A modified discrepancy principle to attain optimal rates under white noise

We consider a linear ill-posed equation in the Hilbert space setting under white noise. Known convergence results for the discrepancy principle are either restricted to Hilbert-Schmidt operators (and they require a self-similarity condition for the unknown solution additional to a classical source condition) or to polynomially ill-posed operators (excluding exponentially ill-posed problems), with an additional saturation for very smooth solutions.  In this work we show optimal convergence for a modified discrepancy principle for both polynomially and exponentially ill-posed operators (without further restrictions or saturation) solely under either Hölder-type or logarithmic source conditions.