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：在反问题中，目标是从含噪声的观测数据中估计未知模型参数。传统上，反问题的求解基于描述观测模型的固定前向算子假设。本文考虑将这种方法扩展到具有动态前向模型的情形，其应用背景源于科学计算与工程领域。我们特别针对无导数优化器——集成卡尔曼反演（EKI）——进行此类扩展研究。我们提出并论证了一种称为动态EKI的新方法，这是一种前向算子可变的粒子方法。我们通过分析粒子系统的协方差结构来控制该系统，给出了相关结果。该分析包括矩量界和集成坍缩，这些对于证明收敛性结果至关重要。我们建立了期望意义下的收敛性，并通过将动态EKI应用于二维达西流偏微分方程的实验验证了理论结果。