## Sparse polynomial approximation of parametric elliptic PDEs.
Part II: lognormal coefficients^{∗}

^{1} Sorbonne Universités, UPMC Univ Paris 06, CNRS, UMR 7598,
Laboratoire Jacques-Louis Lions, 4 place Jussieu, 75005 Paris, France.

bachmayr@ljll.math.upmc.fr; cohen@ljll.math.upmc.fr;
migliorati@ljll.math.upmc.fr
^{2} Department of Mathematics, Texas A&M University, College Station, TX 77840, USA.

rdevore@math.tamu.edu

Received:
27
September
2015

Revised:
9
June
2016

Accepted:
16
July
2016

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}) ∈ ℝ^{N}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 *u*(*y*) = ∑
_{ν ∈
ℱ}*u*_{ν}*H*_{ν}(*y*),
where
*H*_{ν}(*y*) = ∏_{j≥1}*H*_{νj}(*y*_{j})
are the tensorized Hermite polynomials indexed by the
set ℱ 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 *ℓ*^{p} summability of the
sequence (*j* ∥*ψ*_{j} ∥_{L∞})_{j ≥
1} implies *ℓ*^{p} summability of the
sequence (∥ *u*_{ν}∥_{V})_{ν ∈ ℱ}.
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}(ℝ^{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 *ℓ*^{p} summability of
(∥*u*_{ν}∥_{V})_{ν ∈ ℱ}
expressed in terms of the pointwise summability properties of the sequence (|*ψ*_{j}|)_{j ≥ 1}. 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
*ℓ*^{p} summability of
(∥*u*_{ν}∥_{V})_{ν ∈
ℱ}follows from the weaker assumption that (∥*ψ*_{j}∥_{L∞})_{j ≥
1}is *ℓ*^{q} summable for *q* := 2*p*/(2−*p*)
. In the case of arbitrary supports, our results imply
that the *ℓ*^{p} summability of
(∥*u*_{ν}∥_{V})_{ν ∈ ℱ}
follows from the *ℓ*^{p} summability of
(*j*^{β}∥*ψ*_{j}∥_{L∞})_{j ≥
1} for some *β*>1/2
, which 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 *b* might be suboptimal
compared to other representations, in terms of the resulting summability properties of
(∥*u*_{ν}∥_{V})_{ν ∈ ℱ}.
While we focus on the diffusion equation, our analysis applies to other type of linear
PDEs with similar lognormal dependence in the coefficients.

Mathematics Subject Classification: 41A10 / 41A58 / 41A63 / 65N15 / 65T60

Key words: Stochastic PDEs / lognormal coefficients / *n*-term approximation / Hermite polynomials

*© EDP Sciences, SMAI 2016*