A non-adapted sparse approximation of PDEs with stochastic inputs

Journal of Computational Physics

Volumn 230, Issue 8, 2011, Pages 3015-3034

  Doostan, Alireza ^a   Owhadi, Houman ^b  

^a UCB 450   (United States)

^b CALIFORNIA INSTITUTE OF TECHNOLOGY   (United States)

Author keywords

Compressive sampling; Polynomial chaos; Sparse approximation; Stochastic PDE; Uncertainty quantification

Indexed keywords

ERROR ANALYSIS; POLYNOMIAL APPROXIMATION; STOCHASTIC SYSTEMS;

COMPRESSIVE SAMPLING; DETERMINISTIC PROBLEMS; LEGACY CODE; POLYNOMIAL CHAOS; SPARSE APPROXIMATIONS; STOCHASTIC COEFFICIENTS; STOCHASTIC INPUTS; STOCHASTIC PDE'S; STOCHASTIC PDES; UNCERTAINTY QUANTIFICATIONS;

COMPRESSED SENSING;

EID: 79951768991     PISSN: 00219991     EISSN: 10902716     Source Type: Journal    
DOI: 10.1016/j.jcp.2011.01.002     Document Type: Article

References (69)

1. Babuška I., Nobile F., Tempone R. A stochastic collocation method for elliptic partial differential equations with random input data. SIAM J. Numer. Anal. 2007, 45(3):1005-1034.
2. Babuška I., Tempone R., Zouraris G. Galerkin finite element approximations of stochastic elliptic partial differential equations. SIAM J. Numer. Anal. 2004, 42(2):800-825.
3. Beck A., Teboulle M. A fast iterative shrinkage-threshold algorithm for linear inverse problems. SIAM J. Imaging Sciences 2009, 2:183-202.
4. S. Becker, J. Bobin, E.J. Candes, NESTA: a fast and accurate first-order method for sparse recovery, ArXiv e-prints, 2009. Available from http://arxiv.org/abs/0904.3367.
5. E. van den Berg, M.P. Friedlander, SPGL1: A solver for large-scale sparse reconstruction, June 2007. Available from http://www.cs.ubc.ca/labs/scl/spgl1.
6. M. Bieri, A sparse composite collocation finite element method for elliptic spdes, Technical Report Research Report No. 2009-08, Seminar für Angewandte Mathematik, SAM, Zürich, Switzerland, 2009.
7. M. Bieri, R. Andreev, C. Schwab, Sparse tensor discretization of elliptic spdes, Technical Report Research Report No. 2009-07, Seminar für Angewandte Mathematik, SAM, Zürich, Switzerland, 2009.
8. Bieri M., Schwab C. Sparse high order FEM for elliptic sPDEs. Comput. Methods Appl. Mech. Eng. 2009, 198:1149-1170.
9. Bioucas-Dias J.M., Figueiredo M.A.T. A new TwIST: two-step iterative shrinking/thresholding algorithms for image restoration. IEEE Trans. Image Process. 2007, 16(12):2992-3004.
10. G. Blatman, Adaptive sparse polynomial chaos expansions for uncertainty propagation and sensitivity analysis, Ph.D. Thesis, Université Blaise, Pascal - Clermont -Ferrand II, 2009.
11. Blatman G., Sudret B. An adaptive algorithm to build up sparse polynomial chaos expansions for stochastic finite element analysis. Probabilist. Eng. Mech. 2010, 25(2):183-197.
12. Boufounos P., Duarte M.F., Baraniuk R.G. Sparse signal reconstruction from noisy compressive measurements using cross validation. SSP '07: Proceedings of the 2007 IEEE/SP 14th Workshop on Statistical Signal Processing 2007, 299-303. IEEE Computer Society.
13. Bredies K., Lorenz D.A. Linear convergence of iterative soft-thresholding. SIAM J. Sci. Comp. 2008, 30(2):657-683.
14. Bruckstein A.M., Donoho D.L., Elad M. From sparse solutions of systems of equations to sparse modeling of signals and images. SIAM Rev. 2009, 51(1):34-81.
15. Candes E.J., Romberg J. Quantitative robust uncertainty principles and optimally sparse decompositions. Found. Comput. Math. 2006, 6(2):227-254.
16. Candes E.J., Romberg J. Sparsity and incoherence in compressive sampling. Inverse Probl. 2007, 23(3):969-985.
17. Candes E.J., Romberg J., Tao T. Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information. IEEE Trans. Inform. Theory 2006, 52(2):489-509.
18. Candes E.J., Tao T. Near optimal signal recovery from random projections: universal encoding strategies?. IEEE Trans. Inform. Theory 2006, 52(12):5406-5425.
19. 1 minimization. J. Fourier Anal. Appl. 2008, 14(5):877-905.
