  In Bayesian inverse problems sampling the posterior distribution is often a
challenging task when the underlying models are computationally intensive. To
this end, surrogates or reduced models are often used to accelerate the
computation. However, in many practical problems, the parameter of interest can
be of high dimensionality, which renders standard model reduction techniques
infeasible. In this paper, we present an approach that employs the ANOVA
decomposition method to reduce the model with respect to the unknown
parameters, and the reduced basis method to reduce the model with respect to
the physical parameters. Moreover, we provide an adaptive scheme within the
MCMC iterations, to perform the ANOVA decomposition with respect to the
posterior distribution. With numerical examples, we demonstrate that the
proposed model reduction method can significantly reduce the computational cost
of Bayesian inverse problems, without sacrificing much accuracy.
