Sparse polynomial approximation of parametric elliptic PDEs. Part II: lognormal coefficients
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 1, pp. 341-363.

We consider the linear elliptic equation - div ( a u ) = f on some bounded domain D, where a has the form a = exp ( b ) with b a random function defined as b ( y ) = j 1 y j ψ j where y = ( y j ) are i.i.d. standard scalar Gaussian variables and ( ψ j ) j 1 is a given sequence of functions in L (D). We study the summability properties of Hermite-type expansions of the solution map y u ( y ) V : = H 0 1 ( D ) , that is, expansions of the form(D) , that is, expansions of the form u ( y ) = ν F u ν H ν ( y ) , where H ν ( y ) = j 1 H ν j ( y j ) are the tensorized Hermite polynomials indexed by the set H of finitely supported sequences of nonnegative integers. Previous results [V.H. Hoang and C. Schwab, M3AS 24 (2014) 797−826] have demonstrated that, for any 0 < p 1 , the p summability of the sequence ( j ψ j L ) j 1 implies p summability of the sequence ( u ν V ) ν . Such results ensure convergence rates n -s with s = 1 p - 1 2 of polynomial approximations obtained by best n -term truncation of Hermite series, where the error is measured in the mean-square sense, that is, in L 2 ( N ,V,γ), where γ is the infinite-dimensional Gaussian measure. In this paper we considerably improve these results by providing sufficient conditions for the p summability of ( u ν V ) ν expressed in terms of the pointwise summability properties of the sequence (|ψ j |) j1 . This leads to a refined analysis which takes into account the amount of overlap between the supports of the ψ j . For instance, in the case of disjoint supports, our results imply that, for all 0<p<2 the p summability of (u ν V ) ν follows from the weaker assumption that (ψ j L ) j1 is q summable for q : = 2 p 2 - p > p . In the case of arbitrary supports, our results imply that the p summability of (u ν V ) ν follows from the p summability of ( j β ψ j L ) j 1 for some > 1 2 which ch still represents an improvement over the condition in [V.H. Hoang and C. Schwab, M3AS 24 (2014) 797−826]. We also explore intermediate cases of functions with local yet overlapping supports, such as wavelet bases. One interesting observation following from our analysis is that for certain relevant examples, the use of the Karhunen−Loève basis for the representation of might be suboptimal compared to other representations, in terms of the resulting summability properties of ( u ν V ) ν . While we focus on the diffusion equation, our analysis applies to other type of linear PDEs with similar lognormal dependence in the coefficients.

DOI : 10.1051/m2an/2016051
Classification : 41A10, 41A58, 41A63, 65N15, 65T60
Mots clés : Stochastic PDEs, lognormal coefficients, n-term approximation, Hermite polynomials
Bachmayr, Markus 1 ; Cohen, Albert 1 ; DeVore, Ronald 2 ; Migliorati, Giovanni 1

1 Sorbonne Universités, UPMC Univ Paris 06, CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, 4 place Jussieu, 75005 Paris, France.
2 Department of Mathematics, Texas A&M University, College Station, TX 77840, USA.
@article{M2AN_2017__51_1_341_0,
     author = {Bachmayr, Markus and Cohen, Albert and DeVore, Ronald and Migliorati, Giovanni},
     title = {Sparse polynomial approximation of parametric elliptic {PDEs.} {Part} {II:} lognormal coefficients},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {341--363},
     publisher = {EDP-Sciences},
     volume = {51},
     number = {1},
     year = {2017},
     doi = {10.1051/m2an/2016051},
     mrnumber = {3601011},
     zbl = {1366.41005},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an/2016051/}
}
TY  - JOUR
AU  - Bachmayr, Markus
AU  - Cohen, Albert
AU  - DeVore, Ronald
AU  - Migliorati, Giovanni
TI  - Sparse polynomial approximation of parametric elliptic PDEs. Part II: lognormal coefficients
JO  - ESAIM: Mathematical Modelling and Numerical Analysis 
PY  - 2017
SP  - 341
EP  - 363
VL  - 51
IS  - 1
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/m2an/2016051/
DO  - 10.1051/m2an/2016051
LA  - en
ID  - M2AN_2017__51_1_341_0
ER  - 
%0 Journal Article
%A Bachmayr, Markus
%A Cohen, Albert
%A DeVore, Ronald
%A Migliorati, Giovanni
%T Sparse polynomial approximation of parametric elliptic PDEs. Part II: lognormal coefficients
%J ESAIM: Mathematical Modelling and Numerical Analysis 
%D 2017
%P 341-363
%V 51
%N 1
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/m2an/2016051/
%R 10.1051/m2an/2016051
%G en
%F M2AN_2017__51_1_341_0
Bachmayr, Markus; Cohen, Albert; DeVore, Ronald; Migliorati, Giovanni. Sparse polynomial approximation of parametric elliptic PDEs. Part II: lognormal coefficients. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 1, pp. 341-363. doi : 10.1051/m2an/2016051. http://www.numdam.org/articles/10.1051/m2an/2016051/

