Regularizing linear inverse problems under unknown non-Gaussian white noise allowing repeated measurements.

The article focuses on solving generic linear inverse problems in Hilbert spaces where the exact right-hand side is unknown and only accessible through discretized measurements corrupted by arbitrary unknown white noise. It presents two main approaches: one using a finite-dimensional residuum that guarantees convergence in probability without requiring knowledge of the noise distribution or discretization error, and another using an infinite-dimensional residuum that achieves convergence rates under additional assumptions on the discretization error. Both approaches rely on multiple repeated measurements to estimate and reduce noise via averaging, with the discrepancy principle employed as an a posteriori parameter choice rule. The work establishes necessary conditions for convergence, notably that the ratio of measurement channels to repetitions tends to zero, and discusses the impact of discretization schemes, noise moment conditions, and smoothness assumptions on the solution and convergence behavior. Numerical experiments illustrate the theoretical findings across various ill-posed problems and discretization methods.