High-dimensional Collocation Weighted Approximations For Parametric Elliptic PDEs With Lognormal Inputs

We constructed linear non-adaptive methods of non-fully and fully discrete polynomial interpolation weighted approximation for parametric and stochastic elliptic PDEs with lognormal inputs and proved convergence rates of the approximations by them. Our methods are sparse-grid collocation methods. Moreover, the fully discrete methods can be seen as multilevel methods of approximation. The Smolyak sparse interpolation grids in the parametric domain are constructed from the roots of Hermite polynomials or their improved modifications.