Discretization-invariant Bayesian inversion and Besov space priors

Abstract

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 white noise. The data is a realization $m_k$ of the random variable $M_k=P_{k}AU+P_k\mathcal{E}$, where $P_k$ is a linear, finite dimensional operator related to measurement device. To allow computerized inversion, the unknown is discretized as $U_n=T_{n}U$, where $T_n$ is a finite dimensional projection, leading to the computational measurement model $M_{kn}=P_{k}A{U}_n+P_k\mathcal{E}$. Bayes formula gives then the posterior distribution ${\pi }_{kn}(u_n|m_{kn})\sim {\pi }_n(u_n)\exp (-1/2\Vert m_{kn}-P_{k}A{u}_n{\Vert }_2^2)$ in $R^d$, and the mean $U_{kn}^{CM}:=\int u_n{\pi }_{kn}(u_n|m_k)d{u}_n$ is considered as the reconstruction of $U$. We discuss a systematic way of choosing prior distributions $\prior_n$ for all $n\ge n_0>0$ by achieving them as projections of a distribution in a infinite-dimensional limit case. Such choice of prior distributions is {\em discretization-invariant} in the sense that $\prior_n$ represent the same {\em a priori} information for all $n$ and that the mean $U_{kn}^{CM}$ converges to a limit estimate as $k,n\to \infty$. Gaussian smoothness priors and wavelet-based Besov space priors are shown to be discretization invariant. In particular, Bayesian inversion in dimension two with $B_{11}^1$ prior is related to penalizing the ${\ell }^1$ norm of the wavelet coefficients of $U$.