Filtering Complex Turbulent Systems

Many natural phenomena ranging from climate through to biology are described by complex dynamical systems. Getting information about these phenomena involves filtering noisy data and prediction based on incomplete information (complicated by the sheer number of parameters involved), and often we need to do this in real time, for example for weather forecasting or pollution control. All this is further complicated by the sheer number of parameters involved leading to further problems associated with the 'curse of dimensionality' and the 'curse of small ensemble size'. The authors develop, for the first time in book form, a systematic perspective on all these issues from the standpoint of applied mathematics. The book contains enough background material from filtering, turbulence theory and numerical analysis to make the presentation self-contained and suitable for graduate courses as well as for researchers in a range of disciplines where applied mathematics is required to enlighten observations and models.

Preface
Introduction and overview: mathematical strategies for filtering turbulent systems
Fundamentals
Filtering a stochastic complex scalar: the prototype test problem
The Kalman filter for vector systems: reduced filters and a three-dimensional toy model
Continuous and discrete Fourier series and numerical discretization
Mathematical Guidelines for Filtering Turbulent Signals
Stochastic models for turbulence
Filtering turbulent signals: plentiful observations
Filtering turbulent signals: regularly spaced sparse observations
Filtering linear stochastic PDE models with instability and model error
Filtering Turbulent Nonlinear Dynamical Systems
Strategies for filtering nonlinear systems
Filtering prototype nonlinear slow-fast systems
Filtering turbulent nonlinear dynamical systems by finite ensemble methods
Filtering turbulent nonlinear dynamical systems by linear stochastic models
Stochastic parameterized extended Kalman filter for filtering turbulent signal with model error
Filtering turbulent tracers from partial observations: an exactly solvable test model
The search for efficient skilful particle filters for high dimensional turbulent dynamical systems
References
Index
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Filtering Complex Turbulent Systems

Summary

Table of Contents

Supplemental Materials