Abstract
We present an overview of the stochastic finite element method with an emphasis on the computational tasks involved in its implementation.
Similar content being viewed by others
References
Adler R.J. (1981). The Geometry of Random Fields. Wiley, London
Adler, R.J., Taylor, J.E.: Random Fields and Geometry. Birkhäuser (2005)
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
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
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
Börm S., Grasedyck L. and Hackbusch W. (2003). Introduction to hierarchical matrices with applications.. Eng Anal Bound. Elem. 27: 405–422
Christiakos G. (1992). Random Field Models in Earth Sciences.. Academic, San Diego
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)
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
de Sturler E. (1999). Truncation strategies for optimal Krylov subspace methods. SIAM J. Numer. Anal. 36(3): 864–889
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
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
Ernst, O.G., Ullmann, E.: Efficient iterative solution of stochastic finite element equations. (in press)
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
Ghanem R. and Spanos P.D. (1991). Stochastic Finite Elements: A Spectral Approach. Springer, Berlin Heidelberg New York
Ghanem R.G. and Brzkala V. (1996). Stochastic finite element analysis of randomly layered media.. J. Eng. Mech. 129(3): 289–303
Ghanem R.G. and Kruger R.M. (1996). Numerical solution of spectral stochastic finite element systems.. Comput. Meth. Appl. Mech. Eng. 129: 289–303
Hida T., Potthoff J. and Streit L. (1993). White Noise Analysis—An Infinite Dimensional Calculus. Kluwer, Dordrecht
Holden, H., Øksendal, B., Ubøe, J., Zhang, T.: Stochastic Partial Differential Equations—A Modeling White Noise Functional Approach. Birkhäuser (1996)
Karhunen K. (1947). Über lineare methoden in der wahrscheinlichkeitsrechnung. Ann. Acad. Sci. Fenn. Ser. A. I. 37: 3–79
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)
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)
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
Liu J.S. (2001). Monte Carlo. Springer, Berlin Heidelberg New York
Loève, M.: Fonctions aléatoires de second ordre. In: Processus Stochastic et Mouvement Brownien. Gauthier Villars, Paris (1948)
Loève M. (1977). Probability Theory, vol II. Springer, Berlin Heidelberg New York
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
Matthies H.G. and Bucher C. (1999). Finite elements for stochastic media problems.. Comput. Meth. Appl. Mech. Eng. 168: 3–17
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
Papoulis A. and Pilla S.U. (2002). Probability, Random Variables and Stochastic Processes 4th edn. McGraw-Hill, New York
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)
Parlett B.N. (1998). The Symmetric Eigenvalue Problem. SIAM, Philadelphia
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
Saad Y. (1992). Numerical Methods for Large Eigenvalue Problems. Halsted Press, New York
Schuëller G.I. (1997). A state-of-the-art report on computational stochastic mechanics.. Probab. Eng. Mech. 12(4): 197–321
Schwab C. and Todor R.A. (2003). Sparse finite elements for elliptic problems with stochastic data.. Numer. Math. 95: 707–734
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)
Theting T.G. (2000). Solving Wick–stochastic boundary value problems using a finite element method.. Stoch. Stoch. Rep. 70(3–4): 241–270
Trèves F. (1967). Topological Vector Spaces, Distributions and Kernels. Academic, New York
Våge G. (1998). Variational methods for PDEs applied to stochastic partial differential equations. Math. Scand. 82: 113–137
Vanmarcke E. (1983). Random Fields: Analysis and Synthesis. MIT Press, Cambridge
Whittle, P.: On stationary processes in the plane. Biometrika 41(434–449) (1954)
Wiener, N.: The homogeneous chaos. Am. J. Math. 60(897–936) (1938)
Wu K. and Simon H. (2000). Thick-restart lanczos method for large symmetric eigenvalue problems.. SIAM J. Matrix Anal. Appl. 22(2): 602–616
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
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by K. Mikula.
Rights and permissions
About this article
Cite this article
Eiermann, M., Ernst, O.G. & Ullmann, E. Computational aspects of the stochastic finite element method. Comput. Visual Sci. 10, 3–15 (2007). https://doi.org/10.1007/s00791-006-0047-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00791-006-0047-4
Keywords
- Uncertainty quantification
- Stochastic finite element method
- Hierarchical matrices
- Thick-restart Lanczos method
- Multiple right hand sides