Iterate averaging, the Kalman filter, and 3DVAR for linear inverse problems

It has been proposed that classical filtering methods, like the Kalman filter and 3DVAR, can be used to solve linear statistical inverse problems. In the work of Iglesias, Lin, Lu, and Stuart (Commun. Math. Sci. 15(7):1867–1896, ??), error estimates were obtained for this approach. By optimally tuning a regularization parameter in the filters, the authors were able to show that the mean squared error could be systematically reduced. Building on the aforementioned work of Iglesias, Lin, Lu, and Stuart, we prove that by (i) considering the problem in a weaker norm and (ii) applying simple iterate averaging of the filter output, 3DVAR will converge in mean square, unconditionally on the choice of parameter. Without iterate averaging, 3DVAR cannot converge by running additional iterations with a fixed choice of parameter. We also establish that the Kalman filter’s performance in this setting cannot be improved through iterate averaging. We illustrate our results with numerical experiments that suggest our convergence rates are sharp.