Sparse polynomial approximation of parametric elliptic PDEs. Part I: affine coefficients
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 1, pp. 321-339.

We consider the linear elliptic equation - div ( a u ) = f on some bounded domain D, where a has the affine form a = a ( y ) = a ¯ + Σ j 1 y j ψ j for some parameter vector y = ( y j ) j 1 U = [ - 1 , 1 ] . We study the summability properties of polynomial expansions of the solution map y u ( y ) V : = H 0 1 ( D ) . We consider both Taylor series and Legendre series. Previous results [A. Cohen, R. DeVore and C. Schwab, Anal. Appl. 9 (2011) 11­47] show that, under a uniform ellipticity assuption, for any . We consider both Taylor series and Legendre series. Previous results [A. Cohen, R. DeVore and C. Schwab, Anal. Appl. 9 (2011) 11–47] show that, under a uniform ellipticity assuption, for any 0 < p < 1 , the p summability of the ( ψ j L ) j 1 implies the p summability of the V-norms of the Taylor or Legendre coefficients. Such results ensure convergence rates n - s of polynomial approximations obtained by best n-term truncation of such series, with s = 1 p - 1 in L ( U , V ) or s = 1 p - 1 2 in L 2 ( U , V ) . In this paper we considerably improve these results by providing sufficient conditions of. In this paper we considerably improve these results by providing sufficient conditions of p summability of the coefficient V-norm sequences expressed in terms of the pointwise summability properties of the ( | ψ j | ) j 1 . The approach in the present paper strongly differs from that of [A. Cohen, R. DeVore and C. Schwab, Anal. Appl. 9 (2011) 11–47], which is based on individual estimates of the coefficient norms obtained by the Cauchy formula applied to a holomorphic extension of the solution map. Here, we use weighted summability estimates, obtained by real-variable arguments. While the obtained results imply those of [7] as a particular case, they lead to a refined analysis which takes into account the amount of overlap between the support of the the supports of the ψ j . For instance, in the case of disjoint supports, these results imply that for all 0<p<2, the p summability of the coefficient V-norm sequences follows from the weaker assumption that ( ψ j L ) j 1 is q summable for q = q ( p ) : = 2 p 2 p > p . We provide a simple analytic example showing that this result is in general optimal and illustrate our findings by numerical experiments. The analysis in the present paper applies to other types of linear PDEs with similar affine parametrization of the coefficients, and to more general Jacobi polynomial expansions.

DOI : 10.1051/m2an/2016045
Classification : 41A10, 41A58, 41A63, 65N15, 65T60
Mots clés : Parametric PDEs, affine coefficients, n-term approximation, Legendre polynomials
Bachmayr, Markus 1 ; Cohen, Albert 1 ; Migliorati, Giovanni 1

1 Sorbonne Universités, UPMC Univ. Paris 6, CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, 4 place Jussieu, 75005 Paris, France.
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     title = {Sparse polynomial approximation of parametric elliptic {PDEs.} {Part} {I:} affine coefficients},
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     pages = {321--339},
     publisher = {EDP-Sciences},
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Bachmayr, Markus; Cohen, Albert; Migliorati, Giovanni. Sparse polynomial approximation of parametric elliptic PDEs. Part I: affine coefficients. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 1, pp. 321-339. doi : 10.1051/m2an/2016045. http://archive.numdam.org/articles/10.1051/m2an/2016045/

M. Bachmayr, A. Cohen, R. Devore and G. Migliorati, Sparse polynomial approximation of parametric elliptic PDEs. Part II: Lognormal coefficients. ESAIM: M2AN 51 (2017) 341–363. | DOI | Numdam | MR | Zbl

J. Beck, F. Nobile, L. Tamellini and R. Tempone, On the optimal polynomial approximation of stochastic PDEs by Galerkin and collocation methods. Math. Model. Methods Appl. Sci. 22 (2012) 1–33. | DOI | MR | Zbl

J. Beck, F. Nobile, L. Tamellini and R. Tempone, Convergence of quasi-optimal stochastic Galerkin methods for a class of PDEs with random coefficients. Comput. Math. Appl. 67 (2014) 732–751. | DOI | MR | Zbl

A. Chkifa, A. Cohen, R. Devore and C. Schwab, Sparse adaptive Taylor approximation algorithms for parametric and stochastic elliptic PDEs. ESAIM: M2AN 47 (2013) 253–280. | DOI | Numdam | MR | Zbl

A. Chkifa, A. Cohen and C. Schwab, Breaking the curse of dimensionality in sparse polynomial approximation of parametric PDEs. J. Math. Pures Appl. 103 (2015) 400–428. | DOI | MR | Zbl

A. Cohen, Numerical analysis of wavelet methods. Studies in Mathematics and its Applications. Elsevier, Amsterdam (2003). | MR | Zbl

A. Cohen, R. Devore and C. Schwab, Analytic regularity and polynomial approximation of parametric and stochastic PDEs. Anal. Appl. 9 (2011) 11–47. | DOI | MR | Zbl

A. Cohen and R. Devore, Approximation of high-dimensional parametric PDEs. Acta Numerica 24 (2015) 1–159. | DOI | MR | Zbl

R. Devore, Nonlinear Approximation. Acta Numerica 7 (1998) 51–150. | DOI | MR | Zbl

R.G. Ghanem and P.D. Spanos, Spectral techniques for stochastic finite elements. Arch. Comput. Methods Eng. 4 (1997) 63–100. | DOI | MR

R.G. Ghanem and P.D. Spanos, Stochastic Finite Elements: A Spectral Approach, 2nd edition. Dover (2007). | MR | Zbl

C.J. Gittelson, An adaptive stochastic Galerkin method for random elliptic operators. Math. Comput. 82 (2013) 1515–1541. | DOI | MR | Zbl

O. Knio and O. Le Maitre, Spectral Methods for Uncertainty Quantication: With Applications to Computational Fluid Dynamics. Springer (2010). | MR | Zbl

C. Schwab and R. Stevenson, Space-time adaptive wavelet methods for parabolic evolution equations. Math. Comput. 78 (2009) 1293–1318. | DOI | MR | Zbl

H. Tran, C. Webster and G. Zhang, Analysis of quasi-optimal polynomial approximations for parametric PDEs with deterministic and stochastic coefficients. Preprint (2015). | arXiv | MR

D. Xiu, Numerical methods for stochastic computations: a spectral method approach. Princeton University Press (2010). | MR | Zbl

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