Bayesian approach to ill-posed inverse problems

Abstract: In this talk, we summarise the work done in the thesis. We will focus on Bayesian analysis of ill posed inverse problems with Gaussian priors and additive Gaussian noise in a separable Hilbert space setting. We shall deal with the case when the parameter to be found depends sensitively on the observation.

Bayesian method is one of many ways to deal with ill posedness of inverse problems. It also allows us to incorporate a prior measure which can be used to capture our prior notions about the value of the parameter. The solution in the Bayesian method is given by the posterior measure.

Our main concerns will be well posedness and also contraction rates of the posterior measure as the noise goes to zero. We will present results for mildly and severely ill-posed problems in cases when the eigenbases of the operator to be inverted and prior are not simultaneously diagonalizable.