Discretization-invariant Bayesian inversionand Besov space priors

Bayesian solution of an inverse problem for indirect measurement $M = AU + $ε is considered,where U is a function on a domain of ᵈ.Here A is a smoothing linear operator and ε is Gaussian white noise. The data is a realization mₖ of therandom variable Mₖ = PₖA U+Pₖε , 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 A Uₙ + Pₖε .Bayes formula gives then the posterior distributionπₖₙ(uₙ\|ₖₙ)~ Π n (uₙ)exp(-1/2||ₖₙ - PₖA uₙ||₂²) in ᵈ,and the mean ₖₙ:=ₙ \ πₖₙ(uₙ\|ₖ)\ duₙ is considered as the reconstruction of U. We discuss a systematic way of choosingprior distributions Π n 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 Π n represent the same a priori information for all n and that the mean ₖₙ 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 ¹ norm of the wavelet coefficients of U.