In inverse problems, the goal is to estimate unknown model parameters from noisy observational data. Traditionally, inverse problems are solved under the assumption of a fixed forward operator describing the observation model. In this article, we consider the extension of this approach to situations where we have a dynamic forward model, motivated by applications in scientific computation and engineering. We specifically consider this extension for a derivative-free optimizer, the ensemble Kalman inversion (EKI). We introduce and justify a new methodology called dynamic-EKI, which is a particle-based method with a changing forward operator. We analyze our new method, presenting results related to the control of our particle system through its covariance structure. This analysis includes moment bounds and an ensemble collapse, which are essential for demonstrating a convergence result. We establish convergence in expectation and validate our theoretical findings through experiments with dynamic-EKI applied to a 2D Darcy flow partial differential equation.

翻译：在逆问题中，目标是从含噪声的观测数据中估计未知模型参数。传统上，逆问题是在假设描述观测模型的固定正演算子下求解的。本文考虑将此方法推广到具有动态正演模型的情形，这一研究受科学计算与工程领域的应用驱动。我们特别针对无导数优化器——集合卡尔曼反演（EKI）进行了这一推广。提出并论证了一种称为动态EKI的新方法，这是一种基于粒子且正演算子可变的算法。我们分析了该新方法，展示了与通过协方差结构控制粒子系统相关的结果。该分析包括矩有界性和集合坍塌，这两者对证明收敛性结果至关重要。我们建立了期望意义上的收敛性，并通过将动态EKI应用于二维达西流动偏微分方程的实验验证了理论发现。