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 的反演( EKI) 是一种用数字方法解决反倒问题的技术。 EKI 组合法的一大优势是其实施不需要衍生工具。 但理论上, EKI 组合体大小需要超过问题的范围。 这是因为 EKI 的“ 子空间属性 ”, 也就是说, EKI 解决方案是它最初开始的元素的线性组合。 我们显示,当应用“ 本地化” 时, 数学组合可以打破这个初始子空间。 本质上, 本地化可以强制实施一种假定的关联结构, 并且被大量用于同化 Kalman 过滤和数据同化。 我们描述并分析如何将本地化应用到 EKI 的“ 子空间 属性 ”, 也就是说, EKI ( LEKI) 组合体 组合体的初始组合体是线性组合的线性组合。 当应用“ 本地化” 时, 当应用“ 本地化” 时, 当应用这个初始子空间时, 组合时, 组合体会崩溃时, 就会打破这个初始的子空间空间空间空间空间空间。 LEKIKI 的假设,, 将意味着 和 与全球的精确化 的理论化 将逐渐接近同步化 的理论化, 我们的理论化的模型化过程显示一个精确化过程。