메뉴 건너뛰기




Volumn 198, Issue 13-14, 2009, Pages 1149-1170

Sparse high order FEM for elliptic sPDEs

Author keywords

Karhunen Lo ve expansion; Polynomial chaos; Sparse tensor products; Stochastic collocation; Stochastic finite element methods; Stochastic partial differential equations

Indexed keywords

COMPUTATIONAL FLUID DYNAMICS; COMPUTATIONAL MECHANICS; ERROR ANALYSIS; FINITE ELEMENT METHOD; PARTIAL DIFFERENTIAL EQUATIONS; PROBABILITY DISTRIBUTIONS; RANDOM PROCESSES; STOCHASTIC MODELS; TENSORS;

EID: 60649099548     PISSN: 00457825     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.cma.2008.08.019     Document Type: Article
Times cited : (116)

References (44)
  • 2
    • 34548216854 scopus 로고    scopus 로고
    • A stochastic collocation method for elliptic partial differential equations with random input data
    • Babuška I., Nobile F., and Tempone R. A stochastic collocation method for elliptic partial differential equations with random input data. SIAM J. Numer. Anal. 45 3 (2007) 1005-1034
    • (2007) SIAM J. Numer. Anal. , vol.45 , Issue.3 , pp. 1005-1034
    • Babuška, I.1    Nobile, F.2    Tempone, R.3
  • 3
    • 13844319204 scopus 로고    scopus 로고
    • Galerkin finite element approximations of stochastic elliptic partial differential equations
    • Babuška I., Tempone R., and Zouraris G.E. Galerkin finite element approximations of stochastic elliptic partial differential equations. SIAM J. Numer. Anal. 42 2 (2002) 800-825
    • (2002) SIAM J. Numer. Anal. , vol.42 , Issue.2 , pp. 800-825
    • Babuška, I.1    Tempone, R.2    Zouraris, G.E.3
  • 5
    • 9244261365 scopus 로고    scopus 로고
    • Numerical Analysis of Wavelet Methods
    • Elsevier, Amsterdam
    • Albert Cohen, Numerical Analysis of Wavelet Methods, in: Studies in Mathematics and its Applications, vol. 32, Elsevier, Amsterdam, 2003.
    • (2003) Studies in Mathematics and its Applications , vol.32
    • Cohen, A.1
  • 7
    • 0003699170 scopus 로고
    • Interpolation and approximation
    • Blaisdell Publishing Company
    • Davis P.J. Interpolation and approximation. Introductions to Higher Mathematics (1963), Blaisdell Publishing Company
    • (1963) Introductions to Higher Mathematics
    • Davis, P.J.1
  • 8
    • 0035964634 scopus 로고    scopus 로고
    • Solution of stochastic partial differential equations using Galerkin finite element techniques
    • Deb M.-K., Babuška I., and Oden J.T. Solution of stochastic partial differential equations using Galerkin finite element techniques. Comput. Methods Appl. Mech. Engrg. 190 (2001) 6359-6372
    • (2001) Comput. Methods Appl. Mech. Engrg. , vol.190 , pp. 6359-6372
    • Deb, M.-K.1    Babuška, I.2    Oden, J.T.3
  • 9
    • 0003489634 scopus 로고    scopus 로고
    • Monte Carlo: concepts, algorithms and applications
    • Springer, New York
    • Fishman G.S. Monte Carlo: concepts, algorithms and applications. Springer Series in Operations Research (1996), Springer, New York
    • (1996) Springer Series in Operations Research
    • Fishman, G.S.1
  • 10
    • 18244396022 scopus 로고    scopus 로고
    • Orthogonal polynomials, computation and approximation
    • Oxford University Press Inc.
    • Gautschi W. Orthogonal polynomials, computation and approximation. Numerical Mathematics and Scientific Computation (2004), Oxford University Press Inc.
