusška, I., and Oden, J.T.. Solution of stochastic partial differential equations using galerkin finite element techniques. Comput. Methods Appl. Mech. Eng. 190, 48 (2001), 6359–6372.Google Scholar
[15]Donoho, D.Compressed sensing. IEEE Trans. Infor. Theo. 52, 4 (2006), 1289–1306.Google Scholar
[16]Donoho, D., Elad, M., and Temlyakov, V.Stable recovery of sparse overcomplete representations in the presence of noise. IEEE Trans. Infor. Theo. 52, 1 (2006), 6–18.CrossRefGoogle Scholar
[17]Doostan, A. and Owhadi, H. A sparse approximation of partial differential equations with random inputs. presented at the SIAM Annual Meeting, Pittsburgh, PA, USA, July 2010.Google Scholar
[18]Douglas, C., Haase, G., and Iskandarani, M.An additive schwarz preconditioner for the spectral element ocean model formulation of the shallow water equations. Elec. Trans. Numer. Anal. 15 (2003), 18–28.Google Scholar
[19]Efron, B. and Tibshirani, R.Improvements on cross-validation: the. 632+ bootstrap method. J. Amer. Stat. Assoc. 92, 438 (1997), 548–560.Google Scholar
[20]Figueiredo, M.Nowak, R., and Wright, S. Gradient projection for sparse reconstruction: application to compressed sensing and other inverse problems. IEEE Journal of Selected Topics in Signal Processing 1 (2007), 586–597.CrossRefGoogle Scholar
[21]Foucart, S.A note on guaranteed sparse recovery via ℓ1-minimization. Applied and Computational Harmonic Analysis 29, 1 (2010), 97–103.Google Scholar
[22]Frauenfelder, P., Schwab, C., and Todor, R. Finite elements for elliptic problems with stochastic coefficients. Comput. Meth. Appl. Mech. Engrg. 194, 2–5 (2005), 205–228.Google Scholar
[23]Ganapathysubramanian, B., and Zabaras, N.Sparse grid collocation methods for stochastic natural convection problems. J. Comput. Phys. 225 (2007), 652–685.Google Scholar
[24]Ghanem, R., and Spanos, P.Stochastic finite elements. A spectral approach, rev. ed. Springer Verlag, 1991. 222 p.Google Scholar
[25]Gilbert, J., and Lemaréchal, C. Some numerical experiments with variable-storage quasinewton algorithms. Math. Program. 45 (1989), 407–435.Google Scholar
[26]Iskandarani, M., Haidvogel, D., and Boyd, J. A staggered spectral element model with application to the oceanic shallow water equations. Int. J. Num. Meth. Fluids 20, 5 (1995), 393–414.Google Scholar
[27]Le Maître, O., Knio, O., Najm, H., and Ghanem, R.Uncertainty propagation using WienerHaar expansions. J. Comput. Phys. 197, 1 (2004), 28–57.Google Scholar
[28]Le Maître, O., Najm, H., Ghanem, R., and Knio,O., Multi-resolution analysis of Wiener-type uncertainty propagation schemes. J. Comput. Phys. 197, 2 (2004), 502–531.Google Scholar
[29]Mallat, S., and Zhang, Z.Matching pursuit in a time-frequency dictionary. IEEE Trans. Signal Proc. 41, 12 (1993), 3397–3415.Google Scholar
[30]Mathelin, L., and Gallivan, K.Uncertainty quantification for sparse solution of random pdes. presented at the SIAM Annual Meeting, Pittsburgh, PA, USA, July 2010.Google Scholar
[31]Mathelin, L., Hussaini, M., and Zang, T.Stochastic approaches to uncertainty quantification in CFD simulations. Num. Algo. 38, 1 (2005), 209–239.Google Scholar
[32]Mathelin, L., and Le Maître, O.Dual-based a posteriori error estimate for stochastic finite element methods. Comm. App. Math. Comput. Sci. 2 (2007), 83–115.Google Scholar
[33]Mathelin, L., Pastur, L., and Le Maître, O. A compressed-sensing approach for closed-loop optimal control of nonlinear systems. Theo. Comput. Fluid Dyn. (2011). In press, [DOI: 10.1007/s00162-011-0235-9].Google Scholar
[34]Nobile, F., Tempone, R., and Webster, C.An anisotropic sparse grid stochastic collocation method for partial differential equations with random input data. SIAM J. Numer. Anal. 46, 5 (2007), 2411–2442.Google Scholar
[35]Novak, E., and Ritter, K.Simple cubature formulas with high polynomial exactness. Constructive Approximation 15 (1999), 499–522.Google Scholar
[36]Petras, K.Fast calculation in the smolyak algorithm. Num. Algo. 26 (2001), 93–109.Google Scholar
[37]Rauhut, H.Compressive sensing and structured random matrices. In Theoretical Foundations and Numerical Methods for Sparse Recovery, M. Fornasier, Ed., Vol. 9 of Radon Series Comp. Appl. Math. deGruyter, 2010, pp. 1–92.Google Scholar
[38]Rauhut, H., and Ward, R. Sparse legendre expansions via ℓ1-minimization. preprint (2010).Google Scholar
[39]Smolyak, S.Quadrature and interpolation formulas for tensor products of certain classes of functions. Dokl. Akad. Nauk. SSSR 4 (1963), 240–243.Google Scholar
[40]Sobol, I.Distribution of points in a cube and approximate evaluation of integrals. USSR Comput. Maths. Math. Phys. 7 (1967), 86–112.Google Scholar
[41]Sobol, I.Uniformly distributed sequences with an additional uniform property. USSR Comput. Maths. Math. Phys. 16 (1977), 236–242.Google Scholar
[42]Soize, C., and Ghanem, R.Physical systems with random uncertainties: chaos representations with arbitrary probability measure. SIAM J. Sci. Comput. 26, 2 (2004), 395–410.CrossRefGoogle Scholar
[43]Taylor, H., Bank, S., and McCoy, J.Deconvolution with the l 1-norm. Geophysics 44 (1979), 39–52.Google Scholar
[44]Tibshirani, R.Regression shrinkage and selection via the lasso. J. Royal Statist. Soc. B 58 (1996), 267–288.Google Scholar
[45]van den Berg, E., and Friedlander, MProbing the pareto frontier for basis pursuit solutions. SIAM J. Sci. Comput. 31, 2 (2008), 890–912.Google Scholar
[46]Wan, X., and Karniadakis, G.An adaptive multi-element generalized polynomial chaos method for stochastic diffential equations. J. Comput. Phys. 209 (2005), 617–642.Google Scholar
[47]Wiener, N.The homogeneous chaos. Amer. J. Math. 60, 4 (1938), 897–936.Google Scholar
[48]Wright, S.Primal-Dual Interior-Point Methods. SIAM Publications, 1997.Google Scholar
[49]Xiu, D., and Hesthaven, J.High-order collocation methods for differential equations with random inputs. SIAM J. Sci. Comput. 27 (2005), 1118–1139.Google Scholar
[50]Xiu, D., and Karniadakis, G.The Wiener-Askey polynomial chaos for stochastic differential equations. SIAM J. Sci. Comput. 24, 2 (2002), 619–644.Google Scholar
[51]Xiu, D., and Karniadakis, G.Modeling uncertainty in flow simulations via generalized polynomial chaos. J. Comput. Phys. 187 (2003), 137–167.Google Scholar
[52]Zibulevsky, M., and Elad, M.L1-L2 optimization in signal and image processing. IEEE Signal Proc. Mag. 27, 3 (2010), 76–88.Google Scholar