A multilevel sparse kernel-based stochastic collocation finite element method for elliptic problems with random coefficients. (15th October 2018)

• Record Type: Journal Article
• Authors: Dong, Zhaonan; Georgoulis, Emmanuil H.; Levesley, Jeremy; Usta, Fuat
• Is Part Of: Computers & mathematics with applications. Volume 76:issue 8(2018)
• Page Start: 1950
• Page End: 1965
• Publication Date: 2018-10-15
• Subjects: Stochastic collocation -- PDE with random data -- Sparse grids -- Radial basis functions -- High dimensional approximation -- Interpolation; Electronic data processing -- Periodicals; Mathematics -- Data processing -- Periodicals; 510.28541
• DOI: 10.1016/j.camwa.2018.07.041
• Languages: English
• ISSNs: 0898-1221

Abstract

Abstract: A new stochastic collocation finite element method is proposed for the numerical solution of elliptic boundary value problems (BVP) with random coefficients, assuming that the randomness is well-approximated by a finite number of random variables with given probability distributions. The proposed method consists of a finite element approximation in physical space, along with a stochastic collocation quadrature approach utilizing the recent Multilevel Sparse Kernel-Based Interpolation (MuSIK) technique (Georgoulis et al., 2013). MuSIK is based on a multilevel sparse grid-type algorithm with the basis functions consisting of directionally anisotropic Gaussian radial basis functions (kernels) placed at directionally-uniform grid-points. We prove that MuSIK is interpolatory at these nodes, and, therefore, can be naturally used to define a quadrature scheme. Numerical examples are also presented, assessing the performance of the new algorithm in the context of high-dimensional stochastic collocation finite element methods.