Filtering Complex Turbulent Systems

Andrew J. Majda; John Harlim

doi:10.1017/CBO9781139061308

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.

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Contents

Frontmatter
pp i-iv
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Contents
pp v-viii
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Preface
pp ix-x
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1 - Introduction and overview: Mathematical strategies for filtering turbulent systems
pp 1-12
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Part I - Fundamentals
pp 13-14
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2 - Filtering a stochastic complex scalar: The prototype test problem
pp 15-29
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3 - The Kalman filter for vector systems: Reduced filters and a three-dimensional toy model
pp 30-46
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4 - Continuous and discrete Fourier series and numerical discretization
pp 47-58
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Part II - Mathematical guidelines for filtering turbulent signals
pp 59-60
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5 - Stochastic models for turbulence
pp 61-71
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6 - Filtering turbulent signals: Plentiful observations
pp 72-93
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7 - Filtering turbulent signals: Regularly spaced sparse observations
pp 94-115
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8 - Filtering linear stochastic PDE models with instability and model error
pp 116-130
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Part III - Filtering turbulent nonlinear dynamical systems
pp 131-132
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9 - Strategies for filtering nonlinear systems
pp 133-152
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10 - Filtering prototype nonlinear slow–fast systems
pp 153-191
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11 - Filtering turbulent nonlinear dynamical systems by finite ensemble methods
pp 192-213
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12 - Filtering turbulent nonlinear dynamical systems by linear stochastic models
pp 214-235
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13 - Stochastic parametrized extended Kalman filter for filtering turbulent signals with model error
pp 236-275
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14 - Filtering turbulent tracers from partial observations: An exactly solvable test model
pp 276-315
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15 - The search for efficient skillful particle filters for high-dimensional turbulent dynamical systems
pp 316-348
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References
pp 349-355
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Index
pp 356-357
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Filtering Complex Turbulent Systems

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Contents

Frontmatter
pp i-iv

Contents
pp v-viii

Preface
pp ix-x

1 - Introduction and overview: Mathematical strategies for filtering turbulent systems
pp 1-12

Part I - Fundamentals
pp 13-14

2 - Filtering a stochastic complex scalar: The prototype test problem
pp 15-29

3 - The Kalman filter for vector systems: Reduced filters and a three-dimensional toy model
pp 30-46

4 - Continuous and discrete Fourier series and numerical discretization
pp 47-58

Part II - Mathematical guidelines for filtering turbulent signals
pp 59-60

5 - Stochastic models for turbulence
pp 61-71

6 - Filtering turbulent signals: Plentiful observations
pp 72-93

7 - Filtering turbulent signals: Regularly spaced sparse observations
pp 94-115

8 - Filtering linear stochastic PDE models with instability and model error
pp 116-130

Part III - Filtering turbulent nonlinear dynamical systems
pp 131-132

9 - Strategies for filtering nonlinear systems
pp 133-152

10 - Filtering prototype nonlinear slow–fast systems
pp 153-191

11 - Filtering turbulent nonlinear dynamical systems by finite ensemble methods
pp 192-213

12 - Filtering turbulent nonlinear dynamical systems by linear stochastic models
pp 214-235

13 - Stochastic parametrized extended Kalman filter for filtering turbulent signals with model error
pp 236-275

14 - Filtering turbulent tracers from partial observations: An exactly solvable test model
pp 276-315

15 - The search for efficient skillful particle filters for high-dimensional turbulent dynamical systems
pp 316-348

References
pp 349-355

Index
pp 356-357

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