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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 Levy-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 Levy-Ciesielski expansion for purpose of sparse grid collocation.
This paper studies numerical methods for the approximation of elliptic PDEs with lognormal coefficients of the form $-{rm div}(a abla u)=f$ where $a=exp(b)$ and $b$ is a Gaussian random field. The approximant of the solution $u$ is an $n$-term polyno
Relying on the classical connection between Backward Stochastic Differential Equations (BSDEs) and non-linear parabolic partial differential equations (PDEs), we propose a new probabilistic learning scheme for solving high-dimensional semi-linear par
We give a convergence proof for the approximation by sparse collocation of Hilbert-space-valued functions depending on countably many Gaussian random variables. Such functions appear as solutions of elliptic PDEs with lognormal diffusion coefficients
We examine sparse grid quadrature on weighted tensor products (WTP) of reproducing kernel Hilbert spaces on products of the unit sphere, in the case of worst case quadrature error for rules with arbitrary quadrature weights. We describe a dimension a
Convergence of an adaptive collocation method for the stationary parametric diffusion equation with finite-dimensional affine coefficient is shown. The adaptive algorithm relies on a recently introduced residual-based reliable a posteriori error esti