Discretization-invariant Bayesian inversion and Besov space priors

Bayesian solution of an inverse problem for indirect measurement $M = AU +{\mathcal{E}}$ is considered, where $U$ is a function on a domain of $R^d$.Here $A$ is a smoothing linear operator and $ {\mathcal{E}}$ is Gaussian whitenoise. The data is a realization $m_k$ of the random variable $M_k = P_kA U+P_k{\mathcal{E}}$, where $P_k$ is a linear, finite dimensional operator related tomeasurement device. To allow computerized inversion, the unknown is discretizedas $U_n=T_nU$, where $T_n$ is a finite dimensional projection, leading to thecomputational measurement model $M_{kn}=P_k A U_n + P_k {\mathcal{E}}$. Bayesformula gives then the posterior distribution $\pi_{kn}(u_n |m_{kn})\sim\pi_n(u_n) \exp(-{1/2}\|m_{kn} - P_kA u_n\|_2^2)$ in $R^d$, and themean $U^{CM}_{kn}:=\int u_n \pi_{kn}(u_n | m_k) du_n$ is considered as thereconstruction of $U$. We discuss a systematic way of choosing priordistributions $\prior_n$ for all $n\geq n_0>0$ by achieving them as projectionsof a distribution in a infinite-dimensional limit case. Such choice of priordistributions is {\em discretization-invariant} in the sense that $\prior_n$represent the same {\em a priori} information for all $n$ and that the mean$U^{CM}_{kn}$ converges to a limit estimate as $k,n\to\infty$. Gaussiansmoothness priors and wavelet-based Besov space priors are shown to bediscretization invariant. In particular, Bayesian inversion in dimension twowith $B^1_{11}$ prior is related to penalizing the $\ell^1$ norm of the waveletcoefficients of $U$.