Multilevel higher order Quasi-Monte Carlo Bayesian Estimation

TL;DR

This paper introduces a multilevel deterministic approach combining higher order quasi-Monte Carlo quadrature and Petrov-Galerkin methods for efficient Bayesian inversion in high-dimensional PDE problems, achieving high convergence rates.

Contribution

It extends previous single-level methods to a multilevel framework, enabling arbitrarily high convergence rates independent of parameter space dimension.

Findings

ML HoQMC outperforms MLMC and single-level HoQMC in error vs. computational work.

Convergence rates depend on spatial regularity, discretization order, and sparsity of parameters.

Numerical experiments confirm theoretical predictions for 2D elliptic problems with 1024 parameters.

Abstract

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 Z\"urich (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.