Ensemble Kalman inversion (EKI) is a technique for the numerical solution of inverse problems. A great advantage of the EKI's ensemble approach is that derivatives are not required in its implementation. But theoretically speaking, EKI's ensemble size needs to surpass the dimension of the problem. This is because of EKI's "subspace property", i.e., that the EKI solution is a linear combination of the initial ensemble it starts off with. We show that the ensemble can break out of this initial subspace when ``localization'' is applied. In essence, localization enforces an assumed correlation structure onto the problem, and is heavily used in ensemble Kalman filtering and data assimilation. We describe and analyze how to apply localization to the EKI, and how localization helps the EKI ensemble break out of the initial subspace. Specifically, we show that the localized EKI (LEKI) ensemble will collapse to a single point (as intended) and that the LEKI ensemble mean will converge to the global optimum at a sublinear rate. Under strict assumptions on the localization procedure and observation process, we further show that the data misfit decays uniformly. We illustrate our ideas and theoretical developments with numerical examples with simplified toy problems, a Lorenz model, and an inversion of electromagnetic data, where some of our mathematical assumptions may only be approximately valid.
翻译:嵌入 Kalman Inversion (EKI) 是一种用数字方法解决反向问题的技术。 EKI 组合法的一大优点是其实施不需要衍生物。 但理论上说, EKI 组合体大小需要超过问题的维度。 这是因为 EKI 的“ 子空间属性 ”, 也就是说, EKI 解决方案是最初组合的线性组合。 我们显示,当应用“ 本地化” 时, 组合体可以打破这个初始子空间。 本质上, 本地化使问题不需要使用衍生物。 但是, 从理论上讲, EKI 组合体的组合体大小需要超过问题的维度。 我们描述并分析如何将本地化适用于 EKI 的“ 子空间 属性 ”, 也就是说, EKI( LEKI) 组合体的组合体会崩溃到一个单一点( 原意), 而LEKI 组合体将意味着将一个假定结构结构结构结构与一个精确的理论性模型相融合, 我们用一个精确的模型来说明一个精确的数据模型, 。