B1883
Title: Low-rank posterior approximations for linear Gaussian inverse problems on separable Hilbert spaces
Authors: Giuseppe Carere - University of Potsdam (Germany) [presenting]
Han Cheng Lie - Universitaet Potsdam (Germany)
Abstract: In Bayesian inverse problems, the computation of the posterior distribution can be computationally demanding, especially in many-query settings such as filtering, where a new posterior distribution must be computed many times. Some computationally efficient approximations of the posterior distribution are considered for linear Gaussian inverse problems defined on separable Hilbert spaces. The quality of these approximations is measured using the Kullback-Leibler divergence of the approximate posterior with respect to the true posterior and their optimality properties are investigated. The approximation method exploits low dimensional behaviour of the update from prior to posterior, originating from a combination of prior smoothing, forward smoothing, measurement error and a limited number of observations, analogous to the results of a prior study for finite-dimensional parameter spaces. Since the data is only informative on a low dimensional subspace of the parameter space, the approximation class considered for the posterior covariance consists of suitable low-rank updates of the prior. In the Hilbert space setting, care must be taken, such as when inverting covariance operators. This challenge is addressed by using the Feldman-Hajek theorem for Gaussian measures.
