Analysis of regularized inversion of data corrupted by white Gaussian noise

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) + delta epsilon(x), where delta > 0 is the noise magnitude. If epsilon was an L-2-function, Tikhonov regularization gives an estimate T-alpha(m) = arg min (u epsilon Hr) { vertical bar vertical bar Au - m vertical bar vertical bar(2)(2)(L) + alpha vertical bar vertical bar u vertical bar vertical bar(2) (r)(H) } for u where alpha = alpha(delta) is the regularization parameter. Here penalization of the Sobolev norm vertical bar vertical bar u vertical bar vertical bar H-r 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 delta -> 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.