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：本文提出一种新颖的后验误差估计技术，其目标量涉及高维积分比值，例如出现在含偏微分方程约束的贝叶斯反演以及受熵风险度量约束的最优控制问题中。我们特别考虑参数化椭圆型偏微分方程，其中扩散系数具有仿射参数依赖性，且定义在高维参数空间上。通过融合我们近期发展的后验拟蒙特卡洛(QMC)误差分析与有限元后验误差估计方法，所提方案可生成可计算的后验估计量，该估计量在忽略高阶项意义下具有可靠性。该估计量的可靠性关于偏微分方程离散化是均匀的，且关于不确定偏微分方程输入的参数维度具有鲁棒性。