February  2009, 3(1): 87-122. doi: 10.3934/ipi.2009.3.87

Discretization-invariant Bayesian inversion and Besov space priors

1. 

Department of Mathematics and Statistics, University of Helsinki, P.O. Box 68 (Gustaf Hallstromin katu 2b) FI-00014, Finland, Finland

2. 

Tampere University of Technology,Institute of Mathematics,, P.O. Box 553, 33101 Tampere

Received  February 2008 Revised  November 2008 Published  February 2009

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 the random 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 $\u_{kn}$:$=\int u_n \ \pi_{kn}(u_n\|\m_k)\ du_n$ is considered as the reconstruction of $U$. We discuss a systematic way of choosing prior 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 $U$.

Citation: Matti Lassas, Eero Saksman, Samuli Siltanen. Discretization-invariant Bayesian inversion and Besov space priors. Inverse Problems & Imaging, 2009, 3 (1) : 87-122. doi: 10.3934/ipi.2009.3.87
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