Bayesian posterior contraction rates for linear severely ill-posed inverse problems

Sergios Agapiou 1 , Andrew M. Stuart 2  and Yuan-Xiang Zhang 3
  • 1 Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom
  • 2 Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom
  • 3 School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, P. R. China

Abstract.

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.

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Journal + Issues

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.

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