Linear theory for filtering nonlinear multiscale systems with model error

Abstract

In this paper, we study filtering of multiscale dynamical systems with model error arising from limitations in resolving the smaller scale processes. In particular, the analysis assumes the availability of continuous-time noisy observations of all components of the slow variables. Mathematically, this paper presents new results on higher order asymptotic expansion of the first two moments of a conditional measure. In particular, we are interested in the application of filtering multiscale problems in which the conditional distribution is defined over the slow variables, given noisy observation of the slow variables alone. From the mathematical analysis, we learn that for a continuous time linear model with Gaussian noise, there exists a unique choice of parameters in a linear reduced model for the slow variables which gives the optimal filtering when only the slow variables are observed. Moreover, these parameters simultaneously give the optimal equilibrium statistical estimates of the underlying system, and as a consequence they can be estimated offline from the equilibrium statistics of the true signal. By examining a nonlinear test model, we show that the linear theory extends in this non-Gaussian, nonlinear configuration as long as we know the optimal stochastic parametrization and the correct observation model. However, when the stochastic parametrization model is inappropriate, parameters chosen for good filter performance may give poor equilibrium statistical estimates and vice versa; this finding is based on analytical and numerical results on our nonlinear test model and the two-layer Lorenz-96 model.

Accounting for Model Error from Unresolved Scales in Ensemble Kalman Filters by Stochastic Parameterization

“…A more challenging and realistic setting would be one where the full model is unknown, and one has to use noisy observations to infer a parametrization (Li et al 2009;Berry and Harlim 2014;Harlim 2016). This is the challenging topic of parameter estimation for hidden Markov and non-Markov models (Kantas et al 2009).…”

“…Examples include deterministic and stochastic parametrization methods (Palmer 2001;Meng and Zhang 2007;Berry and Harlim 2014;Mitchell and Carrassi 2015), additive random perturbations (Hamill and Whitaker 2005;Houtekamer et al 2009), low dimensional method (Li et al 2009), and averaging and homogenization methods (Pavliotis and Stuart 2008;Mitchell and Gottwald 2012;Gottwald and Harlim 2013). Representations of the model error can be derived either within data assimilation using the noisy observations, or before data assimilation using noiseless training data.…”

Parametric reduced models for the nonlinear Schrödinger equation

“…We should point out that if we fix these two parameters to be those from the Mori-Zwanzig projection and run the filtering procedure to estimate the remaining parameters, a,b,σ 1 ,R, the resulting estimates are completely innaccurate. This suggests that while the perturbation approach [16] suggests the explicit forms of the reduced model, it is more natural to adaptively estimate all the parameters which reconfirms the results in [37]. Moreover, in general, there can be nonunique parameters that provide the same equilibrium statistics (for e.g., see Proposition 1(d) in [26]).…”

“…In general problems, however, the success of this modeling approach will depend mostly on the choice of the ansatz for modeling the memory terms. As it has been theoretically established in [37], if the ansatz is adequate, then it is possible to obtain, both, accurate climatological statistical forecasting and optimal filtering. Our NLS example in this paper empirically suggested that our ansatz is optimal in this case.…”

“…Other potential issue is in the parameter estimation strategy which can be expensive when more observations are included. While many cheaper parameterization methods are available (such as regression-based or maximum likelihood-based algorithms), these methods are often inferior to the adaptive method applied in the present work even when adequate ansatz is used as shown in [37]. Therefore, improving the numerical efficiency of the adaptive parameter estimation scheme that we used here [26] or its variant (see e.g., [35,36]) will be the key for successful applications in more complex problems.…”