Adaptive Tikhonov strategies for stochastic ensemble Kalman inversion

TL;DR

This paper introduces adaptive Tikhonov strategies for ensemble Kalman inversion, improving regularization by learning parameters adaptively, with theoretical analysis and numerical tests showing superior performance over fixed schemes.

Contribution

It extends Tikhonov EKI with adaptive regularization schemes based on deterministic and Bayesian methods, providing theoretical convergence results and empirical validation.

Findings

Adaptive schemes outperform fixed TEKI and EKI in experiments.

Theoretical analysis confirms convergence with noisy and time-varying data.

Numerical tests on PDEs demonstrate improved recovery of unknowns.

Abstract

Ensemble Kalman inversion (EKI) is a derivative-free optimizer aimed at solving inverse problems, taking motivation from the celebrated ensemble Kalman filter. The purpose of this article is to consider the introduction of adaptive Tikhonov strategies for EKI. This work builds upon Tikhonov EKI (TEKI) which was proposed for a fixed regularization constant. By adaptively learning the regularization parameter, this procedure is known to improve the recovery of the underlying unknown. For the analysis, we consider a continuous-time setting where we extend known results such as well-posdeness and convergence of various loss functions, but with the addition of noisy observations. Furthermore, we allow a time-varying noise and regularization covariance in our presented convergence result which mimic adaptive regularization schemes. In turn we present three adaptive regularization schemes, which are highlighted from both the deterministic and Bayesian approaches for inverse problems, which include bilevel optimization, the MAP formulation and covariance learning. We numerically test these schemes and the theory on linear and nonlinear partial differential equations, where they outperform the non-adaptive TEKI and EKI.