Uniformly convergent adaptive methods for a class of parametric operator equations∗

Claude Jeffrey Gittelson

ESAIM: Mathematical Modelling and Numerical Analysis (2012)

  • Volume: 46, Issue: 6, page 1485-1508
  • ISSN: 0764-583X

Abstract

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We derive and analyze adaptive solvers for boundary value problems in which the differential operator depends affinely on a sequence of parameters. These methods converge uniformly in the parameters and provide an upper bound for the maximal error. Numerical computations indicate that they are more efficient than similar methods that control the error in a mean square sense.

How to cite

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Gittelson, Claude Jeffrey. "Uniformly convergent adaptive methods for a class of parametric operator equations∗." ESAIM: Mathematical Modelling and Numerical Analysis 46.6 (2012): 1485-1508. <http://eudml.org/doc/276377>.

@article{Gittelson2012,
abstract = {We derive and analyze adaptive solvers for boundary value problems in which the differential operator depends affinely on a sequence of parameters. These methods converge uniformly in the parameters and provide an upper bound for the maximal error. Numerical computations indicate that they are more efficient than similar methods that control the error in a mean square sense.},
author = {Gittelson, Claude Jeffrey},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Parametric partial differential equations; partial differential equations with random coefficients; uniform convergence; adaptive methods; operator equations; parametric partial differential equations},
language = {eng},
month = {6},
number = {6},
pages = {1485-1508},
publisher = {EDP Sciences},
title = {Uniformly convergent adaptive methods for a class of parametric operator equations∗},
url = {http://eudml.org/doc/276377},
volume = {46},
year = {2012},
}

TY - JOUR
AU - Gittelson, Claude Jeffrey
TI - Uniformly convergent adaptive methods for a class of parametric operator equations∗
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2012/6//
PB - EDP Sciences
VL - 46
IS - 6
SP - 1485
EP - 1508
AB - We derive and analyze adaptive solvers for boundary value problems in which the differential operator depends affinely on a sequence of parameters. These methods converge uniformly in the parameters and provide an upper bound for the maximal error. Numerical computations indicate that they are more efficient than similar methods that control the error in a mean square sense.
LA - eng
KW - Parametric partial differential equations; partial differential equations with random coefficients; uniform convergence; adaptive methods; operator equations; parametric partial differential equations
UR - http://eudml.org/doc/276377
ER -

