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Title: Adaptive algorithm for stochastic Galerkin method (English)
Author: Pultarová, Ivana
Language: English
Journal: Applications of Mathematics
ISSN: 0862-7940 (print)
ISSN: 1572-9109 (online)
Volume: 60
Issue: 5
Year: 2015
Pages: 551-571
Summary lang: English
Category: math
Summary: We introduce a new tool for obtaining efficient a posteriori estimates of errors of approximate solutions of differential equations the data of which depend linearly on random parameters. The solution method is the stochastic Galerkin method. Polynomial chaos expansion of the solution is considered and the approximation spaces are tensor products of univariate polynomials in random variables and of finite element basis functions. We derive a uniform upper bound to the strengthened Cauchy-Bunyakowski-Schwarz constant for a certain hierarchical decomposition of these spaces. Based on this, an adaptive algorithm is proposed. A simple numerical example illustrates the efficiency of the algorithm. Only the uniform distribution of random variables is considered in this paper, but the results obtained can be modified to any other type of random variables. (English)
Keyword: stochastic Galerkin method
Keyword: a posteriori error estimate
Keyword: strengthened Cauchy-Bunyakowski-Schwarz constant
Keyword: adaptive refinement
MSC: 65C20
MSC: 65N22
idZBL: Zbl 06486925
idMR: MR3396480
DOI: 10.1007/s10492-015-0111-9
Date available: 2015-09-03T10:43:31Z
Last updated: 2020-01-05
Stable URL:
Reference: [1] Ainsworth, M., Oden, J. T.: A Posteriori Error Estimation in Finite Element Analysis.Pure and Applied Mathematics. A Wiley-Interscience Series of Texts, Monographs, and Tracts Wiley, Chichester (2000). Zbl 1008.65076, MR 1885308
Reference: [2] Axelsson, O.: Iterative Solution Methods.Cambridge Univ. Press, Cambridge (1994). Zbl 0813.15021, MR 1276069
Reference: [3] Axelsson, O., Blaheta, R., Neytcheva, M.: Preconditioning of boundary value problems using elementwise Schur complements.SIAM J. Matrix Anal. Appl. 31 (2009), 767-789. Zbl 1194.65047, MR 2530276, 10.1137/070679673
Reference: [4] Babuška, I., Nobile, F., Tempone, R.: A stochastic collocation method for elliptic partial differential equations with random input data.SIAM Rev. 52 (2010), 317-355. Zbl 1226.65004, MR 2646806, 10.1137/100786356
Reference: [5] Babuška, I., Tempone, R., Zouraris, G. E.: Galerkin finite element approximations of stochastic elliptic partial differential equations.SIAM J. Numer. Anal. 42 (2004), 800-825. Zbl 1080.65003, MR 2084236, 10.1137/S0036142902418680
Reference: [6] Bank, R. E., Dupont, T. F., Yserentant, H.: The hierarchical basis multigrid method.Numer. Math. 52 (1988), 427-458. Zbl 0645.65074, MR 0932709, 10.1007/BF01462238
Reference: [7] Bespalov, A., Powell, C. E., Silvester, D.: A priori error analysis of stochastic Galerkin mixed approximations of elliptic PDEs with random data.SIAM J. Numer. Anal. 50 (2012), 2039-2063. Zbl 1253.35228, MR 3022209, 10.1137/110854898
Reference: [8] Bosq, D.: Linear Processes in Function Spaces. Theory and Applications.Lecture Notes in Statistics 149 Springer, New York (2000). Zbl 0962.60004, MR 1783138, 10.1007/978-1-4612-1154-9_8
Reference: [9] Bryant, C. M., Prudhomme, S., Wildey, T.: A posteriori error control for partial differential equations with random data.ICES Report 13-08, 2013,
Reference: [10] Butler, T., Constantine, P., Wildey, T.: A posteriori error analysis of parameterized linear systems using spectral methods.SIAM J. Matrix Anal. Appl. 33 (2012), 195-209. Zbl 1248.65021, MR 2902678, 10.1137/110840522
Reference: [11] Deb, M. K., Babuška, I. M., Oden, J. T.: Solution of stochastic partial differential equations using Galerkin finite element techniques.Comput. Methods Appl. Mech. Eng. 190 (2001), 6359-6372. Zbl 1075.65006, MR 1870425, 10.1016/S0045-7825(01)00237-7
Reference: [12] Eigel, M., Gittelson, C. J., Schwab, C., Zander, E.: A convergent adaptive stochastic Galerkin finite element method with quasi-optimal spatial meshes.Preprint No. 1911. Weierstrass Institute für Angewadte Analysis und Stochastic, Berlin, 2013.
Reference: [13] Eigel, M., Gittelson, C. J., Schwab, C., Zander, E.: Adaptive stochastic Galerkin FEM.Comput. Methods Appl. Mech. Eng. 270 (2014), 247-269. Zbl 1296.65157, MR 3154028, 10.1016/j.cma.2013.11.015
Reference: [14] Ernst, O. G., Ullmann, E.: Stochastic Galerkin matrices.SIAM J. Matrix Anal. Appl. 31 (2010), 1848-1872. Zbl 1205.65021, MR 2644740, 10.1137/080742282
Reference: [15] Frauenfelder, P., Schwab, C., Todor, R. A.: Finite elements for elliptic problems with stochastic coefficients.Comput. Methods Appl. Mech. Eng. 194 (2005), 205-228. Zbl 1143.65392, MR 2105161, 10.1016/j.cma.2004.04.008
Reference: [16] Maître, O. P. Le, Knio, O. M., Najm, H. N., Ghanem, R. G.: Uncertainty propagation using Wiener-Haar expansions.J. Comput. Phys. 197 (2004), 28-57. Zbl 1052.65114, MR 2061240, 10.1016/
Reference: [17] Powell, C. E., Elman, H. C.: Block-diagonal preconditioning for spectral stochastic finite-element systems.IMA J. Numer. Anal. 29 (2009), 350-375. Zbl 1169.65007, MR 2491431, 10.1093/imanum/drn014
Reference: [18] Pultarová, I.: Hierarchical preconditioning for the stochastic Galerkin method: upper bounds to the strengthened CBS constants.Submitted. Available in ERC-CZ project LL1202 database,
Reference: [19] Ralston, A.: A First Course in Numerical Analysis.McGraw-Hill Book, New York (1965). Zbl 0139.31603, MR 0191070
Reference: [20] Sousedík, B., Ghanem, R. G.: Truncated hierarchical preconditioning for the stochastic Galerkin FEM.Int. J. Uncertain. Quantif. 4 (2014), 333-348. MR 3260480, 10.1615/Int.J.UncertaintyQuantification.2014007353
Reference: [21] Sousedík, B., Ghanem, R. G., Phipps, E. T.: Hierarchical Schur complement preconditioner for the stochastic Galerkin finite element methods.Numer. Linear Algebra Appl. 21 (2014), 136-151. MR 3150614, 10.1002/nla.1869
Reference: [22] Ullmann, E., Elman, H. C., Ernst, O. G.: Efficient iterative solvers for stochastic Galerkin discretizations of log-transformed random diffusion problems.SIAM J. Sci. Comput. 34 (2012), A659--A682. Zbl 1251.35200, MR 2914299, 10.1137/110836675
Reference: [23] Xiu, D.: Numerical Methods for Stochastic Computations. A Spectral Method Approach.Princeton University Press, Princeton (2010). Zbl 1210.65002, MR 2723020


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