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.

翻译：我们建议采用一种新的外在误差估计技术,在这种技术中,利息目标数量是高维元件的比率,例如PDE受约束的Bayesian反转和PDE受限制的最佳控制,但必须经过一种昆虫风险测量。我们特别考虑在高维参数空间的参数性、椭圆式PDE和偏差分法扩散系数。我们将我们最近进行的子虚线 Quasi-Monte Carlo(QMC)误差分析与Finite Element apositerial错误估计结合起来。提议的方法产生一个可靠的可比较的otheri估计器,最高可达更高的顺序条件。估计器的可靠性在PDE的分解方面是统一的,在不确定的PDE输入的参数性方面是稳健健的。