This paper contributes to the study of optimal experimental design for Bayesian inverse problems governed by partial differential equations (PDEs). We derive estimates for the parametric regularity of multivariate double integration problems over high-dimensional parameter and data domains arising in Bayesian optimal design problems. We provide a detailed analysis for these double integration problems using two approaches: a full tensor product and a sparse tensor product combination of quasi-Monte Carlo (QMC) cubature rules over the parameter and data domains. Specifically, we show that the latter approach significantly improves the convergence rate, exhibiting performance comparable to that of QMC integration of a single high-dimensional integral. Furthermore, we numerically verify the predicted convergence rates for an elliptic PDE problem with an unknown diffusion coefficient in two spatial dimensions, offering empirical evidence supporting the theoretical results and highlighting practical applicability.

翻译：本文致力于研究由偏微分方程（PDE）约束的贝叶斯逆问题中的最优实验设计。我们推导了在贝叶斯最优设计问题中出现的高维参数域和数据域上的多元双重积分问题的参数正则性估计。针对这些双重积分问题，我们提供了详细分析，采用两种方法：参数域和数据域上拟蒙特卡洛（QMC）求积规则的完全张量积与稀疏张量积组合。具体而言，我们证明后者显著提高了收敛速度，其表现可与单个高维积分的QMC积分相媲美。此外，我们通过一个二维空间中含有未知扩散系数的椭圆PDE问题数值验证了所预测的收敛速率，为理论结果提供了实证支持，并突显了实际应用价值。