Convergence of Sparse Collocation for Functions of Countably Many Gaussian Random Variables (with Application to Elliptic PDEs)

Convergence of sparse collocation for functions of countably many Gaussian random variables (with application to elliptic PDEs)

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 outline a general L2-convergence theory based on previous work by Bachmayr et al. (2016) and Chen (2016) and establish an algebraic convergenc...

Convergence of quasi-optimal sparse-grid approximation of Hilbert-space-valued functions: application to random elliptic PDEs

In this work we provide a convergence analysis for the quasi-optimal version of the Stochastic Sparse Grid Collocation method we had presented in our previous work “On the optimal polynomial approximation of Stochastic PDEs by Galerkin and Collocation methods” [6]. Here the construction of a sparse grid is recast into a knapsack problem: a profit is assigned to each hierarchical surplus and onl...

Sparse tensor discretizations of elliptic PDEs with random input data

We consider a stochastic Galerkin and collocation discretization scheme for solving elliptic PDEs with random coefficients and forcing term, which are assumed to depend on a finite, but possibly large number of random variables. Both methods consist of a hierarchic wavelet discretization in space and a sequence of hierarchic approximations to the law of the random solution in probability space....

Dominated Convergence for Fuzzy Random Variables

Stochastic Collocation for Elliptic PDEs with random data - the lognormal case

We investigate the stochastic collocation method for parametric, elliptic partial differential equations (PDEs) with lognormally distributed random parameters in mixed formulation. Such problems arise, e.g., in uncertainty quantification studies for flow in porous media with random conductivity. We show the analytic dependence of the solution of the PDE w.r.t. the parameters and use this to sho...