A Gaussian Mixture Model Smoother for Continuous Nonlinear Stochastic Dynamical Systems: Theory and Scheme

Abstract

Retrospective inference through Bayesian smoothing is indispensable in geophysics, with crucial applications in ocean and numerical weather estimation, climate dynamics, and Earth system modeling. However, dealing with the high-dimensionality and nonlinearity of geophysical processes remains a major challenge in the development of Bayesian smoothers. Addressing this issue, a novel subspace smoothing methodology for high-dimensional stochastic fields governed by general nonlinear dynamics is obtained. Building on recent Bayesian filters and classic Kalman smoothers, the fundamental equations and forwardbackward algorithms of new Gaussian Mixture Model (GMM) smoothers are derived, for both the full state space and dynamic subspace. For the latter, the stochastic Dynamically Orthogonal (DO) field equations and their time-evolving stochastic subspace are employed to predict the prior subspace probabilities. Bayesian inference, both forward and backward in time, is then analytically carried out in the dominant stochastic subspace, after fitting semiparametric GMMs to joint subspace realizations. The theoretical properties, varied forms, and computational costs of the new GMM smoother equations are presented and discussed.

The Extrinsic Geometry of Dynamical Systems Tracking Nonlinear Matrix Projections

“…Their relevance and numerical implementations for tracking smooth decompositions was discussed. Possible future applications of the derived dynamic matrix equations abound over a rich spectrum of needs, from dynamic reduced-order modeling [57,22] and data sciences [41] to adaptive data assimilation [45,7,51] and adaptive path planning and sampling [50,63,46,47]. Downloaded 02/11/20 to 18.10.29.253.…”

Stochastic acoustic ray tracing with dynamically orthogonal equations

“…The stochastic Dynamically Orthogonal differential equations have been derived to evolve stochastic fields while preserving its dominant statistics [78,97,25]. With such stochastic predictions, we can use our gained knowledge of the forecast probability distributions to complete non-Gaussian Bayesian data assimilation [80,58] and optimize the data collection using information-based adaptive sampling and principled model learning [44,50,48]. turing the stochastic variation in an ocean's acoustic propagation due to an uncertain SSP, using a dynamic reduced oder representation of the stochastic ray field.…”

“…• It is amenable to already developed non-Gaussian data-assimilation or Bayesian-inference algorithms [80,81,58,57].…”

Optimal Planning and Sampling Predictions for Autonomous and Lagrangian Platforms and Sensors in the Northern Arabian Sea

“…It is the advances in these disciplines that allow the present results, especially the progress in multiresolution numerical ocean modeling Deleersnijder and Lermusiaux, 2008;Deleersnijder et al, 2010;Haley and Lermusiaux, 2010;Cushman-Roisin and Beckers, 2011;Ringler et al, 2013;Haley et al, 2015;Burchard et al, 2017) and in ensemble uncertainty prediction and Bayesian data assimilation (Lermusiaux and Robinson, 1999;Lermusiaux et al, 2006a,b;Lermusiaux, 2006;Bocquet et al, 2010;Särkkä, 2013;Sondergaard and Lermusiaux, 2013a;Reich and Cotter, 2015;Lolla and Lermusiaux, 2017a). These advances will be employed next.…”