线性反问题的迭代平均、卡尔曼滤波和3DVAR
Iterate Averaging, the Kalman Filter, and 3DVAR for Linear Inverse Problem
摘要：
已经提出，经典的滤波方法，如卡尔曼滤波器和3DVAR，可以用来解决线性统计逆问题。在Iglesias，Lin，Lu，& Stuart（2017）的工作中，获得了这种方法的误差估计。通过优化调整滤波器中的正则化参数，作者能够证明均方误差可以系统地降低。在Iglesias，Lin，Lu和Stuart的上述工作的基础上，我们证明了通过（i）在较弱的范数下考虑问题以及（ii）对滤波器输出应用简单的平均值，3DVAR将无条件地在参数的选择上以均方收敛。如果不进行连续平均，3DVAR无法通过使用固定参数选择运行额外迭代来收敛。我们还建立了卡尔曼滤波器的性能，在这种情况下，不能通过平均值的平均值来提高。我们用数值实验说明了我们的结果，表明我们的收敛速度是尖锐的。
英文摘要：
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.
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