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Volumn 248, Issue , 2013, Pages 87-108

Reweighted ℓ1 minimization method for stochastic elliptic differential equations

Author keywords

Chebyshev probability measure; Compressive sensing; Generalized polynomial chaos; High dimensions

Indexed keywords

PARTIAL DIFFERENTIAL EQUATIONS; POLYNOMIALS; STOCHASTIC SYSTEMS;

EID: 84877881686     PISSN: 00219991     EISSN: 10902716     Source Type: Journal    
DOI: 10.1016/j.jcp.2013.04.004     Document Type: Article
Times cited : (129)

References (40)
  • 2
    • 0037249229 scopus 로고    scopus 로고
    • The Wiener-Askey polynomial chaos for stochastic differential equations
    • Xiu D., Karniadakis G.E. The Wiener-Askey polynomial chaos for stochastic differential equations. SIAM J. Sci. Comput. 2002, 24(2):619-644.
    • (2002) SIAM J. Sci. Comput. , vol.24 , Issue.2 , pp. 619-644
    • Xiu, D.1    Karniadakis, G.E.2
  • 3
    • 33749326978 scopus 로고    scopus 로고
    • Multi-element generalized polynomial chaos for arbitrary probability measures
    • Wan X., Karniadakis G.E. Multi-element generalized polynomial chaos for arbitrary probability measures. SIAM J. Sci. Comput. 2006, 28(3):901-928.
    • (2006) SIAM J. Sci. Comput. , vol.28 , Issue.3 , pp. 901-928
    • Wan, X.1    Karniadakis, G.E.2
  • 4
    • 21444449245 scopus 로고    scopus 로고
    • An adaptive multi-element generalized polynomial chaos method for stochastic differential equations
    • Wan X., Karniadakis G.E. An adaptive multi-element generalized polynomial chaos method for stochastic differential equations. J. Comput. Phys. 2005, 209(2):617-642.
    • (2005) J. Comput. Phys. , vol.209 , Issue.2 , pp. 617-642
    • Wan, X.1    Karniadakis, G.E.2
  • 6
    • 33646577797 scopus 로고    scopus 로고
    • High-order collocation methods for differential equations with random inputs
    • Xiu D., Hesthaven J.S. High-order collocation methods for differential equations with random inputs. SIAM J. Sci. Comput. 2005, 27(3):1118-1139.
    • (2005) SIAM J. Sci. Comput. , vol.27 , Issue.3 , pp. 1118-1139
    • Xiu, D.1    Hesthaven, J.S.2
  • 7
    • 52749083629 scopus 로고    scopus 로고
    • A sparse grid stochastic collocation method for partial differential equations with random input data
    • 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.
    • (2008) SIAM J. Numer. Anal. , vol.46 , Issue.5 , pp. 2309-2345
    • Nobile, F.1    Tempone, R.2    Webster, C.G.3
  • 8
    • 52749087886 scopus 로고    scopus 로고
    • The multi-element probabilistic collocation method (ME-PCM): error analysis and applications
    • Foo J., Wan X., Karniadakis G.E. The multi-element probabilistic collocation method (ME-PCM): error analysis and applications. J. Comput. Phys. 2008, 227(22):9572-9595.
    • (2008) J. Comput. Phys. , vol.227 , Issue.22 , pp. 9572-9595
    • Foo, J.1    Wan, X.2    Karniadakis, G.E.3
  • 9
    • 61349191533 scopus 로고    scopus 로고
    • An adaptive hierarchical sparse grid collocation algorithm for the solution of stochastic differential equations
    • Ma X., Zabaras N. An adaptive hierarchical sparse grid collocation algorithm for the solution of stochastic differential equations. J. Comput. Phys. 2009, 228(8):3084-3113.
    • (2009) J. Comput. Phys. , vol.228 , Issue.8 , pp. 3084-3113
    • Ma, X.1    Zabaras, N.2
  • 10
    • 52749093759 scopus 로고    scopus 로고
