We consider the linear elliptic equation on some bounded domain , where has the form with a random function defined as where are i.i.d. standard scalar Gaussian variables and is a given sequence of functions in . We study the summability properties of Hermite-type expansions of the solution map , that is, expansions of the form(D) , that is, expansions of the form , where 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 , the summability of the sequence implies summability of the sequence . Such results ensure convergence rates with of polynomial approximations obtained by best -term truncation of Hermite series, where the error is measured in the mean-square sense, that is, in , where is the infinite-dimensional Gaussian measure. In this paper we considerably improve these results by providing sufficient conditions for the summability of expressed in terms of the pointwise summability properties of the sequence . This leads to a refined analysis which takes into account the amount of overlap between the supports of the . For instance, in the case of disjoint supports, our results imply that, for all the summability of follows from the weaker assumption that is summable for . In the case of arbitrary supports, our results imply that the summability of follows from the summability of for some 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 . While we focus on the diffusion equation, our analysis applies to other type of linear PDEs with similar lognormal dependence in the coefficients.
Mots clés : Stochastic PDEs, lognormal coefficients, n-term approximation, Hermite polynomials
@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/
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