A Gaussian Mixture Model Smoother for Continuous Nonlinear Stochastic Dynamical Systems: Applications

Abstract

The nonlinear Gaussian Mixture Model Dynamically Orthogonal (GMM-DO) smoother for high-dimensional stochastic fields is exemplified and contrasted with other smoothers by applications to three dynamical systems, all of which admit far-from-Gaussian distributions. The capabilities of the smoother are first illustrated using a double-well stochastic diffusion experiment. Comparisons with the original and improved versions of the ensemble Kalman smoother explain the detailed mechanics of GMM-DO smoothing and show that its accuracy arises from the joint GMM distributions across successive observation times. Next, the smoother is validated using the advection of a passive stochastic tracer by a reversible shear flow. This example admits an exact smoothed solution, whose derivation is also provided. Results show that the GMM-DO smoother accurately captures the full smoothed distributions and not just the mean states. The final example showcases the smoother in more complex nonlinear fluid dynamics caused by a barotropic jet flowing through a sudden expansion and leading to variable jets and eddies. The accuracy of the GMM-DO smoother is compared to that of the Error Subspace Statistical Estimation smoother. It is shown that even when the dynamics result in only slightly multimodal joint distributions, Gaussian smoothing can lead to a severe loss of information. The three examples show that the backward inferences of the GMM-DO smoother are skillful and efficient. Accurate evaluation of Bayesian smoothers for nonlinear high-dimensional dynamical systems is challenging in itself. The present three examples-stochastic low dimension, reversible high dimension, and irreversible high dimension-provide complementary and effective benchmarks for such evaluation.