    • An anisotropic sparse grid stochastic collocation method for partial differential equations with random input data
    • 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.
    • (2008) SIAM J. Numer. Anal. , vol.46 , Issue.5 , pp. 2411-2442
    • Nobile, F.1    Tempone, R.2    Webster, C.G.3
  • 11
    • 84861396236 scopus 로고    scopus 로고
    • Error estimates for the ANOVA method with polynomial chaos interpolation: tensor product functions
    • Zhang Z., Choi M., Karniadakis G.E. Error estimates for the ANOVA method with polynomial chaos interpolation: tensor product functions. SIAM J. Sci. Comput. 2012, 34(2):A1165-A1186.
    • (2012) SIAM J. Sci. Comput. , vol.34 , Issue.2
    • Zhang, Z.1    Choi, M.2    Karniadakis, G.E.3
  • 12
    • 72449136652 scopus 로고    scopus 로고
    • Multi-element probabilistic collocation method in high dimensions
    • Foo J., Karniadakis G.E. Multi-element probabilistic collocation method in high dimensions. J. Comput. Phys. 2010, 229(5):1536-1557.
    • (2010) J. Comput. Phys. , vol.229 , Issue.5 , pp. 1536-1557
    • Foo, J.1    Karniadakis, G.E.2
  • 13
    • 78649472968 scopus 로고    scopus 로고
    • An adaptive high-dimensional stochastic model representation technique for the solution of stochastic partial differential equations
    • Ma X., Zabaras N. An adaptive high-dimensional stochastic model representation technique for the solution of stochastic partial differential equations. J. Comput. Phys. 2010, 229(10):3884-3915.
    • (2010) J. Comput. Phys. , vol.229 , Issue.10 , pp. 3884-3915
    • Ma, X.1    Zabaras, N.2
  • 14
    • 84855205230 scopus 로고    scopus 로고
    • Adaptive ANOVA decomposition of stochastic incompressible and compressible flows
    • Yang X., Choi M., Lin G., Karniadakis G.E. Adaptive ANOVA decomposition of stochastic incompressible and compressible flows. J. Comput. Phys. 2012, 231(4):1587-1614.
    • (2012) J. Comput. Phys. , vol.231 , Issue.4 , pp. 1587-1614
    • Yang, X.1    Choi, M.2    Lin, G.3    Karniadakis, G.E.4
  • 15
    • 16244411330 scopus 로고    scopus 로고
    • Adaptive generalized polynomial chaos for nonlinear random oscillators
    • Lucor D., Karniadakis G.E. Adaptive generalized polynomial chaos for nonlinear random oscillators. SIAM J. Sci. Comput. 2004, 26(2):720-735.
    • (2004) SIAM J. Sci. Comput. , vol.26 , Issue.2 , pp. 720-735
    • Lucor, D.1    Karniadakis, G.E.2
  • 16
    • 45449106388 scopus 로고    scopus 로고
    • Stochastic low-dimensional modelling of a random laminar wake past a circular cylinder
    • Venturi D., Wan X., Karniadakis G.E. Stochastic low-dimensional modelling of a random laminar wake past a circular cylinder. J. Fluid Mech. 2008, 606:339-367.
    • (2008) J. Fluid Mech. , vol.606 , pp. 339-367
    • Venturi, D.1    Wan, X.2    Karniadakis, G.E.3
  • 17
    • 34047123580 scopus 로고    scopus 로고
    • Convergence rates for sparse chaos approximations of elliptic problems with stochastic coefficients
    • 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.
    • (2007) IMA J. Numer. Anal. , vol.27 , Issue.2 , pp. 232-261
    • Todor, R.A.1    Schwab, C.2
  • 18
    • 60649099548 scopus 로고    scopus 로고
    • Sparse high order FEM for elliptic SPDEs
    • Bieri M., Schwab C. Sparse high order FEM for elliptic SPDEs. Comput. Methods Appl. Mech. Eng. 2009, 198(13-14):1149-1170.
    • (2009) Comput. Methods Appl. Mech. Eng. , vol.198 , Issue.13-14 , pp. 1149-1170
