Iterative ensemble Kalman methods: A unified perspective with some new variants

Abstract

This paper provides a unified perspective of iterative ensemble Kalman methods, a family of derivative-free algorithms for parameter reconstruction and other related tasks. We identify, compare and develop three subfamilies of ensemble methods that differ in the objective they seek to minimize and the derivative-based optimization scheme they approximate through the ensemble. Our work emphasizes two principles for the derivation and analysis of iterative ensemble Kalman methods: statistical linearization and continuum limits. Following these guiding principles, we introduce new iterative ensemble Kalman methods that show promising numerical performance in Bayesian inverse problems, data assimilation and machine learning tasks.

Hierarchical Ensemble Kalman Methods with Sparsity-Promoting Generalized Gamma Hyperpriors

“…Ensemble Kalman methods, overviewed in [22,14], were first developed as scalable filtering schemes for high-dimensional state estimation in numerical weather forecasting [20,21]. Since then, they have become popular algorithms in data assimilation, inverse problems, and machine learning.…”

“…Similar algorithms were introduced in [30,29] inspired by classical regularization schemes [26]. Ensemble Kalman methods have grown into a rich family of computational tools for the numerical solution of inverse problems; our computational framework incorporates sparsity-promoting regularization into any of the numerous existing variants [14]. For illustration purposes, this paper will consider two implementations based on the iterative ensemble Kalman filter (IEKF) and the iterative ensemble Kalman filter with statistical linearization (IEKF-SL).…”

“…Here 𝒢 : R 𝑑 → R 𝑘 is a given forward map and 𝜀 represents a Gaussian measurement error with a given covariance Γ ≻ 0. Such problem setting arises in numerous applications in data assimilation, inverse problems, machine learning, and statistics -see for instance [41,14,45] and references therein.…”

“…We will use repeatedly the principle of statistical linearization -see [48,14]-which we recall briefly. Notice that if 𝒢 is linear, i.e., 𝒢(𝑢) = 𝐺𝑢, we have…”

Bayesian calibration for large‐scale fluid structure interaction problems under embedded/immersed boundary framework

“…In the present work, each artificial time step represents a Kalman filtering iteration in which the observation y may consist of the time‐series observation data from a full forward simulation. Given its non‐intrusive nature, Kalman inversion has been widely used as an optimization method for parameter estimation , 76‐83 especially for problems where the forward model is expensive and provided as a black box that is impractical to differentiate.…”