SIAM/ASA Journal on Uncertainty Quantification, Volume 9, Issue 1, Page 1-28, January 2021.
We interpret steady linear statistical inverse problems as artificial dynamic systems with white noise and introduce a stochastic differential equation system where the inverse of the ending time $T$ naturally plays the role of the squared noise level. The time-continuous framework then allows us to apply classical methods from data assimilation, namely, the Kalman--Bucy filter and 3DVAR, and to analyze their behavior as a regularization method for the original problem. Such treatment offers some connections to the famous asymptotical regularization method, which has not yet been analyzed in the context of random noise. We derive error bounds for both methods in terms of the mean-squared error under standard assumptions and discuss commonalities and differences between both approaches. If an additional tuning parameter $\alpha$ for the initial covariance is chosen appropriately in terms of the ending time $T$, one of the proposed methods gains order optimality. Our results extend theoretical findings in the discrete setting given in the recent paper by Iglesias et al. [Commun. Math. Sci., 15 (2017), pp. 1867--1895]. Numerical examples confirm our theoretical results.

中文翻译：

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存在白噪声时线性反问题的渐近正则化

SIAM / ASA不确定性量化杂志，第9卷，第1期，第1-28页，2021年1月。
我们将稳定的线性统计逆问题解释为带有白噪声的人工动力系统，并引入一个随机微分方程组，其中结束时间的反演自然起着噪声平方的作用。然后，时间连续框架使我们能够应用来自数据同化的经典方法，即Kalman-Bucy滤波器和3DVAR，并分析它们的行为，作为针对原始问题的正则化方法。这种处理方法与著名的渐近正则化方法有一些联系，该方法尚未在随机噪声的情况下进行分析。我们根据标准假设下的均方误差得出两种方法的误差范围，并讨论两种方法之间的共性和差异。如果用于初始协方差的附加调谐参数$ \阿尔法$在的结束时间$ T $，所提出的方法的增益阶最优的一个方面适当选择。我们的结果扩展了Iglesias等人在最近的论文中给出的离散环境下的理论发现。[公共。数学。Sci。，15（2017），pp.1867--1895]。数值例子证实了我们的理论结果。