We propose alternatives to Bayesian prior distributions that are frequently used in the study of inverse problems. Our aim is to construct priors that have similar good edge-preserving properties as total variation or Mumford-Shah priors but correspond to well-defined infinite-dimensional random variables, and can be approximated by finite-dimensional random vari-ables. We introduce a new wavelet-based model, where the non-zero coefficients are chosen in a systematic way so that prior draws have certain fractal behaviour. We show that realisations of this new prior take values in Besov spaces and have singularities only on a small set τ with a certain Hausdorff dimension. We also introduce an efficient algorithm for calculating the MAP estimator, arising from the the new prior, in the denoising problem.

The Bayesian approach to inverse problems is studied in the case where the forward map is a linear hypoelliptic pseudodifferential operator and measurement error is additive white Gaussian noise. The measurement model for an unknown Gaussian random variable U (x, w) is Mδ (y, w) = A (U (x, w)) + δ>(y, w), where A is a finitely many orders smoothing linear hypoelliptic operator and δ> 0 is the noise magnitude. The covariance operator C_U of U is smoothing of order 2r, self-adjoint, injective and elliptic pseudodifferential operator. If ϵ was taking values in L² then in Gaussian case solving the conditional mean (and maximum a posteriori) estimate is linked to solving the minimisation problem Tδ (mδ) = arg min μϵHr{ Au-mδ;2_L ²+δ2C_U ^-1/2u2_L ²} However, Gaussian white noise does not take values in L²but in H-s where s>0 is big enough. A modification of the above approach to solve the inverse problem is presented, covering the case of white Gaussian measurement noise. Furthermore, the convergence of the conditional mean estimate to the correct solution as δ → 0 is proven in appropriate function spaces using microlocal analysis. Also the frequentist posterior contractions rates are studied.

Tikhonov regularization is studied in the case of linear pseudodifferential operator as the forward map and additive white Gaussian noise as the measurement error. The measurement model for an unknown function u(x) is m(x) = Au(x) + δε(x) where δ > 0 is the noise magnitude. If ε was an L 2 -function, Tikhonov regularization gives an estimate T α (m) = arg min u∈Hr {∥Au - m∥ 2 L2 + α∥u∥ 2 Hr } for u where α = α(δ) is the regularization parameter. Here penalization of the Sobolev norm ∥u∥ Hr covers the cases of standard Tikhonov regularization (r = 0) and first derivative penalty (r = 1). Realizations of white Gaussian noise are almost never in L 2 , but do belong to H s with probability one if s < 0 is small enough. A modification of Tikhonov regularization theory is presented, covering the case of white Gaussian measurement noise. Furthermore, the convergence of regularized reconstructions to the correct solution as δ → 0 is proven in appropriate function spaces using microlocal analysis. The convergence of the related finite-dimensional problems to the infinite-dimensional problem is also analysed.