    • (2004) Numerical Mathematics and Scientific Computation
    • Gautschi, W.1
  • 12
    • 60649108875 scopus 로고    scopus 로고
    • R. Geus, O. Chinellato, Jdbsym 0.14. .
    • R. Geus, O. Chinellato, Jdbsym 0.14. .
  • 14
    • 0001744704 scopus 로고
    • A class of statistics with asymptotically normal distributions
    • Hoeffding W. A class of statistics with asymptotically normal distributions. Ann. Math. Stat. 19 (1948) 293-325
    • (1948) Ann. Math. Stat. , vol.19 , pp. 293-325
    • Hoeffding, W.1
  • 16
    • 60649119923 scopus 로고    scopus 로고
    • G.E. Karniadakis, C.-H. Su, D. Xiu, D. Lucor, C. Schwab, R.-A. Todor, Generalized polynomial chaos solution for differential equations with random inputs, Technical Report 2005-01, Seminar for Applied Mathematics, ETH Zürich, 2005.
    • G.E. Karniadakis, C.-H. Su, D. Xiu, D. Lucor, C. Schwab, R.-A. Todor, Generalized polynomial chaos solution for differential equations with random inputs, Technical Report 2005-01, Seminar for Applied Mathematics, ETH Zürich, 2005.
  • 17
    • 13144262899 scopus 로고    scopus 로고
    • A review of recent developments in the numerical solution of stochastic partial differential equations (stochastic finite elements)
    • Technical Report 2003-06, Institute of Scientific Computing, TU Braunschweig
    • Andreas Keese, A review of recent developments in the numerical solution of stochastic partial differential equations (stochastic finite elements), Technical Report 2003-06, Institute of Scientific Computing, TU Braunschweig, 2003.
    • (2003)
    • Keese, A.1
  • 18
    • 0005341464 scopus 로고
    • Eigenvalues of analytic kernels
    • Little G., and Reade J.B. Eigenvalues of analytic kernels. SIAM J. Math. Anal. 15 1 (1984) 133-136
    • (1984) SIAM J. Math. Anal. , vol.15 , Issue.1 , pp. 133-136
    • Little, G.1    Reade, J.B.2
  • 20
    • 0035449760 scopus 로고    scopus 로고
    • Optimal systems of nodes for Lagrange interpolation on bounded intervals, a survey
    • Mastroianni G., and Occorsio D. Optimal systems of nodes for Lagrange interpolation on bounded intervals, a survey. J. Comput. Appl. Math. 134 (2001) 325-341
    • (2001) J. Comput. Appl. Math. , vol.134 , pp. 325-341
    • Mastroianni, G.1    Occorsio, D.2
  • 21
    • 13844308726 scopus 로고    scopus 로고
    • Galerkin methods for linear and nonlinear elliptic stochastic partial differential equations
    • Matthies H.G., and Keese A. Galerkin methods for linear and nonlinear elliptic stochastic partial differential equations. Comp. Methods Appl. Mech. Engrg. 194 (2005) 1295-1331
    • (2005) Comp. Methods Appl. Mech. Engrg. , vol.194 , pp. 1295-1331
    • Matthies, H.G.1    Keese, A.2
  • 22
    • 52749083629 scopus 로고    scopus 로고
    • A sparse grid stochastic collocation method for elliptic partial differential equations with random input data
    • Nobile F., Tempone R., and Webster C.G. A sparse grid stochastic collocation method for elliptic partial differential equations with random input data. SIAM J. Num. Anal. 46 5 (2008) 2309-2345
    • (2008) SIAM J. Num. Anal. , vol.46 , Issue.5 , pp. 2309-2345
    • Nobile, F.1    Tempone, R.2    Webster, C.G.3
  • 23
    • 52749093759 scopus 로고    scopus 로고
    • An anisotropic sparse grid stochastic collocation method for elliptic partial differential equations with random input data