I. 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. | DOI | MR | Zbl

M. Bachmayr, A. Cohen and G. Migliorati, Sparse polynomial approximation of parametric elliptic PDEs. Part I: Affine coefficients, ESAIM: M2AN 51 (2017) 321–339. | 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. Models 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

J. Charrier, Strong and weak error estimates for elliptic partial differential equations with random coefficients. SIAM J. Numer. Anal. 50 (2012) 216–246. | DOI | 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 and R. Devore, Approximation of high-dimensional parametric PDEs. Acta Numer. 24 (2015) 1–159. | DOI | 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

M. Dashti and A.M. Stuart, The Bayesian Approach to Inverse Problems. Handbook of Uncertainty Quantification, edited by R. Ghanem, D. Higdon and H. Owhadi. Springer (2015).

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

O. Ernst and B. Sprungk, Stochastic Collocation for Elliptic PDEs with Random Data: The Lognormal Case, in Sparse Grids and Applications – Munich (2012), edited by J. Garcke and D. Pflüger. Vol. 97 of Lect. Notes Comput. Sci. Eng. Springer International Publishing Switzerland (2014). | MR

J. Galvis and M. Sarkis, Approximating infinity-dimensional stochastic Darcy’s equations without uniform ellipticity. SIAM J. Numer. Anal. 47 (2009) 3624–3651. | DOI | MR | Zbl

R. Ghanem and P. Spanos, Spectral techniques for stochastic finite elements. Arch. Comput. Methods Engrg. 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. Gittelson, Stochastic Galerkin discretization of the log-normal isotropic diffusion problem. Math. Models Methods Appl. Sci. 20 (2010) 237–263. | DOI | MR | Zbl

I.G. Graham, F.Y. Kuo, J.A. Nichols, R. Scheichl, Ch. Schwab and I. H. Sloan, Quasi-Monte Carlo finite element methods for elliptic PDEs with lognormal random coefficients. Numer. Math. 131 (2015) 329–368. | DOI | MR | Zbl

M. Hairer, An Introduction to Stochastic PDEs. Lecture notes. Available at http://www.hairer.org (2009).

V.H. Hoang and C. Schwab, N-term Galerkin Wiener chaos approximation rates for elliptic PDEs with lognormal Gaussian random inputs. M3AS 24 (2014) 797–826. | MR | Zbl

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

F.Y. Kuo, R. Scheichl, Ch. Schwab, I.H. Sloan and E. Ullmann, Multilevel Quasi-Monte Carlo Methods for Lognormal Diffusion Problems, , to appear in Math. of Comp. (2015). | arXiv | MR

A. Mugler and H.-J. Starkloff, On the convergence of the stochastic Galerkin methods for random elliptic partial differential equations. ESAIM: M2AN 47 (2013) 1237–1263. | DOI | Numdam | MR | Zbl

D. Xiu, Numerical Methods for Stochastic Computations: A Spectral Method Approach. Princeton University Press (2010). | MR | Zbl

Cité par Sources :