An adaptive stochastic Galerkin method for random elliptic operators
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- by Claude Jeffrey Gittelson
- Math. Comp. 82 (2013), 1515-1541
- DOI: https://doi.org/10.1090/S0025-5718-2013-02654-3
- Published electronically: February 12, 2013
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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
- 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, DOI 10.1002/9781118032824
- Ivo 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. MR 2084236, DOI 10.1137/S0036142902418680
- A. Barinka, Fast evaluation tools for adaptive wavelet schemes, Ph.D. thesis, RWTH Aachen, March 2005.
- Heinz Bauer, Wahrscheinlichkeitstheorie, 5th ed., de Gruyter Lehrbuch. [de Gruyter Textbook], Walter de Gruyter & Co., Berlin, 2002 (German). MR 1902050
- 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, DOI 10.1137/090749256
- Marcel Bieri and Christoph Schwab, Sparse high order FEM for elliptic sPDEs, Comput. Methods Appl. Mech. Engrg. 198 (2009), no. 13-14, 1149–1170. MR 2500242, DOI 10.1016/j.cma.2008.08.019
- Peter Binev, Wolfgang Dahmen, and Ron DeVore, Adaptive finite element methods with convergence rates, Numer. Math. 97 (2004), no. 2, 219–268. MR 2050077, DOI 10.1007/s00211-003-0492-7
- Albert Cohen, Wolfgang Dahmen, and Ronald DeVore, Adaptive wavelet methods for elliptic operator equations: convergence rates, Math. Comp. 70 (2001), no. 233, 27–75. MR 1803124, DOI 10.1090/S0025-5718-00-01252-7
- A. Cohen, W. Dahmen, and R. DeVore, Adaptive wavelet methods. II. Beyond the elliptic case, Found. Comput. Math. 2 (2002), no. 3, 203–245. MR 1907380, DOI 10.1007/s102080010027
- 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, DOI 10.1142/S0219530511001728
- Albert Cohen, Ronald 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, DOI 10.1007/s10208-010-9072-2
- 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, DOI 10.1007/s10444-005-7501-6
- 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, DOI 10.1093/imanum/drl035
- 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, DOI 10.1016/S0045-7825(01)00237-7
- Ronald A. DeVore, Nonlinear approximation, Acta numerica, 1998, Acta Numer., vol. 7, Cambridge Univ. Press, Cambridge, 1998, pp. 51–150. MR 1689432, DOI 10.1017/S0962492900002816
- 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, DOI 10.1007/s00365-009-9064-0
- Willy Dörfler, A convergent adaptive algorithm for Poisson’s equation, SIAM J. Numer. Anal. 33 (1996), no. 3, 1106–1124. MR 1393904, DOI 10.1137/0733054
- 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, DOI 10.1016/j.cma.2004.04.008
- 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. MR 2291830, DOI 10.1090/S0025-5718-06-01917-X
- Claude Jeffrey Gittelson, Adaptive Galerkin methods for parametric and stochastic operator equations, Ph.D. thesis, ETH Zürich, 2011, ETH Dissertation No. 19533.
- —, Adaptive stochastic Galerkin methods: Beyond the elliptic case, Tech. Report 2011-12, Seminar for Applied Mathematics, ETH Zürich, 2011.
- —, Stochastic Galerkin approximation of operator equations with infinite dimensional noise, Tech. Report 2011-10, Seminar for Applied Mathematics, ETH Zürich, 2011.
- 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, DOI 10.1016/j.cma.2004.05.027
- Arend Aalberthus Roeland Metselaar, Handling wavelet expansions in numerical methods, ProQuest LLC, Ann Arbor, MI, 2002. Thesis (Dr.)–Universiteit Twente (The Netherlands). MR 2715507
- 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. MR 1770058, DOI 10.1137/S0036142999360044
- Christoph Schwab and Claude Jeffrey Gittelson, Sparse tensor discretizations of high-dimensional parametric and stochastic PDEs, Acta Numer. 20 (2011), 291–467. MR 2805155, DOI 10.1017/S0962492911000055
- Rob Stevenson, Adaptive solution of operator equations using wavelet frames, SIAM J. Numer. Anal. 41 (2003), no. 3, 1074–1100. MR 2005196, DOI 10.1137/S0036142902407988
- 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, DOI 10.1093/imanum/drl025
- R. Verfürth, A Review of a Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques, Teubner Verlag and J. Wiley, Stuttgart, 1996.
- 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, DOI 10.1016/j.jcp.2005.03.023
- Xiaoliang Wan and George Em Karniadakis, Multi-element generalized polynomial chaos for arbitrary probability measures, SIAM J. Sci. Comput. 28 (2006), no. 3, 901–928. MR 2240796, DOI 10.1137/050627630
- 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. MR 1951058, DOI 10.1137/S1064827501387826
Bibliographic 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
- Email: cgittels@purdue.edu
- 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.
- © Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp. 82 (2013), 1515-1541
- MSC (2010): Primary 35R60, 47B80, 60H25, 65C20, 65N12, 65N22, 65N30, 65J10, 65Y20
- DOI: https://doi.org/10.1090/S0025-5718-2013-02654-3
- MathSciNet review: 3042573