  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.
