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Many inverse problems arising in applications come from continuum modelswhere the unknown parameter is a field. In practice the unknown field isdiscretized resulting in a problem in $\mathbb{R}^N$, with an understandingthat refining the discretization, that is increasing $N$, will often bedesirable. In the context of Bayesian inversion this situation suggests theimportance of two issues: (i) defining hyper-parameters in such a way that theyare interpretable in the continuum limit $N \to \infty$ and so that theirvalues may be compared between different discretization levels; (ii)understanding the efficiency of algorithms for probing the posteriordistribution, as a function of large $N.$ Here we address these two issues inthe context of linear inverse problems subject to additive Gaussian noisewithin a hierarchical modelling framework based on a Gaussian prior for theunknown field and an inverse-gamma prior for a hyper-parameter, namely theamplitude of the prior variance. The structure of the model is such that theGibbs sampler can be easily implemented for probing the posterior distribution.Subscribing to the dogma that one should think infinite-dimensionally beforeimplementing in finite dimensions, we present function space intuition andprovide rigorous theory showing that as $N$ increases, the component of theGibbs sampler for sampling the amplitude of the prior variance becomesincreasingly slower. We discuss a reparametrization of the prior variance thatis robust with respect to the increase in dimension; we give numericalexperiments which exhibit that our reparametrization prevents the slowing down.Our intuition on the behaviour of the prior hyper-parameter, with and withoutreparametrization, is sufficiently general to include a broad class ofnonlinear inverse problems as well as other families of hyper-priors.

Analysis of the Gibbs sampler for hierarchical inverse problems