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.
Mots-clés : parametric partial differential equations, partial differential equations with random coefficients, uniform convergence, adaptive methods, operator equations
@article{M2AN_2012__46_6_1485_0, author = {Gittelson, Claude Jeffrey}, title = {Uniformly convergent adaptive methods for a class of parametric operator equations}, journal = {ESAIM: Mathematical Modelling and Numerical Analysis }, pages = {1485--1508}, publisher = {EDP-Sciences}, volume = {46}, number = {6}, year = {2012}, doi = {10.1051/m2an/2012013}, mrnumber = {2996337}, zbl = {1276.65068}, language = {en}, url = {http://www.numdam.org/articles/10.1051/m2an/2012013/} }
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 PY - 2012 SP - 1485 EP - 1508 VL - 46 IS - 6 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2012013/ DO - 10.1051/m2an/2012013 LA - en ID - M2AN_2012__46_6_1485_0 ER -
%0 Journal Article %A Gittelson, Claude Jeffrey %T Uniformly convergent adaptive methods for a class of parametric operator equations %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2012 %P 1485-1508 %V 46 %N 6 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2012013/ %R 10.1051/m2an/2012013 %G en %F M2AN_2012__46_6_1485_0
Gittelson, Claude Jeffrey. Uniformly convergent adaptive methods for a class of parametric operator equations. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 46 (2012) no. 6, pp. 1485-1508. doi : 10.1051/m2an/2012013. http://www.numdam.org/articles/10.1051/m2an/2012013/
[1] On solving elliptic stochastic partial differential equations. Comput. Methods Appl. Mech. Eng. 191 (2002) 4093-4122. | MR | Zbl
and ,[2] Galerkin finite element approximations of stochastic elliptic partial differential equations. SIAM J. Numer. Anal. 42 (2004) 800-825 (electronic). | MR | Zbl
, and ,[3] A stochastic collocation method for elliptic partial differential equations with random input data. SIAM J. Numer. Anal. 45 (2007) 1005-1034 (electronic). | MR | Zbl
, and ,[4] Fast Evaluation Tools for Adaptive Wavelet Schemes. Ph.D. thesis, RWTH Aachen (2005).
,[5] Sparse high order FEM for elliptic sPDEs. Comput. Methods Appl. Mech. Eng. 198 (2009) 1149-1170. | MR | Zbl
and ,[6] Sparse tensor discretization of elliptic SPDEs. SIAM J. Sci. Comput. 31 (2009/2010) 4281-4304. | MR | Zbl
, and ,[7] Adaptive finite element methods with convergence rates. Numer. Math. 97 (2004) 219-268. | MR | Zbl
, and ,[8] Sparse adaptive Taylor approximation algorithms for parametric and stochastic elliptic PDEs. Technical Report 44, SAM, ETHZ (2011).
, , and ,[9] Adaptive wavelet methods for elliptic operator equations : convergence rates. Math. Comput. 70 (2001) 27-75 (electronic). | MR | Zbl
, and ,[10] Adaptive wavelet methods. II. Beyond the elliptic case. Found. Comput. Math. 2 (2002) 203-245. | MR | Zbl
, and ,[11] Convergence rates of best -term Galerkin approximations for a class of elliptic sPDEs. Found. Comput. Math. 10 (2010) 615-646. | MR | Zbl
, and ,[12] Analytic regularity and polynomial approximation of parametric and stochastic elliptic PDE's. Anal. Appl. (Singap.) 9 (2011) 11-47. | MR | Zbl
, and ,[13] Adaptive frame methods for elliptic operator equations. Adv. Comput. Math. 27 (2007) 27-63. | MR | Zbl
, and ,[14] Adaptive frame methods for elliptic operator equations : the steepest descent approach. IMA J. Numer. Anal. 27 (2007) 717-740. | MR | Zbl
, , , and ,[15] Solution of stochastic partial differential equations using Galerkin finite element techniques. Comput. Methods Appl. Mech. Eng. 190 (2001) 6359-6372. | MR | Zbl
, and ,[16] An adaptive wavelet method for solving high-dimensional elliptic PDEs. Constr. Approx. 30 (2009) 423-455. | MR | Zbl
, and ,[17] A convergent adaptive algorithm for Poisson's equation. SIAM J. Numer. Anal. 33 (1996) 1106-1124. | MR | Zbl
,[18] Finite elements for elliptic problems with stochastic coefficients. Comput. Methods Appl. Mech. Eng. 194 (2005) 205-228. | MR | Zbl
, and ,[19] An optimal adaptive wavelet method without coarsening of the iterands. Math. Comput. 76 (2007) 615-629 (electronic). | MR | Zbl
, and ,[20] Orthogonal polynomials : computation and approximation, in Numer. Math. Sci. Comput. Oxford University Press, Oxford Science Publications, New York (2004). | MR | Zbl
,[21] Stochastic finite elements : a spectral approach. Springer-Verlag, New York (1991). | MR | Zbl
and ,[22] Adaptive Galerkin Methods for Parametric and Stochastic Operator Equations. Ph.D. thesis, ETH Dissertation No. 19533. ETH Zürich (2011).
