Computational aspects of the stochastic finite element method

Adler R.J. (1981). The Geometry of Random Fields. Wiley, London MATHGoogle Scholar

Adler, R.J., Taylor, J.E.: Random Fields and Geometry. Birkhäuser (2005)Google Scholar

Babuška I. and Chleboun J. (2001). Effects of uncertainties in the domain on the solution of Neumann boundary value problems in two spatial dimensions.. Math. Comput. 71(240): 1339–1370 Google Scholar

Babuška I., Tempone R. and Zouraris G.E. (2004). Galerkin finite element approximations of stochastic elliptic partial differential equations.. SIAM J. Numer. Anal. 42(2): 800–825 CrossRefMathSciNetGoogle Scholar

Babuška I., Tempone R. and Zouraris G.E. (2005). Solving elliptic boundary value problems with uncertain coefficients by the finite element method: the stochastic formulation.. Comput. Meth. Appl. Mech. Eng. 194(12–16): 1251–1294 CrossRefGoogle Scholar

Börm S., Grasedyck L. and Hackbusch W. (2003). Introduction to hierarchical matrices with applications.. Eng Anal Bound. Elem. 27: 405–422 MATHCrossRefGoogle Scholar

Christiakos G. (1992). Random Field Models in Earth Sciences.. Academic, San Diego Google Scholar

Deb, M.K.: Solution of stochastic partial differential equations (spdes) using Galerkin method: theory and applications. Ph.D. thesis, The University of Texas, Austin (2000)Google Scholar

Deb M.K., Babuška I.M. and Oden J.T. (2001). Solution of stochastic partial differential equations using Galerkin finite element techniques.. Comput. Meth. Appl. Mech. Eng. 190: 6359–6372 MATHCrossRefGoogle Scholar

de Sturler E. (1999). Truncation strategies for optimal Krylov subspace methods. SIAM J. Numer. Anal. 36(3): 864–889 CrossRefMathSciNetGoogle Scholar

Eiermann M., Ernst O.G. and Schneider O. (2000). Analysis of acceleration strategies for restarted minimal residual methods.. J. Comput. Appl. Math. 123: 261–292 MATHCrossRefMathSciNetGoogle Scholar

Elman H.C., Ernst O.G., O’Leary D.P. and Stewart M. (2005). Efficient iterative algorithms for the stochastic finite element method with applications to acoustic scattering.. Comput. Meth. Appl. Mech. Eng. 194: 1037–1055 MATHCrossRefMathSciNetGoogle Scholar

Ernst, O.G., Ullmann, E.: Efficient iterative solution of stochastic finite element equations. (in press)Google Scholar

Frauenfelder P., Schwab C. and Todor R.A. (2005). Finite elements for elliptic problems with stochastic coefficients.. Comput. Meth. Appl. Mech. Eng. 194: 205–228 MATHCrossRefMathSciNetGoogle Scholar

Ghanem R. and Spanos P.D. (1991). Stochastic Finite Elements: A Spectral Approach. Springer, Berlin Heidelberg New York MATHGoogle Scholar

Ghanem R.G. and Brzkala V. (1996). Stochastic finite element analysis of randomly layered media.. J. Eng. Mech. 129(3): 289–303 MATHGoogle Scholar

Ghanem R.G. and Kruger R.M. (1996). Numerical solution of spectral stochastic finite element systems.. Comput. Meth. Appl. Mech. Eng. 129: 289–303 MATHCrossRefGoogle Scholar

Hida T., Potthoff J. and Streit L. (1993). White Noise Analysis—An Infinite Dimensional Calculus. Kluwer, Dordrecht Google Scholar

Holden, H., Øksendal, B., Ubøe, J., Zhang, T.: Stochastic Partial Differential Equations—A Modeling White Noise Functional Approach. Birkhäuser (1996)Google Scholar

Karhunen K. (1947). Über lineare methoden in der wahrscheinlichkeitsrechnung. Ann. Acad. Sci. Fenn. Ser. A. I. 37: 3–79 Google Scholar

Keese, A.: A review of recent developments in the numerical solution of stochastic partial differential equations. Tech. Rep. Informatikbericht Nr.: 2003-6, Universität Braunschweig, Institut für Wissenschaftliches Rechnen (2003)Google Scholar

