Quasi-Monte Carlo Bayesian estimation under Besov priors in elliptic inverse problems

Abstract

We analyze rates of convergence for quasi-Monte Carlo (QMC) integration for Bayesian inversion of linear, elliptic partial differential equations with uncertain input from function spaces. Adopting a Riesz or Schauder basis representation of the uncertain inputs, function space priors are constructed as product measures on spaces of (sequences of) coefficients in the basis representations. The numerical approximation of the posterior expectation, given data, then amounts to a high- or infinite-dimensional numerical integration problem. We consider in particular so-called Besov priors on the admissible uncertain inputs. We extend the QMC convergence theory from the Gaussian case, and establish sufficient conditions on the uncertain inputs for achieving dimension-independent convergence rates greater than $1/2$ of QMC integration with randomly shifted lattice rules. We apply the theory to a concrete class of linear, second order elliptic boundary value problems with log-Besov uncertain diffusion coefficient.