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Discretizationinvariant Bayesian inversion and Besov space priors
1.  Department of Mathematics and Statistics, University of Helsinki, P.O. Box 68 (Gustaf Hallstromin katu 2b) FI00014, Finland, Finland 
2.  Tampere University of Technology,Institute of Mathematics,, P.O. Box 553, 33101 Tampere 
$\pi_{kn}(u_n\\m_{kn})$~ 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 infinitedimensional limit case. Such choice of prior distributions is discretizationinvariant 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 waveletbased 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 $U$.
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