论文标题
在存在白噪声的情况下，对线性反问题的渐近正规化
On the asymptotical regularization for linear inverse problems in presence of white noise
论文作者
论文摘要
我们将稳定的线性统计反问题解释为具有白噪声的人工动态系统，并引入了随机微分方程（SDE）系统，其中结束时间的倒数$ t $自然扮演了平方噪声水平的作用。然后，时间连续的框架使我们能够从数据同化，即Kalman-Bucy滤波器和3DVAR中应用经典方法，并将其行为分析为原始问题的正则化方法。这种处理提供了与著名的渐近正则化方法的一些联系，该方法尚未在随机噪声的背景下进行分析。我们根据标准假设下的均方误差来得出这两种方法的误差界限，并讨论两种方法之间的共同点和差异。如果适当地选择了初始协方差的其他调整参数$α$，以结束时间$ t $进行适当选择，则建议的方法之一获得了订单最佳性。我们的结果在最近的论文Iglesias等人中给出的离散设置中扩展了理论发现。 （2017）。数值示例证实了我们的理论结果。
We interpret steady linear statistical inverse problems as artificial dynamic systems with white noise and introduce a stochastic differential equation (SDE) 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 behaviour 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 $α$ 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 Iglesias et al. (2017). Numerical examples confirm our theoretical results.
