A kronecker product preconditioner for stochastic galerkin finite eleme nt discretizations

SIAM Journal on Scientific Computing

Volumn 32, Issue 2, 2010, Pages 923-946

  Ullmann, Elisabeth ^a  

^a TU BERGAKADEMIE FREIBERG   (Germany)

Author keywords

Karhunen Lo ve expansion; Kronecker product preconditioner; Lognormal random field; Polynomial chaos expansion; Stochastic Galerkin finite element method

Indexed keywords

CODES (SYMBOLS); FINITE ELEMENT METHOD; GALERKIN METHODS; LINEAR SYSTEMS; PARTIAL DIFFERENTIAL EQUATIONS; STOCHASTIC MODELS; STOCHASTIC SYSTEMS;

CHAOS EXPANSIONS; GALERKIN FINITE ELEMENT METHODS; KARHUNEN-LOÈVE EXPANSION; KRONECKE PRODUCT PRECONDITIONER; KRONECKER PRODUCT; LOGNORMAL RANDOM FIELDS; POLYNOMIAL CHAOS; POLYNOMIAL CHAOS EXPANSION; PRECONDITIONERS; STOCHASTIC GALERKIN FINITE ELEMENT METHOD; STOCHASTICS;

MATRIX ALGEBRA;

EID: 77950848558     PISSN: 10648275     EISSN: None     Source Type: Journal    
DOI: 10.1137/080742853     Document Type: Article

References (28)

1. R. J. ADLER, The Geometry of Random Fields, Wiley, Chichester, United Kingdom, 1981.
2. I. BABUŠKA, F. NOBILE, AND R. TEMPONE, A stochastic collocation method for elliptic partial differential equations with random input data, SIAM J . Numer. Anal., 45 (2007), pp. 1005-1034.
3. I. BABUŠKA, R. TEMPONE, AND G. E. ZoURARIS, Galerkin finite element approximations of stochastic elliptic partial differential equations, SIAM J. Numer. Anal., 42 (2004), pp. 800-825. (Pubitemid 40645215)
4. I. BABUŠKA, R. TEMPONE, AND G. E. ZoURARIS, Solving elliptic boundary value problems with uncertain coefficients by the finite element method: The stochastic formulation, Comput. Methods Appl. Mech. Engrg., 194 (2005), pp. 1251-1294. (Pubitemid 40262931)
5. J. BOYLE, M. D. MlHAJLOVIĆ, AND J. A. SCOTT, HSL-MI20: An efficient AMG Preconditioned Technical report RAL-TR-2007-021, SFTC Rutherford Appleton Laboratory, Did-cot, United Kingdom, 2007.
6. S. C. BRENNER AND L. R. SCOTT, The Mathematical Theory of Finite Element Methods,2nd ed., Springer-Verlag, New York, 2002.
7. C. CANUTO, M. Y. HUSSAINI, A. QUARTERONI, AND T. A. ZANG, Spectral Methods. Fundamentals in Single Domains, Springer-Verlag, Berlin, 2006.
8. R. COURANT AND D. HILBERT, Methods of Mathematical Physics, Vol. I, Interscience, New York, 1953.
9. M. DEB, Solution of Stochastic Partial Differential Equations (SPDEs) Using Galerkin Method: Theory and Applications, Ph.D. thesis, The University of Texas, Austin, TX, 2000.
10. O. G. ERNST, C. E. POWELL, D. J. SILVESTER, AND E. UL LMANN, Efficient solvers for a linear stochastic Galerkin mixed formulation of diffusion problems with random data,SIAM J. Sci. Comput., 31 (2009), pp. 1424-1447.
11. P. FRAUENFELDER, C. SCHWAB, AND R. A. TODOR, Finite elements for elliptic problems with stochastic coefficients, Comput. Methods Appl. Mech . Engrg., 194 (2005), pp. 205-228.
12. R. GHANEM AND P. SPANOS, Stochastic finite elements: A spectral approach, Springer-Verlag, New York, 1991.
13. R. G. GHANEM AND R. M. KRUGER, Numerical solution of spectral stochastic finite element systems, Comput. Methods Appl. Mech. Engrg., 129 (1996), pp. 289-303.
14. M. R. HESTENES AND E. STIEFEL, Methods of conjugate gradients for solving linear systems, J. Res. Natl. Bur. Stand., 49 (1952), pp. 409-436.
15. R. A. HORN AND C. R. JOHNSON, Matrix Analysis, reprinted paperback ed., Cambridge University Press, Cambridge, 1999.
16. R. A. HORN AND C. R. JOHNSON, Topics in Matrix Ana lysis, 7th paperback ed., Cambridge University Press, New York, 2006.
17. S. JANSON, Gaussian Hilbert Spaces, Cambridge Universit y Press, Cambridge, 1997.
18. A. KEESE, Numerical Solution of Systems with Stochastic Uncertainties-A General Purpose Framework for Stochastic Finite Elements, Ph.D. thesis, Fachbereich für Mathematik und Informatik, Technische Universität Braunschweig, Braunschweig, Germany, 2004.
19. M. LOÈVE, Probability Theory, vol. II, 4th ed., Springer-Verlag, New York, Heidelberg, Berlin, 1978.
20. P. MALLIAVIN, Stochastic Analysis, Springer-Verlag, Berlin, 1997.
21. H. G. MATTHIES AND A. KEESE, Galerkin methods for lin ear and nonlinear elliptic stochastic partial differential equations, Comput. Methods Appl. Mech. Engrg., 194 (2005), pp. 1295-1331.
22. C. D. MORAVITZ MARTIN AND C. F. VAN LOAN, Shifted Kronecker product systems,SIAMJ. Matrix Anal. Appl., 29 (2006), pp. 184-198.
23. E. J. NYSTRÖM, Über die praktische Auflösung von Integralgleichungen mit Anwendungen auf Randwertaufgaben, Acta Math., 54 (1930), pp. 185-204.
24. M. F. PELLISSETTI AND R. G. GHANEM, Iterative solution of linear systems arising in the context of stochastic finite elements, Adv. Eng. Softw., 31 (2000), pp. 607-616.
25. C. E. POWELL AND H. C. ELMAN, Block-diagonal preconditioning for spectral stochastic finite-element systems, IMA J. Numer. Anal., 29 (2009), pp . 350-375.
26. E. ULLMANN, Solution Strategies for Stochastic Finite Element Discretizations,Ph.D. thesis, Fakultuät für Mathematik und Informatik, Technische Universituät Bergakademie Freiberg, Freiberg, Germany, 2008.
27. C. F. VAN LOAN AND N. PlTSIANIS, Approximation with Kronecker products, in Linear algebra for large scale and real-time applications, Proceedings of the NATO Advanced Study Institute, Leuven, Belgium, 1992, M. S. Moonen, G. H . Golub, and B. L. R. de Moor, eds., Kluwer, Dordrecht, 1993, pp. 293-314.
28. D. Xiu AND G. E. KARNIADAKIS, The Wiener-Askey polynomial chaos for stochastic differential equations, SIAM J. Sci. Comput., 24 (2002), pp. 619-644.