Towards adaptivity via a new discrepancy principle for Poisson inverse problems

Abstract

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