  This work is a follow-up to our previous contribution ("Convergence of sparse
collocation for functions of countably many Gaussian random variables (with
application to elliptic PDEs)", SIAM J. Numer. Anal., 2018), and contains
further insights on some aspects of the solution of elliptic PDEs with
lognormal diffusion coefficients using sparse grids. Specifically, we first
focus on the choice of univariate interpolation rules, advocating the use of
Gaussian Leja points as introduced by Narayan and Jakeman ("Adaptive Leja
sparse grid constructions for stochastic collocation and high-dimensional
approximation", SIAM J. Sci. Comput., 2014) and then discuss the possible
computational advantages of replacing the standard Karhunen-Lo\`eve expansion
of the diffusion coefficient with the L\'evy-Ciesielski expansion, motivated by
theoretical work of Bachmayr, Cohen, DeVore, and Migliorati ("Sparse polynomial
approximation of parametric elliptic PDEs. part II: lognormal coefficients",
ESAIM: M2AN, 2016). Our numerical results indicate that, for the problem under
consideration, Gaussian Leja collocation points outperform Gauss-Hermite and
Genz-Keister nodes for the sparse grid approximation and that the
Karhunen-Lo\`eve expansion of the log diffusion coefficient is more appropriate
than its L\'evy-Ciesielski expansion for purpose of sparse grid collocation.
