First order $k$-th moment finite element analysis of nonlinear operator equations with stochastic data

Abstract:

We develop and analyze a class of efficient Galerkin approximation methods for uncertainty quantification of nonlinear operator equations. The algorithms are based on sparse Galerkin discretizations of tensorized linearizations at nominal parameters. Specifically, we consider abstract, nonlinear, parametric operator equations $J(\alpha ,u)=0$ for random input $\alpha (\omega )$ with almost sure realizations in a neighborhood of a nominal input parameter $\alpha _0$. Under some structural assumptions on the parameter dependence, we prove existence and uniqueness of a random solution, $u(\omega ) = S(\alpha (\omega ))$.

We derive a multilinear, tensorized operator equation for the deterministic computation of $k$-th order statistical moments of the random solution’s fluctuations $u(\omega ) - S(\alpha _0)$. We introduce and analyse sparse tensor Galerkin discretization schemes for the efficient, deterministic computation of the $k$-th statistical moment equation. We prove a shift theorem for the $k$-point correlation equation in anisotropic smoothness scales and deduce that sparse tensor Galerkin discretizations of this equation converge in accuracy vs. complexity which equals, up to logarithmic terms, that of the Galerkin discretization of a single instance of the mean field problem. We illustrate the abstract theory for nonstationary diffusion problems in random domains.