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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. [ABSTRACT FROM AUTHOR]

The presence of magnetized anomalies in the shell of the Earth interrupts its geomagnetic field. We consider the inverse problem of identifying the anomalies by monitoring the variation of the geomagnetic field. Motivated by the theoretical unique identifiability result in [7] , we develop a novel numerical scheme of locating multiple magnetized anomalies. In our study, we do not assume the source that generates the geomagnetic field, and the medium configurations of the Earth's core and the magnetized anomalies are a-priori known. The core of the reconstruction scheme is a novel imaging functional whose quantitative behaviours can be used to identify the anomalies. Both rigorous analysis and extensive numerical experiments are provided to verify the effectiveness and promising features of the proposed reconstruction scheme. • Magnetic anomalies detection from geomagnetic monitoring is practically important and challenging. • Based on a sophisticated geomagnetic model, a novel reconstruction scheme is proposed and developed. • In our method, the geomagnetic source and the medium configuration of the Earth's core are both not a-priori known. • The core is a novel imaging functional whose promising features are both theoretically and computationally verified. [ABSTRACT FROM AUTHOR]