A-posteriori QMC-FEM error estimation for Bayesian inversion and optimal control with entropic risk measure

TL;DR

This paper introduces a new a-posteriori error estimation method combining QMC and FEM techniques for high-dimensional Bayesian inversion and optimal control problems involving entropic risk measures, ensuring reliable and robust error bounds.

Contribution

It develops a novel a-posteriori error estimator for ratios of high-dimensional integrals in PDE-constrained Bayesian and control problems, integrating recent QMC error analysis with FEM error estimation.

Findings

Estimator is reliable up to higher order terms.

Estimator's reliability is uniform across PDE discretizations.

Estimator is robust to high parametric dimensions.

Abstract

We propose a novel a-posteriori error estimation technique where the target quantities of interest are ratios of high-dimensional integrals, as occur e.g. in PDE constrained Bayesian inversion and PDE constrained optimal control subject to an entropic risk measure. We consider in particular parametric, elliptic PDEs with affine-parametric diffusion coefficient, on high-dimensional parameter spaces. We combine our recent a-posteriori Quasi-Monte Carlo (QMC) error analysis, with Finite Element a-posteriori error estimation. The proposed approach yields a computable a-posteriori estimator which is reliable, up to higher order terms. The estimator's reliability is uniform with respect to the PDE discretization, and robust with respect to the parametric dimension of the uncertain PDE input.