  In this work we propose and analyze a Hessian-based adaptive sparse
quadrature to compute infinite-dimensional integrals with respect to the
posterior distribution in the context of Bayesian inverse problems with
Gaussian prior. Due to the concentration of the posterior distribution in the
domain of the prior distribution, a prior-based parametrization and sparse
quadrature may fail to capture the posterior distribution and lead to erroneous
evaluation results. By using a parametrization based on the Hessian of the
negative log-posterior, the adaptive sparse quadrature can effectively allocate
the quadrature points according to the posterior distribution. A
dimension-independent convergence rate of the proposed method is established
under certain assumptions on the Gaussian prior and the integrands.
Dimension-independent and faster convergence than $O(N^{-1/2})$ is demonstrated
for a linear as well as a nonlinear inverse problem whose posterior
distribution can be effectively approximated by a Gaussian distribution at the
MAP point.
