  Wavelet (Besov) priors are a promising way of reconstructing indirectly
measured fields in a regularized manner. We demonstrate how wavelets can be
used as a localized basis for reconstructing permeability fields with sharp
interfaces from noisy pointwise pressure field measurements in the context of
the elliptic inverse problem. For this we derive the adjoint method of
minimizing the Besov-norm-regularized misfit functional (this corresponds to
determining the maximum a posteriori point in the Bayesian point of view) in
the Haar wavelet setting. As it turns out, choosing a wavelet--based prior
allows for accelerated optimization compared to established
trigonometrically--based priors.
