Ensemble Kalman Inversion: mean-field limit and convergence analysis

Abstract

Ensemble Kalman Inversion (EKI) has been a very popular algorithm used in Bayesian inverse problems. It samples particles from a prior distribution, and introduces a motion to move the particles around in pseudo-time. As the pseudo-time goes to infinity, the method finds the minimizer of the objective function, and when the pseudo-time stops at $1$, the ensemble distribution of the particles resembles, in some sense, the posterior distribution in the linear setting. The ideas trace back further to Ensemble Kalman Filter and the associated analysis, but to today, when viewed as a sampling method, why EKI works, and in what sense with what rate the method converges is still largely unknown. In this paper, we analyze the continuous version of EKI, a coupled SDE system, and prove the mean field limit of this SDE system. In particular, we will show that 1. as the number of particles goes to infinity, the empirical measure of particles following SDE converges to the solution to a Fokker-Planck equation in Wasserstein 2-distance with an optimal rate, for both linear and weakly nonlinear case; 2. the solution to the Fokker-Planck equation reconstructs the target distribution in finite time in the linear case.

References in corpus (7)

• The Ensemble Kalman Filter for Inverse Problems
• Well Posedness and Convergence Analysis of the Ensemble Kalman Inversion
• Data Assimilation: The Schrödinger Perspective
• On the continuous time limit of the Ensemble Kalman Filter
• Accelerating Langevin Sampling with Birth-death
• Ensemble Kalman Inversion for nonlinear problems: weights, consistency, and variance bounds
• The data-driven Schroedinger bridge

Cited by in corpus (15)

• Propagation of chaos: a review of models, methods and applications. II. Applications
• Ensemble Kalman Sampler: mean-field limit and convergence analysis
• Localization in Ensemble Kalman inversion
• Adaptive Tikhonov strategies for stochastic ensemble Kalman inversion
• Non-Asymptotic Analysis of Ensemble Kalman Updates: Effective Dimension and Localization
• Neural Dynamical Operator: Continuous Spatial-Temporal Model with Gradient-Based and Derivative-Free Optimization Methods
• Gradient flow structure and convergence analysis of the ensemble Kalman inversion for nonlinear forward models
• Subsampling in ensemble Kalman inversion
• Nesterov Acceleration for Ensemble Kalman Inversion and Variants
• Rough McKean-Vlasov dynamics for robust ensemble Kalman filtering
• Bayesian Experimental Design for Model Discrepancy Calibration: A Rivalry between Kullback--Leibler Divergence and Wasserstein Distance
• Bayesian Experimental Design for Model Discrepancy Calibration: An Auto-Differentiable Ensemble Kalman Inversion Approach
• Uniform-in-time propagation of chaos for the Cucker--Smale model
• Constrained Ensemble Langevin Monte Carlo
• EnKSGD: A Class Of Preconditioned Black Box Optimization And Inversion Algorithms