We consider a class of linear ill-posed inverse problems
arising from inversion of a compact operator with singular
values which decay exponentially to zero. We adopt a
Bayesian approach, assuming a Gaussian prior on the unknown
function. The observational noise is assumed to be
Gaussian; as a consequence the prior is conjugate to the likelihood so that
the posterior distribution is also Gaussian. We study
Bayesian posterior consistency in the small observational noise
limit. We assume that the forward operator and the
prior and noise covariance operators commute with one another.
We show how, for given smoothness
assumptions on the truth, the scale parameter of the
prior, which is a constant multiplier of the prior covariance operator, can be adjusted to optimize the rate of posterior
contraction to the truth, and we explicitly compute
the logarithmic rate.
This journal presents original articles on the theory, numerics and applications of inverse and ill-posed problems. These inverse and ill-posed problems arise in mathematical physics and mathematical analysis, geophysics, acoustics, electrodynamics, tomography, medicine, ecology, financial mathematics etc. Articles on the construction and justification of new numerical algorithms of inverse problem solutions are also published.