Discretization-invariant Bayesian inversionand 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 $\R^d$.Here $A$ is a smoothing linear operator and ε is Gaussian white noise. The data is a realization $m_k$ of therandom variable $M_k = P_kA U+P_k$ε , where $P_k$ is a linear, finite dimensional operator related to measurement device.To allow computerized inversion,the unknown is discretized as $U_n=T_nU$, where $T_n$ is a finite dimensional projection,leading to the computational measurement model $M_{kn}=P_k A U_n + P_k$ε .Bayes formula gives then the posterior distribution$\pi_{kn}(u_n\|\m_{kn})$~ Π n $(u_n)\exp(-\frac{1}{2}$||$\m_{kn} - P_kA u_n$||$\_2^2)$ in $\R^d$,and the mean $\u_{kn}$:$=\intu_n \ \pi_{kn}(u_n\|\m_k)\ du_n$ is considered as the reconstruction of $U$. We discuss a systematic way of choosingprior distributions Π n for all $n\geq n_0>0$ 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 $\u_{kn}$ converges to a limit estimate as $k,n$→$\infty$.Gaussian smoothness priors and wavelet-based Besov space priors are shown to be discretization invariant. In particular, Bayesian inversion in dimension two with $B^1_11$ prior is related to penalizing the $\l^1$ norm of the wavelet coefficients of $U$.