  We propose and analyze a Stein variational reduced basis method (SVRB) to
solve large-scale PDE-constrained Bayesian inverse problems. To address the
computational challenge of drawing numerous samples requiring expensive PDE
solves from the posterior distribution, we integrate an adaptive and
goal-oriented model reduction technique with an optimization-based Stein
variational gradient descent method (SVGD). The samples are drawn from the
prior distribution and iteratively pushed to the posterior by a sequence of
transport maps, which are constructed by SVGD, requiring the evaluation of the
potential---the negative log of the likelihood function---and its gradient with
respect to the random parameters, which depend on the solution of the PDE. To
reduce the computational cost, we develop an adaptive and goal-oriented model
reduction technique based on reduced basis approximations for the evaluation of
the potential and its gradient. We present a detailed analysis for the reduced
basis approximation errors of the potential and its gradient, the induced
errors of the posterior distribution measured by Kullback--Leibler divergence,
as well as the errors of the samples. To demonstrate the computational accuracy
and efficiency of SVRB, we report results of numerical experiments on a
Bayesian inverse problem governed by a diffusion PDE with random parameters
with both uniform and Gaussian prior distributions. Over 100X speedups can be
achieved while the accuracy of the approximation of the potential and its
gradient is preserved.
