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We consider a linear ill-posed equation in the Hilbert space setting underwhite noise. Known convergence results for the discrepancy principle are eitherrestricted to Hilbert-Schmidt operators (and they require a self-similaritycondition for the unknown solution $\hat{x}$, additional to a classical sourcecondition) or to polynomially ill-posed operators (excluding exponentiallyill-posed problems). In this work we show optimal convergence for a modifieddiscrepancy principle for both polynomially and exponentially ill-posedoperators (without further restrictions) solely under either H\"older-type orlogarithmic source conditions. In particular, the method includes only a singlesimple hyper parameter, which does not need to be adapted to the type ofill-posedness.

Optimal Convergence of the Discrepancy Principle for polynomially and exponentially ill-posed Operators under White Noise