Bayesian solution of an inverse problem for indirect measurement M = AU +E is considered, where U is a function on a domain of R d . Here A is a smoothing linear operator and E is Gaussian white noise. The data is a realization m k of the random variable M k = P k AU + P k 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 AU n + P k E. Bayes formula gives then the posterior distribution π kn (u n | m kn ) ∼ Π n (u n ) exp(− 1 2 m kn − P k Au n 2 2 ) in R d , and the mean u kn := u n π 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 ≥ 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 → ∞. 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.
The problem this paper addresses is how to use the two-dimensional D-bar method for electrical impedance tomography with experimental data collected on finitely many electrodes covering a portion of the boundary of a body. This requires an approximation of the Dirichlet-to-Neumann, or voltage-to-current density map, defined on the entire boundary of the region, from a finite number of matrix elements of the current-to-voltage map. Reconstructions from experimental data collected on a saline filled tank containing agar heart and lung phantoms are presented, and the results are compared to reconstructions by the NOSER algorithm on the same data.
In theorem 3.1 of this paper we presented an estimate for t(k) and µ(x, k) for k near zero. The statement of theorem 3.1 holds, but the proof contains two errors.• We incorrectly stated that S 0 = R 1 for a general C 2 domain . This led to an erroneous formula (24). We prove below that the identity S 0 = 1 2 R 1 holds when is the unit disc. The proof of theorem 3.1 can then be corrected by reducing it to that case.C|k| for small |k|. However, in the original proof H k is an operator from H −1/2 (∂ ) to H 1/2 (∂ ). We provide a new argument showing that the above estimate can be used.
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