Convergence rates of stationary and non-stationary asymptotical regularization methods for statistical inverse problems in Banach spaces

Abstract

We investigate stationary and non-stationary asymptotical regularization methods for statistical inverse problems in Banach spaces. The mean-squared errors (MSE) of both methods are estimated without the conventional assumption of the commutativity between the solution variance and the noise variance. Moreover, the a priori smoothness of solution is characterized in terms of real interpolation, which is strictly weaker than the classical settings. The sufficient and necessary conditions of optimal convergence rates of MSE are proven. Our results extend theoretical findings in Hilbert settings given in the recent paper by Lu et al. [SIAM/ASA J. Uncertain., 9 (2021), pp. 1488-1525]. Two examples of integral equations with white noise and inverse source problems in elliptic partial differential equations are demonstrated to show the usefulness of our theoretical results.