In this chapter, performance comparison between the adaptive filter (AF) and other estimation methods, especially with the variational method (VM), is given in the context of data assimilation problem in dynamical systems with (very) high dimension. The emphasis is put on the importance of innovation approach which is a basis for construction of the AF as well as the choice of a set of tuning parameters in the filter gain. It will be shown that the innovation representation for the initial dynamical system plays essential role in providing stability of the assimilation algorithms for stable and unstable system dynamics. Numerical experiments will be given to illustrate the performance of the AF.
This chapter is devoted to different types of optimal perturbations (OP), deterministic, stochastic, OP in an invariant subspace, and simultaneous stochastic perturbations (SSP). The definitions of OPs are given. It will be shown how the OPs are important for the study on the predictability of behavior of system dynamics, generating ensemble forecasts as well as in the design of a stable filter. A variety of algorithm-based SSP methodology for estimation and decomposition of very high-dimensional (Hd) matrices are presented. Numerical experiments will be presented to illustrate the efficiency and benefice of the perturbation technique.
Part of the book: Perturbation Methods with Applications in Science and Engineering
This chapter proposes a new approach for the design of an adaptive filter (AF), which is based on an artificial neural network (NN) structure for estimating the system state. The NNs are now widely used as a technology offering a way to solve complex and nonlinear problems such as time-series forecasting, process control, parameter state estimation, and fault diagnosis. The proposed NN-based adaptive filtering (NNAF) is designed by considering the filtering algorithm as an input–output system and two-stage optimization procedure. The first concerns a learning process where the weights of the NNAF are estimated to minimize the error between the filtered state and the state samples generated by a numerical model. The adaptation is carried out next to minimize the mean prediction error (MPE) of the system outputs (error between the observations and the system output forecast) subject to the coefficients associated with the estimated NN weights. Simulation results for different numerical models, especially for state estimation of the chaotic Lorenz system as well as for the ocean current at different deep layers which is important for renewable energy device placements, are presented to show the efficiency of the NNAF.
In this contribution, the problem of data assimilation as state estimation for dynamical systems under uncertainties is addressed. This emphasize is put on high-dimensional systems context. Major difficulties in the design of data assimilation algorithms is a concern for computational resources (computational power and memory) and uncertainties (system parameters, statistics of model, and observational errors). The idea of the adaptive filter will be given in detail to see how it is possible to overcome uncertainties as well as to explain the main principle and tools for implementation of the adaptive filter for complex dynamical systems. Simple numerical examples are given to illustrate the principal differences of the AF with the Kalman filter and other methods. The simulation results are presented to compare the performance of the adaptive filter with the Kalman filter.
Part of the book: Dynamic Data Assimilation