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：本文研究由偏微分方程（PDE）控制的贝叶斯反问题中的最优实验设计问题。针对贝叶斯最优设计问题中产生的高维参数域和数据域上的多元二重积分问题，我们推导了其参数正则性的估计。通过两种方法对这些二重积分问题进行了详细分析：在参数域和数据域上采用全张量积与稀疏张量积组合的拟蒙特卡洛（QMC）求积规则。具体而言，我们证明后一种方法显著提升了收敛速度，其表现可媲美单一高维积分的QMC积分。此外，针对二维空间中有未知扩散系数的椭圆型PDE问题，我们对预测的收敛速率进行了数值验证，为理论结果提供了经验证据，并凸显了其实用适用性。