The Ensemble Kalman inversion (EKI), proposed by Iglesias et al. for the solution of Bayesian inverse problems of type $y=A u^\dagger +\varepsilon$, with $u^\dagger$ being an unknown parameter and $y$ a given datum, is a powerful tool usually derived from a sequential Monte Carlo point of view. It describes the dynamics of an ensemble of particles $\{u^j(t)\}_{j=1}^J$, whose initial empirical measure is sampled from the prior, evolving over an artificial time $t$ towards an approximate solution of the inverse problem. Using spectral techniques, we provide a complete description of the deterministic dynamics of EKI and their asymptotic behavior in parameter space. In particular, we analyze the dynamics of deterministic EKI and mean-field EKI. We demonstrate that the Bayesian posterior can only be recovered with the mean-field limit and not with finite sample sizes or deterministic EKI. Furthermore, we show that -- even in the deterministic case -- residuals in parameter space do not decrease monotonously in the Euclidean norm and suggest a problem-adapted norm, where monotonicity can be proved. Finally, we derive a system of ordinary differential equations governing the spectrum and eigenvectors of the covariance matrix.
翻译:由Iglesias et al等提出,旨在解决Bayesian反问题(美元=A u ⁇ dagger ⁇ varepsilon$,美元是一个未知参数,美元是给定数据值,美元是一个未知的参数,美元是给定数据值)的复文(EKI),这是一个强有力的工具,通常来自相继的Monte Carlo观点。它描述了一个粒子集合的动态。它最初的经验性衡量标准是从先前的样本中抽取的,在人为时间里,美元逐渐演变为对反问题的一种近似解决办法。我们使用光谱技术,完整地描述了EKI的确定性动态及其在参数空间的无症状行为。特别是,我们分析了确定性 EKI和平均场EKI的动态。我们证明,Bayesian后方的后方的动态只能通过中位差异来恢复,而不是以有限的样本大小或确定性埃基为基。此外,我们表明,即使在确定性标准中,在常规的公式中,也能够证明,一等式的一等式标准。