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. In the Galerkin setting, the stochastic approximations are conducted by a best-N -term polynomial chaos approximation while in the collocation setting we use interpolation operators based on a Smolyak grid of Gauss points. In both approaches, we cover the case of bounded random variables as well as unbounded Gaussian random variables as input parameters. In a sparse tensor product fashion, we then compose the levels of spatial and stochastic approximations, resulting in a substantial reduction of overall degrees of freedom. Numerical analysis is then used to estimate the convergence rates of the sparse tensor stochastic Galerkin and collocation methods, depending on the regularity of the random inputs. Numerical examples illustrate the theoretical results and indicate superiority of this novel sparse tensor product approximation compared to the ‘full tensor’ approaches used so far and the Monte Carlo method.