    • Nobile F., Tempone R., and Webster C.G. An anisotropic sparse grid stochastic collocation method for elliptic partial differential equations with random input data. SIAM J. Num. Anal. 46 5 (2008) 2411-2442
    • (2008) SIAM J. Num. Anal. , vol.46 , Issue.5 , pp. 2411-2442
    • Nobile, F.1    Tempone, R.2    Webster, C.G.3
  • 24
    • 0030524566 scopus 로고    scopus 로고
    • High dimensional integration of smooth functions over cubes
    • Novak E., and Ritter K. High dimensional integration of smooth functions over cubes. Numer. Math. 75 (1996) 79-97
    • (1996) Numer. Math. , vol.75 , pp. 79-97
    • Novak, E.1    Ritter, K.2
  • 25
    • 0041055781 scopus 로고
    • Eigenvalues of positive definite kernels
    • Reade J.B. Eigenvalues of positive definite kernels. SIAM J. Math. Anal. 14 3 (1983) 152-157
    • (1983) SIAM J. Math. Anal. , vol.14 , Issue.3 , pp. 152-157
    • Reade, J.B.1
  • 26
    • 33750051668 scopus 로고
    • Eigenvalues of positive definite kernels II
    • Reade J.B. Eigenvalues of positive definite kernels II. SIAM J. Math. Anal. 15 1 (1984) 137-142
    • (1984) SIAM J. Math. Anal. , vol.15 , Issue.1 , pp. 137-142
    • Reade, J.B.1
  • 27
    • 0003554096 scopus 로고    scopus 로고
    • Iterative methods for sparse linear systems
    • PWS
    • Yousef Saad, Iterative methods for sparse linear systems, Series in Computer Science, PWS, 1996.
    • (1996) Series in Computer Science
    • Saad, Y.1
  • 28
    • 10444240549 scopus 로고    scopus 로고
    • Introduction to positive definite functions and to unconditional simulation of random fields
    • Technical Report ST-99-10, Department of Mathematics and Statistics, Lancaster University UK
    • Martin Schlather, Introduction to positive definite functions and to unconditional simulation of random fields. Technical Report ST-99-10, Department of Mathematics and Statistics, Lancaster University UK, 1999.
    • (1999)
    • Schlather, M.1
  • 29
    • 0242370956 scopus 로고    scopus 로고
    • Sparse finite elements for stochastic elliptic problems
    • Schwab Ch., and Todor R.-A. Sparse finite elements for stochastic elliptic problems. Numer. Math. 95 (2003) 707-734
    • (2003) Numer. Math. , vol.95 , pp. 707-734
    • Schwab, Ch.1    Todor, R.-A.2
  • 30
    • 0141957096 scopus 로고    scopus 로고
    • Sparse finite elements for stochastic elliptic problems - higher order moments
    • Schwab Ch., and Todor R.-A. Sparse finite elements for stochastic elliptic problems - higher order moments. Computing 71 (2003) 43-63
    • (2003) Computing , vol.71 , pp. 43-63
    • Schwab, Ch.1    Todor, R.-A.2
  • 31
    • 33746346426 scopus 로고    scopus 로고
    • Karhunen-Loève approximation of random fields by generalized fast multipole methods
    • Schwab Ch., and Todor R.-A. Karhunen-Loève approximation of random fields by generalized fast multipole methods. J. Comput. Phys. 217 1 (2006) 100-122
    • (2006) J. Comput. Phys. , vol.217 , Issue.1 , pp. 100-122
    • Schwab, Ch.1    Todor, R.-A.2
  • 33
    • 16244409214 scopus 로고    scopus 로고
    • Physical systems with random uncertainties: chaos representations with arbitrary probability measure
    • 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
    • (2004) SIAM J. Sci. Comput. , vol.26 , Issue.2 , pp. 395-410
    • Soize, C.1    Ghanem, R.2