    • Bieri, M.1    Schwab, C.2
  • 19
    • 85190774115 scopus 로고    scopus 로고
    • Analytic regularity and best N-term approximation of high dimensional parametric initial value problems
    • Seminar for Applied Mathematics, ETHZ
    • M. Hansen, C. Schwab, Analytic regularity and best N-term approximation of high dimensional parametric initial value problems, Tech. Rep. 2011-64, Seminar for Applied Mathematics, ETHZ, 2011.
    • (2011) Tech. Rep. 2011-64
    • Hansen, M.1    Schwab, C.2
  • 20
    • 84877908213 scopus 로고    scopus 로고
    • N-term Galerkin Wiener chaos approximations of elliptic PDEs with lognormal Gaussian random inputs
    • Seminar for Applied Mathematics, ETHZ
    • V.H. Hoang, C. Schwab, N-term Galerkin Wiener chaos approximations of elliptic PDEs with lognormal Gaussian random inputs, Tech. Rep. 2011-59, Seminar for Applied Mathematics, ETHZ, 2011.
    • (2011) Tech. Rep. 2011-59
    • Hoang, V.H.1    Schwab, C.2
  • 21
    • 79951768991 scopus 로고    scopus 로고
    • A non-adapted sparse approximation of PDEs with stochastic inputs
    • Doostan A., Owhadi H. A non-adapted sparse approximation of PDEs with stochastic inputs. J. Comput. Phys. 2011, 230(8):3015-3034.
    • (2011) J. Comput. Phys. , vol.230 , Issue.8 , pp. 3015-3034
    • Doostan, A.1    Owhadi, H.2
  • 22
    • 57349174008 scopus 로고    scopus 로고
    • Enhancing sparsity by reweighted l1 minimization
    • Candès E.J., Wakin M.B., Boyd S.P. Enhancing sparsity by reweighted l1 minimization. J. Fourier Anal. Appl. 2008, 14(5-6):877-905.
    • (2008) J. Fourier Anal. Appl. , vol.14 , Issue.5-6 , pp. 877-905
    • Candès, E.J.1    Wakin, M.B.2    Boyd, S.P.3
  • 25
    • 84858165576 scopus 로고    scopus 로고
    • Sparse legendre expansions via l1-minimization
    • Rauhut H., Ward R. Sparse legendre expansions via l1-minimization. J. Approx. Theory 2012, 164(5):517-533.
    • (2012) J. Approx. Theory , vol.164 , Issue.5 , pp. 517-533
    • Rauhut, H.1    Ward, R.2
  • 26
    • 59749104367 scopus 로고    scopus 로고
    • From sparse solutions of systems of equations to sparse modeling of signals and images
    • 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.
    • (2009) SIAM Rev. , vol.51 , Issue.1 , pp. 34-81
    • Bruckstein, A.M.1    Donoho, D.L.2    Elad, M.3
  • 27
    • 0039439070 scopus 로고    scopus 로고
    • Some remarks on greedy algorithms
    • DeVore R.A., Temlyakov V.N. Some remarks on greedy algorithms. Adv. Comput. Math. 1996, 5(2-3):173-187.
    • (1996) Adv. Comput. Math. , vol.5 , Issue.2-3 , pp. 173-187
    • DeVore, R.A.1    Temlyakov, V.N.2
  • 28
    • 0033130204 scopus 로고    scopus 로고
    • Greedy algorithms and m-term approximation with regard to redundant dictionaries
    • Temlyakov V.N. Greedy algorithms and m-term approximation with regard to redundant dictionaries. J. Approx. Theory 1999, 98(1):117-145.
    • (1999) J. Approx. Theory , vol.98 , Issue.1 , pp. 117-145
    • Temlyakov, V.N.1
  • 29
    • 22844454658 scopus 로고    scopus 로고
    • Weak greedy algorithms
    • Temlyakov V.N. Weak greedy algorithms. Adv. Comput. Math. 2000, 12(2-3):213-227.
    • (2000) Adv. Comput. Math. , vol.12 , Issue.2-3 , pp. 213-227
    • Temlyakov, V.N.1
  • 30
    • 33144483155 scopus 로고    scopus 로고
    • Stable recovery of sparse overcomplete representations in the presence of noise
    • 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.
