Ensemble Collapse Analysis of Tikhonov-regularized Ensemble Kalman Inversion in Parameter Estimation for Nonlinear Models

We investigate the efficacy of Tikhonov-regularized ensemble Kalman inversion (TEKI) in identifying unknown parameters within partial differential equations (PDEs), under the assumption of a nonlinear mapping from the parameter space to the observation space and with a fixed ensemble size. We conduct a comprehensive analysis of the coupled stochastic differential equations (SDEs) derived from continuous-time formulations, characterizing the ensemble collapse in terms of second moment and almost sure convergence with given rate. Furthermore, this collapse serves as an indicator of the confidence level of our parameter estimator, which is the empirical mean of the particle ensemble.