We consider linear inverse problems under white noise. These types of problems can be tackled with, e.g., iterative regularisation methods and the main challenge is to determine a suitable stopping index for the iteration. Convergence results for popular adaptive methods to determine the stopping index often come along with restrictions, e.g. concerning the type of ill-posedness of the problem, the unknown solution or the error distribution. In the recent work \cite{jahn2021optimal} a modification of the discrepancy principle, one of the most widely used adaptive methods, applied to spectral cut-off regularisation was presented which provides excellent convergence properties in general settings. Here we investigate the performance of the modified discrepancy principle with other filter based regularisation methods and we hereby focus on the iterative Landweber method. We show that the method yields optimal convergence rates and present some numerical experiments confirming that it is also attractive in terms of computational complexity. The key idea is to incorporate and modify the discretisation dimension in an adaptive manner.