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Sparse adaptive Taylor approximation algorithms for parametric and stochastic elliptic PDEs∗

Published online by Cambridge University Press:  29 November 2012

Abdellah Chkifa ,
Albert Cohen ,
Ronald DeVore  and
Christoph Schwab
Abdellah Chkifa
Affiliation:
UPMC Univ. Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, 75005 Paris, France.. chkifa@ann.jussieu.fr; cohen@ann.jussieu.fr CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, 75005 Paris, France.
Albert Cohen
Affiliation:
UPMC Univ. Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, 75005 Paris, France.. chkifa@ann.jussieu.fr; cohen@ann.jussieu.fr CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, 75005 Paris, France.
Ronald DeVore
Affiliation:
Department of Mathematics, Texas A&M University, College Station, 77843 TX, USA.; rdevore@math.tamu.ed
Christoph Schwab
Affiliation:
Seminar for Applied Mathematics, ETH Zürich, 8092 Zürich, Switzerland.; schwab@math.ethz.ch
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Abstract

The numerical approximation of parametric partial differential equations is a computational challenge, in particular when the number of involved parameter is large. This paper considers a model class of second order, linear, parametric, elliptic PDEs on a bounded domain D with diffusion coefficients depending on the parameters in an affine manner. For such models, it was shown in [9, 10] that under very weak assumptions on the diffusion coefficients, the entire family of solutions to such equations can be simultaneously approximated in the Hilbert space V = H01(D) by multivariate sparse polynomials in the parameter vector y with a controlled number N of terms. The convergence rate in terms of N does not depend on the number of parameters in V, which may be arbitrarily large or countably infinite, thereby breaking the curse of dimensionality. However, these approximation results do not describe the concrete construction of these polynomial expansions, and should therefore rather be viewed as benchmark for the convergence analysis of numerical methods. The present paper presents an adaptive numerical algorithm for constructing a sequence of sparse polynomials that is proved to converge toward the solution with the optimal benchmark rate. Numerical experiments are presented in large parameter dimension, which confirm the effectiveness of the adaptive approach.

Keywords

Parametric and stochastic PDE’ssparse polynomial approximationhigh dimensional problemsadaptive algorithms
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 47 , Issue 1 , January 2013 , pp. 253 - 280
Copyright
© EDP Sciences, SMAI, 2012

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References

Références

Babuska, I., Tempone, R. and Zouraris, G.E., Galerkin finite element approximations of stochastic elliptic partial differential equations. SIAM J. Numer. Anal. 42 (2004) 800−825. Google Scholar
Babuška, I., Nobile, F. and Tempone, R., A stochastic collocation method for elliptic partial differential equations with random input data. SIAM J. Numer. Anal. 45 (2007) 1005−1034. Google Scholar
Bernardi, C. and Verfürth, R., Adaptive finite element methods for elliptic equations with non-smooth coefficients. Numer. Math. 85 (2000) 579−608. Google Scholar
Binev, P., Dahmen, W., and DeVore, R., Adaptive finite element methods with convergence rates. Numer. Math. 97 (2004) 219−268. Google Scholar
Buffa, A., Maday, Y., Patera, A.T., Prudhomme, C. and Turinici, G., A priori convergence of the greedy algorithm for the parameterized reduced basis. ESAIM : M2AN 3 (2012) 595–603. Google Scholar
A. Cohen, Numerical analysis of wavelet methods. Elsevier, Amsterdam (2003).
Cohen, A., Dahmen, W. and DeVore, R., Adaptive wavelet methods for elliptic operator equations – Convergence rates. Math. Comput. 70 (2000) 27−75. Google Scholar
Cohen, A., Dahmen, W. and DeVore, R., Adaptive wavelet methods for operator equations – Beyond the elliptic case. J. FoCM 2 (2002) 203−245. Google Scholar
Cohen, A., DeVore, R. and Schwab, C., Convergence rates of best N-term Galerkin approximations for a class of elliptic sPDEs. Found. Comput. Math. 10 (2010) 615–646. Google Scholar
Cohen, A., DeVore, R. and Schwab, C., Analytic regularity and polynomial approximation of parametric and stochastic PDE’s. Anal. Appl. 9 (2011) 11–47. Google Scholar
DeVore, R., Nonlinear approximation. Acta Numer. 7 (1998) 51–150. Google Scholar
Dörfler, W., A convergent adaptive algorithm for Poisson’s equation. SIAM J. Numer. Anal. 33 (1996) 1106–1124. Google Scholar
Frauenfelder, Ph., Schwab, Ch. and Todor, R.A., Finite elements for elliptic problems with stochastic coefficients. Comput. Methods Appl. Mech. Eng. 194 (2005) 205–228. Google Scholar
Gantumur, T., Harbrecht, H. and Stevenson, R., An optimal adaptive wavelet method without coarsening of the iterands. Math. Comput. 76 (2007) 615–629. Google Scholar
Ghanem, R. and Spanos, P., Spectral techniques for stochastic finite elements. Arch. Comput. Methods Eng. 4 (1997) 63–100. Google Scholar
P. Grisvard, Elliptic problems on non-smooth domains. Pitman (1983).
V.H. Hoang and Ch. Schwab, Sparse tensor Galerkin discretizations for parametric and random parabolic PDEs I : Analytic regularity and gpc-approximation. Report 2010-11, Seminar for Applied Mathematics, ETH Zürich (in review).
V.H. Hoang and Ch. Schwab, Analytic regularity and gpc approximation for parametric and random 2nd order hyperbolic PDEs. Report 2010-19, Seminar for Applied Mathematics, ETH Zürich (to appear in Anal. Appl. (2011)).
M. Kleiber and T.D. Hien, The stochastic finite element methods. John Wiley & Sons, Chichester (1992).
Milani, R., Quarteroni, A. and Rozza, G., Reduced basis methods in linear elasticity with many parameters. Comput. Methods Appl. Mech. Eng. 197 (2008) 4812–4829. Google Scholar
Morin, P., Nochetto, R.H. and Siebert, K.G., Data oscillation and convergence of adaptive FEM. SIAM J. Numer. Anal. 38 (2000) 466–488. Google Scholar
Nobile, F., Tempone, R. and Webster, C.G., A sparse grid stochastic collocation method for elliptic partial differential equations with random input data. SIAM J. Numer. Anal. 46 (2008) 2309–2345. Google Scholar
Nobile, F., Tempone, R. and Webster, C.G., An anisotropic sparse grid stochastic collocation method for elliptic partial differential equations with random input data. SIAM J. Numer. Anal. 46 (2008) 2411–2442. Google Scholar
Ch. Schwab and A.M. Stuart Sparse deterministic approximation of Bayesian inverse problems. Report 2011-16, Seminar for Applied Mathematics, ETH Zürich (to appear in Inverse Probl.).
Schwab, Ch. and Todor, R.A., Karhúnen-Loève, Approximation of random fields by generalized fast multipole methods. J. Comput. Phys. 217 (2000) 100–122. Google Scholar
Stevenson, R., Optimality of a standard adaptive finite element method. Found. Comput. Math. 7 (2007) 245–269. Google Scholar