Higher order Quasi-Monte Carlo integration for Bayesian PDE Inversion

Abstract

We analyze combined Quasi-Monte Carlo quadrature and Finite Element approximations in Bayesian estimation of solutions to countably-parametric operator equations with holomorphic dependence on the parameters as considered in Schillings and Schwab (2014). Such problems arise in numerical uncertainty quantification and in Bayesian inversion of operator equations with distributed uncertain inputs, such as uncertain coefficients, uncertain domains or uncertain source terms and boundary data. We show that the parametric Bayesian posterior densities belong to a class of weighted Bochner spaces of functions of countably many variables, with a particular structure of the QMC quadrature weights: up to a (problem-dependent, and possibly large) finite dimension S product weights can be used, and beyond this dimension, weighted spaces with so-called SPOD weights, recently introduced in Dick et al. (2014), are used to describe the solution regularity. We establish error bounds for higher order Quasi-Monte Carlo quadrature for the Bayesian estimation based on Dick et al. (2016). It implies, in particular, regularity of the parametric solution and of the countably-parametric Bayesian posterior density in SPOD (“Smoothness driven, Product and Order Dependent”) weighted spaces of integrand functions. This, in turn, implies that the Quasi-Monte Carlo quadrature methods in Dick et al. (2014) are applicable to these problem classes, with dimension-independent convergence rates O(N−1∕p) of N-point HoQMC approximated Bayesian estimates, where 0<p<1 depends only on the sparsity class of the uncertain input in the Bayesian estimation. Fast component-by-component (CBC for short) construction Gantner and Schwab (2016) allow efficient deterministic Bayesian estimation with up to 104 parameters.