2019/06/04 by Oliver G. Ernst, Björn Sprungk, Ernst, Oliver G. +3 · 1 citation
Economics, Econometrics and Finance · Environmental Science · Mathematics · #65C30 #65D05 #65D15 #FOS: Mathematics #Hydrology and Drought Analysis #Numerical Analysis (math.NA) #Probability (math.PR) #Statistical Methods and Inference #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.1906.01252
openalex publication_date 2019/06/04 · openalex created_date 2022/07/29 · openalex updated_date 2026/07/28
This work is a follow-up to our previous contribution ("Convergence of sparse\ncollocation for functions of countably many Gaussian random variables (with\napplication to elliptic PDEs)", SIAM J. Numer. Anal., 2018), and contains\nfurther insights on some aspects of the solution of elliptic PDEs with\nlognormal diffusion coefficients using sparse grids. Specifically, we first\nfocus on the choice of univariate interpolation rules, advocating the use of\nGaussian Leja points as introduced by Narayan and Jakeman ("Adaptive Leja\nsparse grid constructions for stochastic collocation and high-dimensional\napproximation", SIAM J. Sci. Comput., 2014) and then discuss the possible\ncomputational advantages of replacing the standard Karhunen-Lo `eve expansion\nof the diffusion coefficient with the L 'evy-Ciesielski expansion, motivated by\ntheoretical work of Bachmayr, Cohen, DeVore, and Migliorati ("Sparse polynomial\napproximation of parametric elliptic PDEs. part II: lognormal coefficients",\nESAIM: M2AN, 2016). Our numerical results indicate that, for the problem under\nconsideration, Gaussian Leja collocation points outperform Gauss-Hermite and\nGenz-Keister nodes for the sparse grid approximation and that the\nKarhunen-Lo `eve expansion of the log diffusion coefficient is more appropriate\nthan its L 'evy-Ciesielski expansion for purpose of sparse grid collocation.\n