References

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  1. I. Babuška and P. Chatzipantelidis, On solving elliptic stochastic partial differential equations. Comput. Methods Appl. Mech. Eng.191 (2002) 4093–4122.  Zbl1019.65010
  2. I.M. Babuška, R. Tempone and G.E. Zouraris, Galerkin finite element approximations of stochastic elliptic partial differential equations. SIAM J. Numer. Anal.42 (2004) 800–825 (electronic).  Zbl1080.65003
  3. I.M. 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) 1005–1034 (electronic).  Zbl1151.65008
  4. A. Barinka, Fast Evaluation Tools for Adaptive Wavelet Schemes. Ph.D. thesis, RWTH Aachen (2005).  
  5. M. Bieri and C. Schwab, Sparse high order FEM for elliptic sPDEs. Comput. Methods Appl. Mech. Eng.198 (2009) 1149–1170.  Zbl1157.65481
  6. M. Bieri, R. Andreev and C. Schwab, Sparse tensor discretization of elliptic SPDEs. SIAM J. Sci. Comput.31 (2009/2010) 4281–4304.  Zbl1205.35346
  7. P. Binev, W. Dahmen and R.A. DeVore, Adaptive finite element methods with convergence rates. Numer. Math.97 (2004) 219–268.  Zbl1063.65120
  8. A. Chkifa, A. Cohen, R. DeVore and C. Schwab, Sparse adaptive Taylor approximation algorithms for parametric and stochastic elliptic PDEs. Technical Report 44, SAM, ETHZ (2011).  Zbl1273.65009
  9. A. Cohen, W. Dahmen and R.A. DeVore, Adaptive wavelet methods for elliptic operator equations : convergence rates. Math. Comput.70 (2001) 27–75(electronic).  Zbl0980.65130
  10. A. Cohen, W. Dahmen and R.A. DeVore, Adaptive wavelet methods. II. Beyond the elliptic case. Found. Comput. Math.2 (2002) 203–245.  Zbl1025.65056
  11. A. Cohen, R.A. DeVore and C. Schwab, Convergence rates of best -term Galerkin approximations for a class of elliptic sPDEs. Found. Comput. Math.10 (2010) 615–646.  Zbl1206.60064
  12. A. Cohen, R. DeVore and C. Schwab, Analytic regularity and polynomial approximation of parametric and stochastic elliptic PDE’s. Anal. Appl. (Singap.)9 (2011) 11–47.  Zbl1219.35379
  13. S. Dahlke, M. Fornasier and T. Raasch, Adaptive frame methods for elliptic operator equations. Adv. Comput. Math.27 (2007) 27–63.  Zbl1122.65103
  14. S. Dahlke, T. Raasch, M. Werner, M. Fornasier and R. Stevenson, Adaptive frame methods for elliptic operator equations : the steepest descent approach. IMA J. Numer. Anal.27 (2007) 717–740.  Zbl1153.65050
  15. M.K. Deb, I.M. Babuška and J.T. Oden, Solution of stochastic partial differential equations using Galerkin finite element techniques. Comput. Methods Appl. Mech. Eng.190 (2001) 6359–6372.  Zbl1075.65006
  16. T.J. Dijkema, C. Schwab and R. Stevenson, An adaptive wavelet method for solving high-dimensional elliptic PDEs. Constr. Approx.30 (2009) 423–455.  Zbl1205.65313
  17. W. Dörfler, A convergent adaptive algorithm for Poisson’s equation. SIAM J. Numer. Anal.33 (1996) 1106–1124.  Zbl0854.65090
  18. P. Frauenfelder, C. Schwab and R.A. Todor, Finite elements for elliptic problems with stochastic coefficients. Comput. Methods Appl. Mech. Eng.194 (2005) 205–228.  Zbl1143.65392
  19. T. Gantumur, H. Harbrecht and R. Stevenson, An optimal adaptive wavelet method without coarsening of the iterands. Math. Comput.76 (2007) 615–629 (electronic).  Zbl1115.41023
  20. W. Gautschi, Orthogonal polynomials : computation and approximation, in Numer. Math. Sci. Comput. Oxford University Press, Oxford Science Publications, New York (2004).  Zbl1130.42300
  21. R.G. Ghanem and P.D. Spanos, Stochastic finite elements : a spectral approach. Springer-Verlag, New York (1991).  Zbl0722.73080
  22. C.J. Gittelson, Adaptive Galerkin Methods for Parametric and Stochastic Operator Equations. Ph.D. thesis, ETH Dissertation No. 19533. ETH Zürich (2011).  
  23. C.J. Gittelson, An adaptive stochastic Galerkin method for random elliptic operators. Math. Comput. (2011). To appear.  Zbl1268.35131
  24. C.J. Gittelson, Convergence Rates of Multilevel and Sparse Tensor Approximations for a Random Elliptic PDE (2012). Submitted.  Zbl1274.35438
  25. I.G. Graham, F.Y. Kuo, D. Nuyens, R. Scheichl and I.H. Sloan, Quasi-Monte Carlo methods for elliptic PDEs with random coefficients and applications. J. Comput. Phys.230 (2011) 3668–3694.  Zbl1218.65009
  26. R.V. Kadison and J.R. Ringrose, Fundamentals of the theory of operator algebras I, Elementary theory, Reprint of the 1983 original, in Graduate Studies in Mathematics. Amer. Math. Soc.15 (1997).  Zbl0888.46039
  27. H.G. Matthies and A. Keese, Galerkin methods for linear and nonlinear elliptic stochastic partial differential equations. Comput. Methods Appl. Mech. Eng.194 (2005) 1295–1331.  Zbl1088.65002
  28. A. Metselaar, Handling Wavelet Expansions in Numerical Methods. Ph.D. thesis, University of Twente (2002).  
  29. P. Morin, R.H. Nochetto and K.G. Siebert, Data oscillation and convergence of adaptive FEM. SIAM J. Numer. Anal.38 (2000) 466–488 (electronic).  Zbl0970.65113
  30. F. Nobile, R. Tempone and C.G. Webster, An anisotropic sparse grid stochastic collocation method for partial differential equations with random input data. SIAM J. Numer. Anal.46 (2008) 2411–2442.  Zbl1176.65007
  31. W. Rudin, Functional analysis, 2nd edition. International Series in Pure Appl. Math. McGraw-Hill Inc., New York (1991).  
  32. C. Schwab and C.J. Gittelson, Sparse tensor discretization of high-dimensional parametric and stochastic PDEs. Acta Numer.20 (2011) 291–467.  Zbl1269.65010
  33. R. Stevenson, Adaptive solution of operator equations using wavelet frames. SIAM J. Numer. Anal.41 (2003) 1074–1100 (electronic).  Zbl1057.41010
  34. M.H. Stone, The generalized Weierstrass approximation theorem. Math. Mag.21 (1948) 237–254.  
  35. G. Szegő, Orthogonal polynomials, 4th edition, in Colloq. Publ. XXIII. Amer. Math. Soc. (1975).  Zbl0305.42011
  36. R.A. Todor and C. Schwab, Convergence rates for sparse chaos approximations of elliptic problems with stochastic coefficients. IMA J. Numer. Anal.27 (2007) 232–261.  Zbl1120.65004
  37. X. Wan and G.E. Karniadakis, An adaptive multi-element generalized polynomial chaos method for stochastic differential equations. J. Comput. Phys.209 (2005) 617–642.  Zbl1078.65008
  38. X. Wan and G.E. Karniadakis, Multi-element generalized polynomial chaos for arbitrary probability measures. SIAM J. Sci. Comput.28 (2006) 901–928 (electronic).  Zbl1128.65009
  39. X. Wan and G.E. Karniadakis, Solving elliptic problems with non-Gaussian spatially-dependent random coefficients. Comput. Methods Appl. Mech. Eng.198 (2009) 1985–1995.  Zbl1227.65014
  40. D. Xiu, Efficient collocational approach for parametric uncertainty analysis. Commun. Comput. Phys.2 (2007) 293–309.  Zbl1164.65302
  41. D. Xiu, Numerical methods for stochastic computations : A spectral method approach. Princeton University Press, Princeton, NJ (2010).  Zbl1210.65002
  42. D. Xiu and J.S. Hesthaven, High-order collocation methods for differential equations with random inputs. SIAM J. Sci. Comput.27 (2005) 1118–1139 (electronic).  Zbl1091.65006
  43. D. Xiu and G.E. Karniadakis, The Wiener-Askey polynomial chaos for stochastic differential equations. SIAM J. Sci. Comput.24 (2002) 619–644(electronic).  Zbl1014.65004

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