  • 36
    • 34047123580 scopus 로고    scopus 로고
    • Convergence rates for sparse chaos approximations of elliptic problems with stochastic coefficients
    • Todor R.-A., and Schwab Ch. Convergence rates for sparse chaos approximations of elliptic problems with stochastic coefficients. IMA J. Numer. Anal. 27 (2007) 232-261
    • (2007) IMA J. Numer. Anal. , vol.27 , pp. 232-261
    • Todor, R.-A.1    Schwab, Ch.2
  • 37
    • 0004117231 scopus 로고
    • The MIT press, Cambridge - Mass. & London
    • Vanmarcke E. Random Fields (1983), The MIT press, Cambridge - Mass. & London
    • (1983) Random Fields
    • Vanmarcke, E.1
  • 38
    • 84867925855 scopus 로고    scopus 로고
    • Sparse finite element methods for operator equations with stochastic data
    • von Petersdorff T., and Schwab C. Sparse finite element methods for operator equations with stochastic data. Appl. Math. 52 (2006) 145-180
    • (2006) Appl. Math. , vol.52 , pp. 145-180
    • von Petersdorff, T.1    Schwab, C.2
  • 39
    • 60649095419 scopus 로고    scopus 로고
    • Error control in multi-element generalized polynomial chaos method for elliptic problems with random coefficients
    • Wan X., and Karniadakis G.E. Error control in multi-element generalized polynomial chaos method for elliptic problems with random coefficients. Commun. Comput. Phys. 5 2-4 (2009) 793-820
    • (2009) Commun. Comput. Phys. , vol.5 , Issue.2-4 , pp. 793-820
    • Wan, X.1    Karniadakis, G.E.2
  • 40
    • 33749326978 scopus 로고    scopus 로고
    • Multi-element generalized polynomial chaos for arbitrary probability measures
    • Wan X., and Karniadakis G.E. Multi-element generalized polynomial chaos for arbitrary probability measures. SIAM J. Sci. Comput. 28 3 (2006) 901-928
    • (2006) SIAM J. Sci. Comput. , vol.28 , Issue.3 , pp. 901-928
    • Wan, X.1    Karniadakis, G.E.2
  • 41
    • 0000786435 scopus 로고
    • The homogeneous chaos
    • Wiener N. The homogeneous chaos. Amer. J. Math. 60 (1930) 897-936
    • (1930) Amer. J. Math. , vol.60 , pp. 897-936
    • Wiener, N.1
  • 42
    • 33646577797 scopus 로고    scopus 로고
    • High-order collocation methods for differential equations with random inputs
    • Xiu D., and Hesthaven J.S. High-order collocation methods for differential equations with random inputs. SIAM J. Sci. Comput. 27 3 (2005) 1118-1139
    • (2005) SIAM J. Sci. Comput. , vol.27 , Issue.3 , pp. 1118-1139
    • Xiu, D.1    Hesthaven, J.S.2
  • 43
    • 0037184252 scopus 로고    scopus 로고
    • Modeling uncertainty in steady state diffusion problems via generalized polynomial chaos
    • Xiu D., and Karniadakis G.E. Modeling uncertainty in steady state diffusion problems via generalized polynomial chaos. Comput. Methods Appl. Mech. Engrg. 191 43 (2002) 4927-4948
    • (2002) Comput. Methods Appl. Mech. Engrg. , vol.191 , Issue.43 , pp. 4927-4948
    • Xiu, D.1    Karniadakis, G.E.2
  • 44
    • 0037249229 scopus 로고    scopus 로고
    • The Wiener-Askey polynomial chaos for stochastic differential equations
    • Xiu D., and Karniadakis G.E. The Wiener-Askey polynomial chaos for stochastic differential equations. SIAM J. Sci. Comput. 24 2 (2002) 619-644
    • (2002) SIAM J. Sci. Comput. , vol.24 , Issue.2 , pp. 619-644
    • Xiu, D.1    Karniadakis, G.E.2


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.