20. Chen S.S., Donoho D.L., Saunders M. Atomic decomposition by basis pursuit. SIAM J. Sci. Comput. 1998, 20:33-61.
21. Chen S.S., Donoho D.L., Saunders M. Atomic decomposition by basis pursuit. SIAM Rev. 2001, 43(1):129-159.
22. Cohen A., Dahmen W., DeVore R. Compressed sensing and best k-term approximation. J. Am. Math. Soc. 2009, 22:211-231.
23. Combettes P.L., Wajs V.R. Signal recovery by proximal forward-backward splitting. Multiscale Model. Simul. 2005, 4(4):1168-1200.
24. Dai W., Milenkovic O. Subspace pursuit for compressive sensing signal reconstruction. IEEE Trans. Inform. Theor. 2009, 55(5):2230-2249.
25. Daubechies I., Defrise M., De Mol C. An iterative thresholding algorithm for linear inverse problems with a sparsity constraint. Commun. Pure Appl. Math. 2004, 57(11):1413-1457.
26. Blumensath T., Davies M.E. Iterative hard thresholding for compressed sensing. Appl. Comp. Harm. Anal. 2009, 27(3):265-274.
27. Davis G., Mallat S., Avellaneda M. Adaptive greedy approximation. J. Constr. Approx. 1997, 13:57-98.
28. Deb M.K., Babuska I., Oden J.T. Solution of stochastic partial differential equations using Galerkin finite element techniques. Comput. Methods Appl. Mech. Eng. 2001, 190:6359-6372.
29. D.L. Donoho, I. Drori, V.C. Stodden, Y. Tsaig, Available from http://www-stat.stanford.edu/~sparselab/.
30. Donoho D.L. Compressed sensing. IEEE Trans. Inform. Theory 2006, 52(4):1289-1306.
31. D.L. Donoho, I. Drori, Y. Tsaig, J.L. Starck, Sparse solution of underdetermined linear equations by stagewise orthogonal matching pursuit, Technical report, 2006.
32. Donoho D.L., Elad M., Temlyakov V.N. Stable recovery of sparse overcomplete representations in the presence of noise. IEEE Trans. Inform. Theory 2006, 52(1):6-18.
33. Doostan A., Ghanem R., Red-Horse J. Stochastic model reduction for chaos representations. Comput. Methods Appl. Mech. Eng. 2007, 196(37-40):3951-3966.
34. Doostan A., Iaccarino G. A least-squares approximation of partial differential equations with high-dimensional random inputs. J. Comput. Phys. 2009, 228(12):4332-4345.
35. A. Doostan, H. Owhadi, A. Lashgari, G. Iaccarino, Non-adapted sparse approximation of PDEs with stochastic inputs, Technical report, 2009. Center for Turbulence Research, Annual Research Brief, Stanford University (available from). http://ctr.stanford.edu/ResBriefs09/10_doostan.pdf.
36. Efron B., Hastie T., Johnstone L., Tibshirani R. Least angle regression. Ann. Stat. 2004, 32:407-499.
37. T Figueiredo M.A., Nowak R.D., Wright S.J. Gradient projection for sparse reconstruction: application to compressed sensing and other inverse problems. IEEE J. Sel. Top. Signal Process. 2007, 1(4):586-597.
38. Ghanem R., Spanos P. Stochastic Finite Elements: A Spectral Approach 2002, Dover.
39. 1-minimization: methodology and convergence. SIAM J. Optim. 2008, 19(3):1107-1130.
40. S. Hosder, R.W. Walters, R. Perez, A non-intrusive polynomial chaos method for uncertainty propagation in CFD simulations, in: 44th AIAA Aerospace Sciences Meeting and Exhibit, AIAA-2006-891, Reno (NV), 2006.
41. M.A. Khajehnejad, W. Xu, A.S. Avestimehr, B. Hassibi, Improved sparse recovery thresholds with two-step reweighted ℓ1 minimization, ArXiv e-prints, 2010. Available from http://arxiv.org/abs/1004.0402.
42. Kim S.-J., Koh K., Lustig M., Boyd S., Gorinevsky D. An interior-point method for large-scale l1-regularized least squares. IEEE J. Sel. Top. Signal Process. 2007, 1(4):606-617.
43. Le Maitre O.P., Najm H., Ghanem R., Knio O. Multi-resolution analysis of Wiener-type uncertainty propagation schemes. J. Comput. Phys. 2004, 197(2):502-531.
44. Ma X., Zabaras N. An adaptive hierarchical sparse grid collocation algorithm for the solution of stochastic differential equations. J. Comput. Phys. 2009, 228:3084-3113.
45. L. Mathelin, M.Y. Hussaini, A stochastic collocation algorithm for uncertainty analysis, Technical Report NAS 1.26:212153; NASA/CR-2003-212153, NASA Langley Research Center, 2003.
