Hessian-based adaptive sparse quadrature for infinite-dimensional Bayesian inverse problems

摘要

In this work we propose and analyze a Hessian-based adaptive sparsequadrature to compute infinite-dimensional integrals with respect to theposterior distribution in the context of Bayesian inverse problems withGaussian prior. Due to the concentration of the posterior distribution in thedomain of the prior distribution, a prior-based parametrization and sparsequadrature may fail to capture the posterior distribution and lead to erroneousevaluation results. By using a parametrization based on the Hessian of thenegative log-posterior, the adaptive sparse quadrature can effectively allocatethe quadrature points according to the posterior distribution. Adimension-independent convergence rate of the proposed method is establishedunder certain assumptions on the Gaussian prior and the integrands.Dimension-independent and faster convergence than $O(N^{-1/2})$ is demonstratedfor a linear as well as a nonlinear inverse problem whose posteriordistribution can be effectively approximated by a Gaussian distribution at theMAP point.