Discretization-invariant bayesian inversion and besov space priors

Abstract

Bayesian solution of an inverse problem for indirect measurement M = AU + ε is considered, where U is a function on a domain of Rd. Here A is a smoothing linear operator and ε is Gaussian white noise. The data is a realization mk of the random variable Mk = Pk AU + Pk ε, where Pk is a linear, finite dimensional operator related to measurement device. To allow computerized inversion, the unknown is discretized as Un = TnU, where Tn is a finite dimensional projection, leading to the computational measurement model Mkn = Pk AUn + Pk ε. Bayes formula gives then the posterior distribution πkn (un | mkn) ∼ Πn(un) exp(−12‖mkn − Pk Aun‖22) in Rd, and the mean ukn:=∫ un πkn (un | mk) dun is considered as the reconstruction of U. We dis-cuss a systematic way of choosing prior distributions Πn for all n ≥ n0 > 0 by achieving them as projections of a distribution in a infinite-dimensional limit case. Such choice of prior distributions is discretization-invariant in the sense that Πn represent the same a priori information for all n and that the mean ukn converges to a limit estimate as k, n → ∞. Gaussian smoothness priors and wavelet-based Besov space priors are shown to be discretization invariant. In particular, Bayesian inversion in dimension two with B111 prior is related to penalizing the ℓ1 norm of the wavelet coefficients of U.