Direct and Inverse Problems for PDEs with Random Coefficients - Abstract

Schwab, Christoph

Higher order quasi Monte-Carlo FEM for Bayesian inverse problems with distributed input data

We analyze Quasi-Monte Carlo quadratures in Bayesian estimation of solutions to countably-parametric operator equations with holomorphic dependence on the parameters considered in [Cl. Schillings and Ch. Schwab: Sparsity in Bayesian Inversion of Parametric Operator Equations. Inverse Problems, bf 30, (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 the so-called SPOD weights recently introduced in [F.Y. Kuo, Ch. Schwab, I.H. Sloan, Quasi-Monte Carlo finite element methods for a class of elliptic partial differential equations with random coefficients. SIAM J. Numer. Anal. 50 (2012), 3351--3374] are used to describe the solution regularity. We establish error bounds for higher order, Quasi-Monte Carlo quadrature for the Bayesian estimation based on [J. Dick, Q.T. LeGia and Ch. Schwab, Higher order Quasi-Monte Carlo integration for holomorphic, parametric operator equations, Report 2014-23, SAM, ETH Zürich]. It implies, in particular, regularity of the parametric solution and of the countably-parametric Bayesian posterior density in SPOD weighted spaces. This, in turn, implies that the Quasi-Monte Carlo quadrature methods in [J. Dick, F.Y. Kuo, Q.T. Le Gia, D. Nuyens, Ch. Schwab, Higher order QMC Galerkin discretization for parametric operator equations, SINUM (2014)] are applicable to these problem classes, with dimension-independent convergence rates $mathcal (N^-1/p)$ 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. Hybridized versions of the fast component-by-component (CBC for short) construction [R. N. Gantner and Ch. Schwab, Computational Higher Order Quasi-Monte Carlo Integration, Report 2014-25, SAM, ETH Zürich] allow efficient Bayesian estimation on parametric input with up $10^4$ dimensional parameter vectors. Joint work with J. Dick and T. LeGia (University of New South Wales, Sydney, Australia) and with R. N. Ganter (ETH Zürich, Switzerland). Work supported by Swiss National Science Foundation (SNF) and by the European Research Council (ERC).