The Fundamental Subspaces of Ensemble Kalman Inversion

TL;DR

This paper analyzes the convergence behavior of Ensemble Kalman Inversion (EKI) methods for linear inverse problems by introducing fundamental subspaces, providing spectral decompositions, and verifying convergence rates through numerical experiments.

Contribution

It introduces a new analysis of EKI's convergence using fundamental subspaces, offering spectral decompositions and insights into deterministic and stochastic EKI behavior.

Findings

Verifies convergence rates for EKI in linear settings.

Defines six fundamental subspaces for EKI analysis.

Provides spectral decompositions linking EKI to linear algebra fundamentals.

Abstract

Ensemble Kalman Inversion (EKI) methods are a family of iterative methods for solving weighted least-squares problems, especially those arising in scientific and engineering inverse problems in which unknown parameters or states are estimated from observed data by minimizing the weighted square norm of the data misfit. Implementation of EKI requires only evaluation of the forward model mapping the unknown to the data, and does not require derivatives or adjoints of the forward model. The methods therefore offer an attractive alternative to gradient-based optimization approaches in inverse problem settings where evaluating derivatives or adjoints of the forward model is computationally intractable. This work presents a new analysis of the behavior of both deterministic and stochastic versions of basic EKI for linear observation operators, resulting in a natural interpretation of EKI's convergence properties in terms of ``fundamental subspaces'' analogous to Strang's fundamental subspaces of linear algebra. Our analysis directly examines the discrete EKI iterations instead of their continuous-time limits considered in previous analyses, and provides spectral decompositions that define six fundamental subspaces of EKI spanning both observation and state spaces. This approach verifies convergence rates previously derived for continuous-time limits, and yields new results describing both deterministic and stochastic EKI convergence behavior with respect to the standard minimum-norm weighted least squares solution in terms of the fundamental subspaces. Numerical experiments illustrate our theoretical results.