Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



An adaptive stochastic Galerkin method for random elliptic operators

Author: Claude Jeffrey Gittelson
Journal: Math. Comp. 82 (2013), 1515-1541
MSC (2010): Primary 35R60, 47B80, 60H25, 65C20, 65N12, 65N22, 65N30, 65J10, 65Y20
Published electronically: February 12, 2013
MathSciNet review: 3042573
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We derive an adaptive solver for random elliptic boundary value problems, using techniques from adaptive wavelet methods. Substituting wavelets by polynomials of the random parameters leads to a modular solver for the parameter dependence of the random solution, which combines with any discretization on the spatial domain. In addition to selecting active polynomial modes, this solver can adaptively construct a separate spatial discretization for each of their coefficients. We show convergence of the solver in this general setting, along with a computable bound for the mean square error, and an optimality property in the case of a single spatial discretization. Numerical computations demonstrate convergence of the solver and compare it to a sparse tensor product construction.

References [Enhancements On Off] (What's this?)

  • 1. Mark Ainsworth and J. Tinsley Oden, A posteriori error estimation in finite element analysis, Pure and Applied Mathematics (New York), Wiley-Interscience [John Wiley & Sons], New York, 2000. MR 1885308 (2003b:65001)
  • 2. Ivo M. Babuška, Raúl Tempone, and Georgios E. Zouraris, Galerkin finite element approximations of stochastic elliptic partial differential equations, SIAM J. Numer. Anal. 42 (2004), no. 2, 800-825 (electronic). MR 2084236 (2005h:65012)
  • 3. A. Barinka, Fast evaluation tools for adaptive wavelet schemes, Ph.D. thesis, RWTH Aachen, March 2005.
  • 4. Heinz Bauer, Wahrscheinlichkeitstheorie, Fifth ed., de Gruyter Lehrbuch. [de Gruyter Textbook], Walter de Gruyter & Co., Berlin, 2002. MR 1902050 (2003b:60001)
  • 5. Marcel Bieri, Roman Andreev, and Christoph Schwab, Sparse tensor discretization of elliptic SPDEs, SIAM J. Sci. Comput. 31 (2009/10), no. 6, 4281-4304. MR 2566594
  • 6. Marcel Bieri and Christoph Schwab, Sparse high order FEM for elliptic sPDEs, Comput. Methods Appl. Mech. Engrg. 198 (2009), no. 37-40, 1149-1170. MR 2500242 (2010g:65205)
  • 7. Peter Binev, Wolfgang Dahmen, and Ronald A. DeVore, Adaptive finite element methods with convergence rates, Numer. Math. 97 (2004), no. 2, 219-268. MR 2050077 (2005d:65222)
  • 8. Albert Cohen, Wolfgang Dahmen, and Ronald A. DeVore, Adaptive wavelet methods for elliptic operator equations: convergence rates, Math. Comp. 70 (2001), no. 233, 27-75 (electronic). MR 1803124 (2002h:65201)
  • 9. -, Adaptive wavelet methods. II. Beyond the elliptic case, Found. Comput. Math. 2 (2002), no. 3, 203-245. MR 1907380 (2003f:65212)
  • 10. Albert Cohen, Ronald DeVore, and Christoph Schwab, Analytic regularity and polynomial approximation of parametric and stochastic elliptic PDE's, Anal. Appl. (Singap.) 9 (2011), no. 1, 11-47. MR 2763359
  • 11. Albert Cohen, Ronald A. DeVore, and Christoph Schwab, Convergence rates of best $ N$-term Galerkin approximations for a class of elliptic sPDEs, Found. Comput. Math. 10 (2010), no. 6, 615-646. MR 2728424
  • 12. Stephan Dahlke, Massimo Fornasier, and Thorsten Raasch, Adaptive frame methods for elliptic operator equations, Adv. Comput. Math. 27 (2007), no. 1, 27-63. MR 2317920 (2008c:65366)
  • 13. Stephan Dahlke, Thorsten Raasch, Manuel Werner, Massimo Fornasier, and Rob Stevenson, Adaptive frame methods for elliptic operator equations: the steepest descent approach, IMA J. Numer. Anal. 27 (2007), no. 4, 717-740. MR 2371829 (2008i:65239)
  • 14. Manas K. Deb, Ivo M. Babuška, and J. Tinsley Oden, Solution of stochastic partial differential equations using Galerkin finite element techniques, Comput. Methods Appl. Mech. Engrg. 190 (2001), no. 48, 6359-6372. MR 1870425 (2003g:65009)
  • 15. Ronald A. DeVore, Nonlinear approximation, Acta Numerica, 1998, Acta Numer., vol. 7, Cambridge Univ. Press, Cambridge, 1998, pp. 51-150. MR 1689432 (2001a:41034)
