Linear theory for filtering nonlinear multiscale systems with model error

We study filtering of multiscale dynamical systems with model error arising from unresolved smaller scale processes. The analysis assumes continuous-time noisy observations of all components of the slow variables alone. For a linear model with Gaussian noise, we prove existence of a unique choice of parameters in a linear reduced model for the slow variables. The linear theory extends to to a non-Gaussian, nonlinear test problem, where we assume we know the optimal stochastic parameterization and the correct observation model. We show that when the parameterization is inappropriate, parameters chosen for good filter performance may give poor equilibrium statistical estimates and vice versa. Given the correct parameterization, it is imperative to estimate the parameters simultaneously and to account for the nonlinear feedback of the stochastic parameters into the reduced filter estimates. In numerical experiments on the two-layer Lorenz-96 model, we find that parameters estimated online, as part of a filtering procedure, produce accurate filtering and equilibrium statistical prediction. In contrast, a linear regression based offline method, which fits the parameters to a given training data set independently from the filter, yields filter estimates which are worse than the observations or even divergent when the slow variables are not fully observed

Data-assisted reduced-order modeling of extreme events in complex dynamical systems

Dynamical systems with high intrinsic dimensionality are often characterized by extreme events having the form of rare transitions several standard deviations away from the mean. For such systems, order-reduction methods through projection of the governing equations have limited applicability due to the large intrinsic dimensionality of the underlying attractor but also the complexity of the transient events. An alternative approach is data-driven techniques that aim to quantify the dynamics of specific modes utilizing data-streams. Several of these approaches have improved performance by expanding the state representation using delayed coordinates. However, such strategies are limited in regions of the phase space where there is a small amount of data available, as is the case for extreme events. In this work, we develop a blended framework that integrates an imperfect model, obtained from projecting equations into a subspace that still contains crucial dynamical information, with data-streams through a recurrent neural network (RNN) architecture. In particular, we employ the long-short-term memory (LSTM), to model portions of the dynamics which cannot be accounted by the equations. The RNN is trained by analyzing the mismatch between the imperfect model and the data-streams, projected in the reduced-order space. In this way, the data-driven model improves the imperfect model in regions where data is available, while for locations where data is sparse the imperfect model still provides a baseline for the prediction of the system dynamics. We assess the developed framework on two challenging prototype systems exhibiting extreme events and show that the blended approach has improved performance compared with methods that use either data streams or the imperfect model alone. The improvement is more significant in regions associated with extreme events, where data is sparse.Comment: Submitted to PLOS ONE on March 8, 201

LEMDA: A Lagrangian-Eulerian Multiscale Data Assimilation Framework

Lagrangian trajectories are widely used as observations for recovering the underlying flow field via Lagrangian data assimilation (DA). However, the strong nonlinearity in the observational process and the high dimensionality of the problems often cause challenges in applying standard Lagrangian DA. In this paper, a Lagrangian-Eulerian multiscale DA (LEMDA) framework is developed. It starts with exploiting the Boltzmann kinetic description of the particle dynamics to derive a set of continuum equations, which characterize the statistical quantities of particle motions at fixed grids and serve as Eulerian observations. Despite the nonlinearity in the continuum equations and the processes of Lagrangian observations, the time evolutions of the posterior distribution from LEMDA can be written down using closed analytic formulae. This offers an exact and efficient way of carrying out DA, which avoids using ensemble approximations and the associated tunings. The analytically solvable properties also facilitate the derivation of an effective reduced-order Lagrangian DA scheme that further enhances computational efficiency. The Lagrangian DA within the framework has advantages when a moderate number of particles is used, while the Eulerian DA can effectively save computational costs when the number of particle observations becomes large. The Eulerian DA is also valuable when particles collide, such as using sea ice floe trajectories as observations. LEMDA naturally applies to multiscale turbulent flow fields, where the Eulerian DA recovers the large-scale structures, and the Lagrangian DA efficiently resolves the small-scale features in each grid cell via parallel computing. Numerical experiments demonstrate the skilful results of LEMDA and its two components