A dynamic mode decomposition-based Kalman filter for Bayesian inverse problem of nonlinear dynamical systems

Abstract

Ensemble Kalman filter (EnKF) method has been widely used in parameter estimation of the dynamic models, which needs to compute the forward model repeatedly. For nonlinear parameterized PDEs, constructing an accurate reduced order model is extremely challenging. To accelerate the posterior exploration efficiently, building surrogates of the forward models is necessary. In this paper, the dynamic mode decomposition (DMD) coupled with the weighted and interpolated nearest-neighbors (wiNN) algorithm is introduced to construct the surrogates for nonlinear dynamical systems. This extends the applicability of DMD to parameterized problems. Moreover, a low rank approximation of Kalman gain is used to EnKF, which can avoid the ensemble degenerate from the singularity of the covariance matrix. Finally, we apply the proposed method to nonlinear parameterized PDEs for the two-dimensional fluid flow and investigate their Bayesian inverse problems. The results are presented to show the applicability and efficiency of the proposed EnKF with DMD-wiNN method by taking account of parameters in nonlinear diffusion functions, nonlinear reaction functions and source functions.