Towards adaptivity via a new discrepancy principle for Poisson inverse problems

A new form of the discrepancy principle for Poisson inverse problems with compact operators is proposed and discussed in relation to various other proposals. It shown that filter-induced spectral regularization a priori chosen smoothing parameter produces estimators are rate-minimax under source conditions on estimated function. With used posteriori choice parameter, solutions consistent (in probability), but convergence rates suboptimal, at least finitely case, which often happens when principles stochastic problems. Finite sample performance procedure applied stereological problem Spektor, Lord Willis illustrated simulation experiment.