  • 16. Tammo Jan Dijkema, Christoph Schwab, and Rob Stevenson, An adaptive wavelet method for solving high-dimensional elliptic PDEs, Constr. Approx. 30 (2009), no. 3, 423-455. MR 2558688
  • 17. Willy Dörfler, A convergent adaptive algorithm for Poisson's equation, SIAM J. Numer. Anal. 33 (1996), no. 3, 1106-1124. MR 1393904 (97e:65139)
  • 18. Philipp Frauenfelder, Christoph Schwab, and Radu Alexandru Todor, Finite elements for elliptic problems with stochastic coefficients, Comput. Methods Appl. Mech. Engrg. 194 (2005), no. 2-5, 205-228. MR 2105161 (2005i:65186)
  • 19. Tsogtgerel Gantumur, Helmut Harbrecht, and Rob Stevenson, An optimal adaptive wavelet method without coarsening of the iterands, Math. Comp. 76 (2007), no. 258, 615-629 (electronic). MR 2291830 (2008i:65310)
  • 20. Claude Jeffrey Gittelson, Adaptive Galerkin methods for parametric and stochastic operator equations, Ph.D. thesis, ETH Zürich, 2011, ETH Dissertation No. 19533.
  • 21. -, Adaptive stochastic Galerkin methods: Beyond the elliptic case, Tech. Report 2011-12, Seminar for Applied Mathematics, ETH Zürich, 2011.
  • 22. -, Stochastic Galerkin approximation of operator equations with infinite dimensional noise, Tech. Report 2011-10, Seminar for Applied Mathematics, ETH Zürich, 2011.
  • 23. Hermann G. Matthies and Andreas Keese, Galerkin methods for linear and nonlinear elliptic stochastic partial differential equations, Comput. Methods Appl. Mech. Engrg. 194 (2005), no. 12-16, 1295-1331. MR 2121216 (2005j:65146)
  • 24. A. Metselaar, Handling wavelet expansions in numerical methods, Ph.D. thesis, University of Twente, 2002. MR 2715507
  • 25. Pedro Morin, Ricardo H. Nochetto, and Kunibert G. Siebert, Data oscillation and convergence of adaptive FEM, SIAM J. Numer. Anal. 38 (2000), no. 2, 466-488 (electronic). MR 1770058 (2001g:65157)
  • 26. Christoph Schwab and Claude Jeffrey Gittelson, Sparse tensor discretization of high-dimensional parametric and stochastic PDEs, Acta Numerica, Acta Numer., vol. 20, Cambridge Univ. Press, Cambridge, 2011, pp. 291-467. MR 2805155
  • 27. Rob Stevenson, Adaptive solution of operator equations using wavelet frames, SIAM J. Numer. Anal. 41 (2003), no. 3, 1074-1100 (electronic). MR 2005196 (2004e:42062)
  • 28. Radu Alexandru Todor and Christoph Schwab, Convergence rates for sparse chaos approximations of elliptic problems with stochastic coefficients, IMA J. Numer. Anal. 27 (2007), no. 2, 232-261. MR 2317004 (2008b:65016)
  • 29. R. Verfürth, A Review of a Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques, Teubner Verlag and J. Wiley, Stuttgart, 1996.
  • 30. Xiaoliang Wan and George Em Karniadakis, An adaptive multi-element generalized polynomial chaos method for stochastic differential equations, J. Comput. Phys. 209 (2005), no. 2, 617-642. MR 2151997 (2006e:65007)
  • 31. -, Multi-element generalized polynomial chaos for arbitrary probability measures, SIAM J. Sci. Comput. 28 (2006), no. 3, 901-928 (electronic). MR 2240796 (2007d:65008)
  • 32. Dongbin Xiu and George Em Karniadakis, The Wiener-Askey polynomial chaos for stochastic differential equations, SIAM J. Sci. Comput. 24 (2002), no. 2, 619-644 (electronic). MR 1951058 (2003m:60174)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2010): 35R60, 47B80, 60H25, 65C20, 65N12, 65N22, 65N30, 65J10, 65Y20

Retrieve articles in all journals with MSC (2010): 35R60, 47B80, 60H25, 65C20, 65N12, 65N22, 65N30, 65J10, 65Y20

Additional Information

Claude Jeffrey Gittelson
Affiliation: Seminar for Applied Mathematics, ETH Zurich, Rämistrasse 101, CH-8092 Zurich, Switzerland
Address at time of publication: Department of Mathematics, Purdue University, 150 N. University Street, West Lafayette, Indiana 47907

Received by editor(s): March 2, 2011
Received by editor(s) in revised form: September 24, 2011
Published electronically: February 12, 2013
Additional Notes: This research was supported in part by the Swiss National Science Foundation grant No. 200021-120290/1.
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society