A New Look at the Ensemble Kalman Filter for Inverse Problems: Duality, Non-Asymptotic Analysis and Convergence Acceleration

TL;DR

This paper introduces a duality-based perspective on the Ensemble Kalman Filter (EnKF) for inverse problems, providing new non-asymptotic convergence results and proposing an adaptive covariance correction strategy that accelerates convergence and improves solution quality.

Contribution

It presents a novel duality framework for EnKF, derives non-asymptotic convergence guarantees, and introduces an adaptive covariance correction method (EnKI-MC) that enhances convergence speed and solution accuracy.

Findings

EnKF can be derived from the sample average approximation of the dual function.

The proposed EnKI-MC algorithms accelerate convergence compared to existing methods.

Numerical experiments demonstrate faster convergence and better solutions with the new strategies.

Abstract

This work presents new results and understanding of the Ensemble Kalman filter (EnKF) for inverse problems. In particular, using a Lagrangian dual perspective we show that EnKF can be derived from the sample average approximation (SAA) of the Lagrangian dual function. The beauty of this new duality perspective is that it facilitates us to prove and numerically verify a novel non-asymptotic convergence result for the EnKF. Motivated by the new perspective, we also present a new convergence improvement strategy for the Ensemble Kalman Inversion Algorithm (EnKI), which is an iterative version of the EnKF for inverse problems. In particular, we propose an adaptive multiplicative correction to the sample covariance matrix at each iteration and we call this new algorithm as EnKI-MC (I). Based on the new duality perspective, we derive an expression for the optimal correction factor at each iteration of the EnKI algorithm to accelerate the convergence. In addition, we also consider an ensemble specific multiplicative covariance correction strategy (EnKI-MC (II)) where a different correction is employed for each ensemble. By viewing EnKI through the lens of fixed-point iteration, we also provide theoretical results that guarantees the convergence of EnKI-MC (I) algorithm. Numerical investigations for the deconvolution problem, initial condition inversion in advection-convection problem, initial condition inversion in a Lorenz 96 model, and inverse problem constrained by elliptic partial differential equation are conducted to verify the non-asymptotic results for EnKF and to assess the performance of convergence improvement strategies for EnKI. The numerical results suggest that the proposed strategies for EnKI not only led to faster convergence in comparison to the currently employed techniques but also better quality solutions at termination of the algorithm.