  By now Bayesian methods are routinely used in practice for solving inverse
problems. In inverse problems the parameter or signal of interest is observed
only indirectly, as an image of a given map, and the observations are typically
further corrupted with noise. Bayes offers a natural way to regularize these
problems via the prior distribution and provides a probabilistic solution,
quantifying the remaining uncertainty in the problem. However, the
computational costs of standard, sampling based Bayesian approaches can be
overly large in such complex models. Therefore, in practice variational Bayes
is becoming increasingly popular. Nevertheless, the theoretical understanding
of these methods is still relatively limited, especially in context of inverse
problems. In our analysis we investigate variational Bayesian methods for
Gaussian process priors to solve linear inverse problems. We consider both
mildly and severely ill-posed inverse problems and work with the popular
inducing variables variational Bayes approach proposed by Titsias in 2009. We
derive posterior contraction rates for the variational posterior in general
settings and show that the minimax estimation rate can be attained by correctly
tunned procedures. As specific examples we consider a collection of inverse
problems including the heat equation, Volterra operator and Radon transform and
inducing variable methods based on population and empirical spectral features.
