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We interpret steady linear statistical inverse problems as artificial dynamicsystems with white noise and introduce a stochastic differential equation (SDE)system where the inverse of the ending time $T$ naturally plays the role of thesquared noise level. The time-continuous framework then allows us to applyclassical methods from data assimilation, namely the Kalman-Bucy filter and3DVAR, and to analyze their behaviour as a regularization method for theoriginal problem. Such treatment offers some connections to the famousasymptotical regularization method, which has not yet been analyzed in thecontext of random noise. We derive error bounds for both methods in terms ofthe mean-squared error under standard assumptions and discuss commonalities anddifferences between both approaches. If an additional tuning parameter $\alpha$for the initial covariance is chosen appropriately in terms of the ending time$T$, one of the proposed methods gains order optimality. Our results extendtheoretical findings in the discrete setting given in the recent paper Iglesiaset al. (2017). Numerical examples confirm our theoretical results.

On the asymptotical regularization for linear inverse problems in presence of white noise