,[23] An adaptive stochastic Galerkin method for random elliptic operators. Math. Comput. (2011). To appear. | MR | Zbl
,[24] Convergence Rates of Multilevel and Sparse Tensor Approximations for a Random Elliptic PDE (2012). Submitted. | MR | Zbl
,[25] Quasi-Monte Carlo methods for elliptic PDEs with random coefficients and applications. J. Comput. Phys. 230 (2011) 3668-3694. | MR | Zbl
, , , and ,[26] 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). | MR | Zbl
and ,[27] Galerkin methods for linear and nonlinear elliptic stochastic partial differential equations. Comput. Methods Appl. Mech. Eng. 194 (2005) 1295-1331. | MR | Zbl
and ,[28] Handling Wavelet Expansions in Numerical Methods. Ph.D. thesis, University of Twente (2002). | MR
,[29] Data oscillation and convergence of adaptive FEM. SIAM J. Numer. Anal. 38 (2000) 466-488 (electronic). | MR | Zbl
, and ,[30] An anisotropic sparse grid stochastic collocation method for partial differential equations with random input data. SIAM J. Numer. Anal. 46 (2008) 2411-2442. | MR | Zbl
, and ,[31] Functional analysis, 2nd edition. International Series in Pure Appl. Math. McGraw-Hill Inc., New York (1991). | MR | Zbl
,[32] Sparse tensor discretization of high-dimensional parametric and stochastic PDEs. Acta Numer. 20 (2011) 291-467. | MR | Zbl
and ,[33] Adaptive solution of operator equations using wavelet frames. SIAM J. Numer. Anal. 41 (2003) 1074-1100 (electronic). | MR | Zbl
,[34] The generalized Weierstrass approximation theorem. Math. Mag. 21 (1948) 237-254. | MR | Zbl
,[35] Orthogonal polynomials, 4th edition, in Colloq. Publ. XXIII. Amer. Math. Soc. (1975).
,[36] Convergence rates for sparse chaos approximations of elliptic problems with stochastic coefficients. IMA J. Numer. Anal. 27 (2007) 232-261. | MR | Zbl
and ,[37] An adaptive multi-element generalized polynomial chaos method for stochastic differential equations. J. Comput. Phys. 209 (2005) 617-642. | MR | Zbl
and ,[38] Multi-element generalized polynomial chaos for arbitrary probability measures. SIAM J. Sci. Comput. 28 (2006) 901-928 (electronic). | MR | Zbl
and ,[39] Solving elliptic problems with non-Gaussian spatially-dependent random coefficients. Comput. Methods Appl. Mech. Eng. 198 (2009) 1985-1995. | MR | Zbl
and ,[40] Efficient collocational approach for parametric uncertainty analysis. Commun. Comput. Phys. 2 (2007) 293-309. | MR | Zbl
,[41] Numerical methods for stochastic computations : A spectral method approach. Princeton University Press, Princeton, NJ (2010). | MR | Zbl
,[42] High-order collocation methods for differential equations with random inputs. SIAM J. Sci. Comput. 27 (2005) 1118-1139 (electronic). | MR | Zbl
and ,[43] The Wiener-Askey polynomial chaos for stochastic differential equations. SIAM J. Sci. Comput. 24 (2002) 619-644 (electronic). | MR | Zbl
and ,Cité par Sources :