Ensemble Kalman Inversion (EKI) has been proposed as an efficient method for solving inverse problems with expensive forward models. However, the method is based on the assumption that we proceed through a sequence of Gaussian measures in moving from the prior to the posterior, and that the forward model is linear. In this work, we introduce Sequential Kalman Monte Carlo (SKMC) samplers, where we exploit EKI and Flow Annealed Kalman Inversion (FAKI) within a Sequential Monte Carlo (SMC) sampling scheme to perform efficient gradient-free inference in Bayesian inverse problems. FAKI employs normalizing flows (NF) to relax the Gaussian ansatz of the target measures in EKI. NFs are able to learn invertible maps between a Gaussian latent space and the original data space, allowing us to perform EKI updates in the Gaussianized NF latent space. However, FAKI alone is not able to correct for the model linearity assumptions in EKI. Errors in the particle distribution as we move through the sequence of target measures can therefore compound to give incorrect posterior moment estimates. In this work we consider the use of EKI and FAKI to initialize the particle distribution for each target in an adaptive SMC annealing scheme, before performing t-preconditioned Crank-Nicolson (tpCN) updates to distribute particles according to the target. We demonstrate the performance of these SKMC samplers on three challenging numerical benchmarks, showing significant improvements in the rate of convergence compared to standard SMC with importance weighted resampling at each temperature level. Code implementing the SKMC samplers is available at https://github.com/RichardGrumitt/KalmanMC.

翻译：集成卡尔曼反演（EKI）已被提出作为解决具有昂贵正演模型的反问题的有效方法。然而，该方法基于以下假设：从先验分布到后验分布的过程中，我们通过一系列高斯测度进行递推，且正演模型是线性的。本文中，我们提出了序贯卡尔曼蒙特卡洛（SKMC）采样器，通过在序贯蒙特卡洛（SMC）采样框架中结合EKI与流退火卡尔曼反演（FAKI），实现了贝叶斯反问题中高效的无梯度推断。FAKI利用归一化流（NF）来松弛EKI中目标测度的高斯假设。NF能够学习高斯隐空间与原始数据空间之间的可逆映射，从而允许我们在高斯化的NF隐空间中进行EKI更新。然而，仅依靠FAKI无法修正EKI中的模型线性假设。当我们在目标测度序列中递推时，粒子分布的误差可能会累积，导致后验矩估计不准确。本文提出在自适应SMC退火方案中，使用EKI和FAKI为每个目标初始化粒子分布，随后执行t-预处理Crank-Nicolson（tpCN）更新，使粒子按目标分布重新配置。我们在三个具有挑战性的数值基准测试中验证了SKMC采样器的性能，结果表明：与在每个温度层级进行重要性加权重采样的标准SMC方法相比，SKMC显著提升了收敛速度。SKMC采样器的实现代码发布于https://github.com/RichardGrumitt/KalmanMC。