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

TL;DR

This paper demonstrates that iterate averaging ensures convergence of 3DVAR in linear inverse problems, while the Kalman filter's performance remains unaffected by averaging, supported by theoretical proofs and numerical experiments.

Contribution

It proves that iterate averaging guarantees mean square convergence of 3DVAR in linear inverse problems, unlike the Kalman filter which does not benefit from averaging.

Findings

3DVAR converges with iterate averaging in mean square

Without averaging, 3DVAR does not converge with fixed parameters

Kalman filter's performance is unaffected by iterate averaging

Abstract

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.