46. C. McDiarmid, On the method of bounded differences, in: Surveys in Combinatorics, Cambridge University Press, pp. 148-188.
47. D. Needell, Noisy signal recovery via iterative reweighted l1-minimization, in: Proceedings of the Asilomar Conference on Signals, Systems, and Computers, Pacific Grove, CA, 2009.
48. Needell D., Tropp J.A. CoSaMP: iterative signal recovery from incomplete and inaccurate samples. Appl. Comput. Harmonic Anal. 2008, 26(3):1-321.
49. D. Needell, R. Vershynin, Signal recovery from incomplete and inaccurate measurements via Regularized Orthogonal Matching Pursuit, ArXiv e-prints, 2007.
50. Nobile F., Tempone R., Webster C.G. An anisotropic sparse grid stochastic collocation method for partial differential equations with random input data. SIAM J. Numer. Anal. 2008, 46(5):2411-2442.
51. Nobile F., Tempone R., Webster C.G. A sparse grid stochastic collocation method for partial differential equations with random input data. SIAM J. Numer. Anal. 2008, 46(5):2309-2345.
52. Nouy A. A generalized spectral decomposition technique to solve a class of linear stochastic partial differential equations. Comput. Methods Appl. Mech. Eng. 2007, 196(37-40):4521-4537.
53. Nouy A. Generalized spectral decomposition method for solving stochastic finite element equations: invariant subspace problem and dedicated algorithms. Comput. Methods Appl. Mech. Eng. 2008, 197:4718-4736.
54. Osborne M.R., Presnell B., Turlach B. A new approach to variable selection in least squares problems. IMA J. Numer. Anal. 2000, 20:389-403.
55. Y.C. Pati, R. Rezaiifar, P.S. Krishnaprasad, Orthogonal Matching Pursuit: Recursive function approximation with applications to wavelet decomposition, in: Proceedings of the 27th Annual Asilomar Conference on Signals, Systems, and Computers, 1993, pp. 40-44.
56. H. Rauhut, R. Ward, Sparse legendre expansions via ℓ1-minimization, ArXiv e-prints, 2010.
57. Schwab C., Todor R.A. Karhunen-Loève approximation of random fields by generalized fast multipole methods. J. Comput. Phys. 2006, 217(1):100-122.
58. M.A. Tatang, Direct incorporation of uncertainty in chemical and environmental engineering systems, PhD thesis, Massachusetts Institute of Technology, 1995.
59. Tibshirani R. Regression shrinkage and selection via the Lasso. J. Roy. Stat. Soc. Ser. B 1996, 58(1):267-288.
60. Todor R.A., Schwab C. Convergence rates for sparse chaos approximations of elliptic problems with stochastic coefficients. IMA J. Numer. Anal. 2007, 27(2):232-261.
61. J.A. Tropp, S.J. Wright, Computational methods for sparse solution of linear inverse problems, in: Proceedings of the IEEE, in press.
62. van den Berg E., Friedlander M.P. Probing the Pareto frontier for basis pursuit solutions. SIAM J. Sci. Comput. 2008, 31(2):890-912.
63. Wan X., Karniadakis G. An adaptive multi-element generalized polynomial chaos method for stochastic differential equations. J. Comput. Phys. 2005, 209:617-642.
64. Wan X., Rozovskii B., Karniadakis G.E. A stochastic modeling methodology based on weighted Wiener chaos and Malliavin calculus. Proc. National Acad. Sci. 2009, 106(34):14189-14194.
65. Ward R. Compressed sensing with cross validation. IEEE Trans. Inform. Theory 2009, 55(12):5773-5782.
66. Xiu D., Hesthaven J.S. High-order collocation methods for differential equations with random inputs. SIAM J. Sci. Comput. 2005, 27(3):1118-1139.
67. Xiu D., Karniadakis G.M. The Wiener-Askey polynomial chaos for stochastic differential equations. SIAM J. Sci. Comput. 2002, 24(2):619-644.
68. W. Xu, M.A. Khajehnejad, A.S. Avestimehr, B. Hassibi, Breaking through the thresholds: an analysis for iterative reweighted ℓ1 minimization via the Grassmann Angle Framework, ArXiv e-prints, 2009. Available from http://arxiv.org/abs/0904.0994.
69. J. Yang, Y. Zhang, Alternating direction algorithms for ℓ1-problems in compressive sensing, ArXiv e-prints, 2009. Available from http://arxiv.org/abs/0912.1185.