  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, & Stuart (2017), 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, & 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.
