Continuous time limit of the stochastic ensemble Kalman inversion: Strong convergence analysis

TL;DR

This paper rigorously proves that the stochastic ensemble Kalman inversion (EKI) method converges to its continuous-time stochastic differential equation limit, validating the use of continuous models for analyzing EKI in inverse problems.

Contribution

It provides a rigorous convergence analysis of stochastic EKI to its continuous-time limit, including proofs in both nonlinear and linear cases.

Findings

Convergence in probability for nonlinear EKI

Convergence in moments for linear EKI

Validation of continuous-time limit approach

Abstract

The Ensemble Kalman inversion (EKI) method is a method for the estimation of unknown parameters in the context of (Bayesian) inverse problems. The method approximates the underlying measure by an ensemble of particles and iteratively applies the ensemble Kalman update to evolve (the approximation of the) prior into the posterior measure. For the convergence analysis of the EKI it is common practice to derive a continuous version, replacing the iteration with a stochastic differential equation. In this paper we validate this approach by showing that the stochastic EKI iteration converges to paths of the continuous-time stochastic differential equation by considering both the nonlinear and linear setting, and we prove convergence in probability for the former, and convergence in moments for the latter. The methods employed can also be applied to the analysis of more general numerical schemes for stochastic differential equations in general.