We propose and analyze deterministic multilevel (ML) approximations for Bayesian inversion of operator equations with uncertain distributed parameters, subject to additive Gaussian measurement data. The algorithms use a 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 (SL) 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)]. Compared to the SL approach, the present convergence analysis of the ML method requires stronger assumptions on holomorphy and regularity of the countably-parametric uncertainty-to-observation maps of the forward problem. As in the SL case and in the affine-parametric case analyzed in [J. Dick, F. Y. Kuo, Q. T. Le Gia and Ch. Schwab, Multi-level higher order QMC Galerkin discretization for affine parametric operator equations, SIAM J. Numer. Anal. 54 (2016) 2541–2568], 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 PG discretization, and by the sparsity of the uncertainty parametrization. We provide detailed numerical experiments for linear elliptic problems in two space dimensions, with [Formula: see text] parameters characterizing the uncertain input, confirming the theory and showing that the ML HoQMC algorithms can outperform, in terms of error versus computational work, both multilevel Monte Carlo (MLMC) methods and SL HoQMC methods, provided the parametric solution maps of the forward problems afford sufficient smoothness and sparsity of the high-dimensional parameter spaces.
Quasi-Monte Carlo tractability of high dimensional integration over products of simplices
