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E-raamat: Nonlinear Data Assimilation

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This volume of Frontiers in Applied Dynamical Systems focuses on two potential solutions to the nonlinear data-assimilation problem in high-dimensional systems. Both contributions focus on so-called particle filters. The contribution of Van Leeuwen focuses on the potential of proposal densities. It discusses the issues with present-day particle filters and explorers new ideas for proposal densities to solve them, converging to particle filters that work well in systems of any dimension, closing the contribution with a high-dimensional example. The contribution by Reich and Cotter discusses a unified framework for ensemble-transform particle filters. This allows one to bridge successful Ensemble Kalman Filters with fully nonlinear particle filters, and allows a proper introduction of localisation in particle filters, which has been lacking up to now.

Nonlinear Data Assimiliatoin for High-Dimensional Systems.- Assimilating Data into Scientific Models: An Optimal Coupling Perspective.

Arvustused

In the present volume two solutions are presented to deal with high-dimensional systems, where both methods start from particle filters, i.e., from sequential Monte Carlo methods in which the samples are called particles. The volume, containing many figures and references, can be recommended to readers interested in the design and application of data assimilation algorithms. (Kurt Marti, zbMATH 1330.62004, 2016)

1 Nonlinear Data Assimilation for high-dimensional systems
1(74)
Peter Jan van Leeuwen
1 Introduction
1(7)
1.1 What is data assimilation?
1(2)
1.2 How do inverse methods fit in?
3(2)
1.3 Issues in geophysical systems and popular present-day data-assimilation methods
5(2)
1.4 Potential nonlinear data-assimilation methods for geophysical systems
7(1)
1.5 Organisation of this paper
7(1)
2 Nonlinear data-assimilation methods
8(11)
2.1 The Gibbs sampler
9(2)
2.2 Metropolis-Hastings sampling
11(2)
2.3 Hybrid Monte-Carlo Sampling
13(4)
2.4 Langevin Monte-Carlo Sampling
17(1)
2.5 Discussion and preview
18(1)
3 A simple Particle filter based on Importance Sampling
19(5)
3.1 Importance Sampling
19(1)
3.2 Basic Importance Sampling
20(4)
4 Reducing the variance in the weights
24(7)
4.1 Resampling
24(4)
4.2 The Auxiliary Particle Filter
28(2)
4.3 Localisation in particle filters
30(1)
5 Proposal densities
31(9)
5.1 Proposal densities: theory
31(2)
5.2 Moving particles at observation time
33(7)
6 Changing the model equations
40(30)
6.1 The `Optimal' proposal density
42(3)
6.2 The Implicit Particle Filter
45(3)
6.3 Variational methods as proposal densities
48(10)
6.4 The Equivalent-Weights Particle Filter
58(12)
7 Conclusions
70(5)
References
71(4)
2 Assimilating data into scientific models: An optimal coupling perspective
75
Yuan Cheng
Sebastian Reich
1 Introduction
75(3)
2 Data assimilation and Feynman-Kac formula
78(5)
3 Monte Carlo methods in path space
83(2)
3.1 Ensemble prediction and importance sampling
83(1)
3.2 Markov chain Monte Carlo (MCMC) methods
84(1)
4 McKean optimal transportation approach
85(8)
5 Linear ensemble transform methods
93(9)
5.1 Sequential Monte Carlo methods (SMCMs)
93(3)
5.2 Ensemble Kalman filter (EnKF)
96(4)
5.3 Ensemble transform particle filter (ETPF)
100(2)
5.4 Quasi-Monte Carlo (QMC) convergence
102(1)
6 Spatially extended dynamical systems and localization
102(4)
7 Applications
106(6)
7.1 Lorenz-63 model
107(2)
7.2 Lorenz-96 model
109(3)
8 Historical comments
112(1)
9 Summary and Outlook
113
References
115
Peter Jan van Leeuwen is a Professor of Data Assimilation at the University of Reading.  His research interests include nonlinear data assimilation, geophysical fluid dynamics, interaction thermohaline and wind-driven ocean circulation, and perturbation theory.

Sebastian Reich is a Professor in the Department of Numerical Mathematics at Universität Potsdam. His research interests include uncertainty quantification, geophysical fluid dynamics, molecular dynamics, and geometric integration.
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