In the Bayesian approach to inverse problems, data are often informative,
relative to the prior, only on a low-dimensional subspace of the parameter
space. Significant computational savings can be achieved by using this subspace
to characterize and approximate the posterior distribution of the parameters.
We first investigate approximation of the posterior covariance matrix as a
low-rank update of the prior covariance matrix. We prove optimality of a
particular update, based on the leading eigendirections of the matrix pencil
defined by the Hessian of the negative log-likelihood and the prior precision,
for a broad class of loss functions. This class includes the F\"{o}rstner
metric for symmetric positive definite matrices, as well as the
Kullback-Leibler divergence and the Hellinger distance between the associated
distributions. We also propose two fast approximations of the posterior mean
and prove their optimality with respect to a weighted Bayes risk under
squared-error loss. These approximations are deployed in an offline-online
manner, where a more costly but data-independent offline calculation is
followed by fast online evaluations. As a result, these approximations are
particularly useful when repeated posterior mean evaluations are required for
multiple data sets. We demonstrate our theoretical results with several
numerical examples, including high-dimensional X-ray tomography and an inverse
heat conduction problem. In both of these examples, the intrinsic
low-dimensional structure of the inference problem can be exploited while
producing results that are essentially indistinguishable from solutions
computed in the full space.

基于贝叶斯方法解决的逆问题，通常相对于先验只在参数空间的低维子空间上有信息。因此，可以利用该子空间对参数的后验分布进行近似计算。本文从近似后验协方差矩阵和后验均值两个角度，提出了两种快速的近似方法，并在多个应用案例中进行了验证。