Adaptive Tikhonov strategies for stochastic ensemble Kalman inversion

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