Andrew J. Majda is the Morse Professor of Arts and Sciences at the Courant Institute of New York University. John Harlim is Assistant Professor in the Department of Mathematics at North Carolina State University.

• Preface; 1. Introduction and overview: mathematical strategies for filtering turbulent systems; Part I. Fundamentals: 2. Filtering a stochastic complex scalar: the prototype test problem; 3. The Kalman filter for vector systems: reduced filters and a three-dimensional toy model; 4. Continuous and discrete Fourier series and numerical discretization; Part II. Mathematical Guidelines for Filtering Turbulent Signals: 5. Stochastic models for turbulence; 6. Filtering turbulent signals: plentiful observations; 7. Filtering turbulent signals: regularly spaced sparse observations; 8. Filtering linear stochastic PDE models with instability and model error; Part III. Filtering Turbulent Nonlinear Dynamical Systems: 9. Strategies for filtering nonlinear systems; 10. Filtering prototype nonlinear slow-fast systems; 11. Filtering turbulent nonlinear dynamical systems by finite ensemble methods; 12. Filtering turbulent nonlinear dynamical systems by linear stochastic models; 13. Stochastic parameterized extended Kalman filter for filtering turbulent signal with model error; 14. Filtering turbulent tracers from partial observations: an exactly solvable test model; 15. The search for efficient skilful particle filters for high dimensional turbulent dynamical systems; References; Index.

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