Robert N. Gantner

Josef Dick, Robert N. Gantner, Quoc T. Le Gia et al.

We propose and analyze deterministic multilevel approximations for Bayesian inversion of operator equations with uncertain distributed parameters, subject to additive Gaussian measurement data. The algorithms use a multilevel (ML) approach based on deterministic, higher order quasi-Monte Carlo (HoQMC) quadrature for approximating the high-dimensional expectations, which arise in the Bayesian estimators, and a Petrov-Galerkin (PG) method for approximating the solution to the underlying partial differential equation (PDE). This extends the previous single-level approach from [J. Dick, R. N. Gantner, Q. T. Le Gia and Ch. Schwab, Higher order Quasi-Monte Carlo integration for Bayesian Estimation. Report 2016-13, Seminar for Applied Mathematics, ETH Zürich (in review)]. We obtain sufficient conditions which allow us to achieve arbitrarily high, algebraic convergence rates in terms of work, which are independent of the dimension of the parameter space. The convergence rates are limited only by the spatial regularity of the forward problem,the discretization order achieved by the Petrov Galerkin discretization, and by the sparsity of the uncertainty parametrization. We provide detailed numerical experiments for linear elliptic problems in two space dimensions, with $s=1024$ parameters characterizing the uncertain input, confirming the theory and showing that the ML HoQMC algorithms outperform, in terms of error vs.~computational work, both multilevel Monte Carlo (MLMC) methods and single-level (SL) HoQMC methods.

Josef Dick, Robert N. Gantner, Quoc T. Le Gia et al.

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 [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 so-called SPOD weights 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 $\calO(N^{-1/p})$ 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.