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It has been proposed that classical filtering methods, like the Kalman filterand 3DVAR, can be used to solve linear statistical inverse problems. In thework of Iglesias, Lin, Lu, & Stuart (2017), error estimates were obtained forthis approach. By optimally tuning a regularization parameter in the filters,the authors were able to show that the mean squared error could besystematically reduced.  Building on the aforementioned work of Iglesias, Lin, Lu, & Stuart, we provethat by (i) considering the problem in a weaker norm and (ii) applying simpleiterate averaging of the filter output, 3DVAR will converge in mean square,unconditionally on the choice of parameter. Without iterate averaging, 3DVARcannot converge by running additional iterations with a fixed choice ofparameter. We also establish that the Kalman filter's performance in thissetting cannot be improved through iterate averaging. We illustrate our resultswith numerical experiments that suggest our convergence rates are sharp.

Iterate Averaging, the Kalman Filter, and 3DVAR for Linear Inverse Problem