Gradient-, Ensemble-, and Adjoint-Free Data-Driven Parameter Estimation

Nonlinear estimation methods, such as the extended Kalman filter, unscented Kalman filter, and ensemble Kalman filter, can be used for parameter estimation by viewing the unknown parameters as constant states. This paper presents an alternative approach to this problem based on retrospective cost parameter estimation (RCPE), which uses the difference between the output of the physical system and the output of the model to update the parameter estimate. The parameter update is based on a retrospective cost function, whose minimizer updates the coefficients of the estimator. The present paper extends RCPE to the case where the model depends nonlinearly on multiple unknown parameters. The main contribution is to demonstrate the need for choosing a permutation matrix that correctly associates each parameter estimate with the corresponding unknown parameter. RCPE is illustrated through several numerical examples, including the Burgers equation.