Keese, A.: Numerical solution of systems with stochastic uncertainties—a general purpose framework for stochastic finite elements. Ph.D. thesis, TU Braunschweig, Germany, Fachbereich Mathematik und Informatik (2004)Google Scholar

Lehoucq R.B., Sorensen D.C. and Yang C. (1998). ARPACK User’s Guide: Solution of Large Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods. SIAM, Philadelphia MATHGoogle Scholar

Liu J.S. (2001). Monte Carlo. Springer, Berlin Heidelberg New York MATHGoogle Scholar

Loève, M.: Fonctions aléatoires de second ordre. In: Processus Stochastic et Mouvement Brownien. Gauthier Villars, Paris (1948)Google Scholar

Loève M. (1977). Probability Theory, vol II. Springer, Berlin Heidelberg New York Google Scholar

Matthies H.G., Brenner C., Bucher C.G. and Soares C.G. (1997). Uncertainties in probabilistic numerical analysis of structures and solids—stochastic finite elements. Struct. Saf. 19: 283–336 CrossRefGoogle Scholar

Matthies H.G. and Bucher C. (1999). Finite elements for stochastic media problems.. Comput. Meth. Appl. Mech. Eng. 168: 3–17 MATHCrossRefMathSciNetGoogle Scholar

Matthies H.G. and Keese A. (2005). Galerkin methods for linear and nonlinear elliptic stochastic partial differential equations.. Comput. Meth. Appl. Mech. Eng. 194: 1295–1331 MATHCrossRefMathSciNetGoogle Scholar

Papoulis A. and Pilla S.U. (2002). Probability, Random Variables and Stochastic Processes 4th edn. McGraw-Hill, New York Google Scholar

Parks, M., de Sturler, E., Mackey, G., Johnson, D., Maiti, S.: Recycling Krylov subspaces for sequences of linear systems. Technical Report UIUCDCS-R-2004-2421(CS), University of Illinois (2004)Google Scholar

Parlett B.N. (1998). The Symmetric Eigenvalue Problem. SIAM, Philadelphia MATHGoogle Scholar

Pellissetti M.F. and Ghanem R.G. (2000). Iterative solution of systems of linear equations arising in the context of stochastic finite elements.. Adv. Eng. Softw. 31: 607–616 MATHCrossRefGoogle Scholar

Saad Y. (1992). Numerical Methods for Large Eigenvalue Problems. Halsted Press, New York MATHGoogle Scholar

Schuëller G.I. (1997). A state-of-the-art report on computational stochastic mechanics.. Probab. Eng. Mech. 12(4): 197–321 CrossRefGoogle Scholar

Schwab C. and Todor R.A. (2003). Sparse finite elements for elliptic problems with stochastic data.. Numer. Math. 95: 707–734 MATHCrossRefMathSciNetGoogle Scholar

Sudret, B., Kiureghian, A.D.: Stochastic finite element methods and reliability: a state-of-the-art report. Technical Report on UCB/SEMM-2000/08, Department of Civil and Environmental Engineering, UC Berkeley (2000)Google Scholar

Theting T.G. (2000). Solving Wick–stochastic boundary value problems using a finite element method.. Stoch. Stoch. Rep. 70(3–4): 241–270 MATHMathSciNetGoogle Scholar

Trèves F. (1967). Topological Vector Spaces, Distributions and Kernels. Academic, New York MATHGoogle Scholar

Våge G. (1998). Variational methods for PDEs applied to stochastic partial differential equations. Math. Scand. 82: 113–137 MathSciNetGoogle Scholar

Vanmarcke E. (1983). Random Fields: Analysis and Synthesis. MIT Press, Cambridge Google Scholar

Whittle, P.: On stationary processes in the plane. Biometrika 41(434–449) (1954)Google Scholar

Wiener, N.: The homogeneous chaos. Am. J. Math. 60(897–936) (1938)Google Scholar

Wu K. and Simon H. (2000). Thick-restart lanczos method for large symmetric eigenvalue problems.. SIAM J. Matrix Anal. Appl. 22(2): 602–616 MATHCrossRefMathSciNetGoogle Scholar

Xiu D. and Karniadakis G.E. (2002). Modeling uncertainty in steady state diffusion problems via generalized polynomial chaos.. Comput. Meth. Appl. Mech. Eng. 191: 4927–4948 MATHCrossRefMathSciNetGoogle Scholar