Gradient flow structure and convergence analysis of the ensemble Kalman inversion for nonlinear forward models (2203.17117v2)

Abstract: The ensemble Kalman inversion (EKI) is a particle based method which has been introduced as the application of the ensemble Kalman filter to inverse problems. In practice it has been widely used as derivative-free optimization method in order to estimate unknown parameters from noisy measurement data. For linear forward models the EKI can be viewed as gradient flow preconditioned by a certain sample covariance matrix. Through the preconditioning the resulting scheme remains in a finite dimensional subspace of the original high-dimensional (or even infinite dimensional) parameter space and can be viewed as optimizer restricted to this subspace. For general nonlinear forward models the resulting EKI flow can only be viewed as gradient flow in approximation. In this paper we discuss the effect of applying a sample covariance as preconditioning matrix and quantify the gradient flow structure of the EKI by controlling the approximation error through the spread in the particle system. The ensemble collapse on the one side leads to an accurate gradient approximation, but on the other side to degeneration in the preconditioning sample covariance matrix. In order to ensure convergence as optimization method we derive lower as well as upper bounds on the ensemble collapse. Furthermore, we introduce covariance inflation without breaking the subspace property intending to reduce the collapse rate of the ensemble such that the convergence rate improves. In a numerical experiment we apply EKI to a nonlinear elliptic boundary-value problem and illustrate the dependence of EKI as derivative-free optimizer on the choice of the initial ensemble.

• The paper establishes a gradient flow approximation of the EKI for nonlinear forward models using a derivative-free optimization approach.
• It rigorously proves convergence with Lyapunov functions and coercivity arguments, demonstrating ensemble collapse rates of order 1/t.
• It introduces covariance inflation to mitigate ensemble collapse, enhancing stability and convergence in inverse problem solutions.

Gradient Flow Structure and Convergence Analysis of the Ensemble Kalman Inversion for Nonlinear Forward Models

The paper by Simon Weissmann provides a detailed analysis of the Ensemble Kalman Inversion (EKI) technique applied to nonlinear forward models, emphasizing its interpretation as a derivative-free optimization method. EKI, originally derived from the Ensemble Kalman Filter used in data assimilation, is utilized to estimate parameters in inverse problems. This paper builds upon the continuous-time formulation of the EKI, which can be traced back to the Underlying Particle System's deterministic ODE representation.

Core Contributions and Theoretical Insights

The primary focus of the paper is to quantify EKI's role as an optimizer within the context of inverse problems with nonlinear forward models. It showcases several pivotal findings:

1. Gradient Flow Approximation: The EKI for nonlinear models can be approximated as a gradient flow despite the challenges arising from the nonlinearity. Weissmann separate the EKI flow into a preconditioned gradient flow and an approximation error, capturing the intricacy of the non-linear dynamics through Taylor expansions.
2. Convergence and Coercivity Analysis: The paper offers a rigorous examination of convergence by deriving lower and upper bounds on the EKI's ensemble collapse. The convergence is ensured through proofs based on Lyapunov functions and coercivity arguments. This includes bounds showing that the degeneracy rate is bounded by $1/t$, suggesting that ensemble collapse contributes to gradient calculation accuracy while maintaining finite sub-space representation.
3. Covariance Inflation: Introducing covariance inflation, Weissmann develops a method to reduce ensemble collapse rates while preserving the subspace constraint property. This inflation serves to stabilize the optimization process, facilitating superior convergence rates.
4. Numerical Experiments and Observations: The theoretical advancements are substantiated through numerical simulations, including solving a one-dimensional elliptic boundary-value problem. The choice of initial ensembles plays a role in optimization performance, as illustrated in these experiments.

Implications and Future Directions

The theoretical expansions presented in this paper allow for broader understandings of EKI's applicability to complex inverse problems, especially where derivative information may not be accessible or practical to compute. The incorporation of covariance inflation presents a fresh angle that might prove instrumental in enhancing convergence speed, thus offering practical value in real-world applications.

For future developments, several pathways emerge. Extending these findings to stochastic EKI formulations remains a significant yet challenging task, as this involves grappling with path-dependent properties of ensemble collapse. Furthermore, exploring this approach's computational efficiency compared to traditional derivative-based methods, such as Quasi-Newton or other gradient-biased optimizers, could provide deeper insights into its broader applicability.

Concluding Thoughts

Simon Weissmann's paper contributes valuable theoretical insights and practical considerations, enhancing the ensemble Kalman inversion technique's utility for nonlinear forward models. By underpinning EKI's operations with rigorous mathematical foundations and productive innovations like covariance inflation, this research offers a significant step forward in utilizing derivative-free approaches within the field of optimization for inverse problems. Through both theoretical exposition and empirical evidence, it underscores the sophistication of EKI when treated with appropriate analytical tools.