    • (2006) IEEE Trans. Inform. Theory , vol.52 , Issue.1 , pp. 6-18
    • Donoho, D.L.1    Elad, M.2    Temlyakov, V.N.3
  • 31
    • 29144439194 scopus 로고    scopus 로고
    • Decoding by linear programming
    • Candès E.J., Tao T. Decoding by linear programming. IEEE Trans. Inform. Theory 2005, 51(12):4203-4215.
    • (2005) IEEE Trans. Inform. Theory , vol.51 , Issue.12 , pp. 4203-4215
    • Candès, E.J.1    Tao, T.2
  • 32
    • 42649140570 scopus 로고    scopus 로고
    • The restricted isometry property and its implications for compressed sensing
    • Candès E.J. The restricted isometry property and its implications for compressed sensing. C.R. Math. Acad. Sci. Paris 2008, 346(9-10):589-592.
    • (2008) C.R. Math. Acad. Sci. Paris , vol.346 , Issue.9-10 , pp. 589-592
    • Candès, E.J.1
  • 33
    • 77955281521 scopus 로고    scopus 로고
    • A note on guaranteed sparse recovery via l1-minimization
    • Foucart S. A note on guaranteed sparse recovery via l1-minimization. Appl. Comput. Harmon. Anal. 2010, 29(1):97-103.
    • (2010) Appl. Comput. Harmon. Anal. , vol.29 , Issue.1 , pp. 97-103
    • Foucart, S.1
  • 34
    • 62549128663 scopus 로고    scopus 로고
    • Sparsest solutions of underdetermined linear systems via lq-minimization for 0 < q ≤ 1
    • Foucart S., Lai M.-J. Sparsest solutions of underdetermined linear systems via lq-minimization for 0 < q ≤ 1. Appl. Comput. Harmon. Anal. 2009, 26(3):395-407.
    • (2009) Appl. Comput. Harmon. Anal. , vol.26 , Issue.3 , pp. 395-407
    • Foucart, S.1    Lai, M.-J.2
  • 36
    • 84877879220 scopus 로고    scopus 로고
    • Stochastic collocation algorithms using l1-minimization
    • Yan L., Guo L., Xiu D. Stochastic collocation algorithms using l1-minimization. Int. J. Uncertainty Quantif. 2012, 2(3):279-293.
    • (2012) Int. J. Uncertainty Quantif. , vol.2 , Issue.3 , pp. 279-293
    • Yan, L.1    Guo, L.2    Xiu, D.3
  • 37
    • 62749175137 scopus 로고    scopus 로고
    • CoSaMP: iterative signal recovery from incomplete and inaccurate samples
    • Needell D., Tropp J.A. CoSaMP: iterative signal recovery from incomplete and inaccurate samples. Appl. Comput. Harmon. Anal. 2009, 26(3):301-321.
    • (2009) Appl. Comput. Harmon. Anal. , vol.26 , Issue.3 , pp. 301-321
    • Needell, D.1    Tropp, J.A.2
  • 38
    • 69949164527 scopus 로고    scopus 로고
    • Iterative hard thresholding for compressed sensing
    • Blumensath T., Davies M.E. Iterative hard thresholding for compressed sensing. Appl. Comput. Harmon. Anal. 2009, 27(3):265-274.
    • (2009) Appl. Comput. Harmon. Anal. , vol.27 , Issue.3 , pp. 265-274
    • Blumensath, T.1    Davies, M.E.2
  • 39
    • 65649137930 scopus 로고    scopus 로고
    • Probing the Pareto frontier for basis pursuit solutions
    • van den Berg E., Friedlander M.P. Probing the Pareto frontier for basis pursuit solutions. SIAM J. Sci. Comput. 2008, 31(2):890-912.
    • (2008) SIAM J. Sci. Comput. , vol.31 , Issue.2 , pp. 890-912
    • van Den Berg, E.1    Friedlander, M.P.2
  • 40
    • 79957444557 scopus 로고    scopus 로고
    • Compressed sensing: how sharp is the restricted isometry property?
    • Blanchard J.D., Cartis C., Tanner J. Compressed sensing: how sharp is the restricted isometry property?. SIAM Rev. 2011, 53(1):105-125.
    • (2011) SIAM Rev. , vol.53 , Issue.1 , pp. 105-125
    • Blanchard, J.D.1    Cartis, C.2    Tanner, J.3


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