Iterate Averaging and Filtering Algorithms for Linear Inverse Problems
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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 Igelsias, Lin, Lu, Stuart (2017), error estimates were obtained for this approach. By optimally tuning a free parameter in the filters, the authors were able to show that the mean squared error can be minimized. In the present work, we prove that by (i) considering the problem in a weaker, weighted, space and (ii) applying simple iterate averaging of the filter output, 3DVAR will converge in mean square, unconditionally on the parameter. Without iterate averaging, 3DVAR cannot converge by running additional iterations with a given, fixed, choice of parameter. We also establish that the Kalman filter's performance cannot be improved through iterate averaging. We illustrate our results with numerical experiments that suggest our convergence rates are sharp.
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