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2015 | Buch

Data Assimilation

A Mathematical Introduction

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This book provides a systematic treatment of the mathematical underpinnings of work in data assimilation, covering both theoretical and computational approaches. Specifically the authors develop a unified mathematical framework in which a Bayesian formulation of the problem provides the bedrock for the derivation, development and analysis of algorithms; the many examples used in the text, together with the algorithms which are introduced and discussed, are all illustrated by the MATLAB software detailed in the book and made freely available online.

The book is organized into nine chapters: the first contains a brief introduction to the mathematical tools around which the material is organized; the next four are concerned with discrete time dynamical systems and discrete time data; the last four are concerned with continuous time dynamical systems and continuous time data and are organized analogously to the corresponding discrete time chapters.

This book is aimed at mathematical researchers interested in a systematic development of this interdisciplinary field, and at researchers from the geosciences, and a variety of other scientific fields, who use tools from data assimilation to combine data with time-dependent models. The numerous examples and illustrations make understanding of the theoretical underpinnings of data assimilation accessible. Furthermore, the examples, exercises and MATLAB software, make the book suitable for students in applied math

ematics, either through a lecture course, or through self-study.

Inhaltsverzeichnis

Frontmatter
Chapter 1. Mathematical Background
Abstract
The purpose of this chapter is to provide a brief overview of the key mathematical ways of thinking that underpin our presentation of the subject of data assimilation. In particular, we touch on the subjects of probability, dynamical systems, probability metrics, and dynamical systems for probability measures, in Sections 1.1, 1.2, 1.3, and 1.4 respectively. Our treatment is necessarily terse and very selective, and the bibliography section 1.5 provides references to the literature.
Kody Law, Andrew Stuart, Konstantinos Zygalakis
Chapter 2. Discrete Time: Formulation
Abstract
In this chapter, we introduce the mathematical framework for discrete-time data assimilation. Section 2.1 describes the mathematical models we use for the underlying signal, which we wish to recover, and for the data, which we use for the recovery.
Kody Law, Andrew Stuart, Konstantinos Zygalakis
Chapter 3. Discrete Time: Smoothing Algorithms
Abstract
The formulation of the data-assimilation problem described in the previous chapter is probabilistic, and its computational resolution requires the probing of a posterior probability distribution on signal-given data.
Kody Law, Andrew Stuart, Konstantinos Zygalakis
Chapter 4. Discrete Time: Filtering Algorithms
Abstract
In this chapter, we describe various algorithms for the filtering problem. Recall from Section 2.​4 that filtering refers to the sequential update of the probability distribution on the state given the data, as data is acquired, and that \(Y _{j} =\{ y_{\ell}\}_{\ell=1}^{j}\) denotes the data accumulated up to time j. The filtering update from time j to time j + 1 may be broken into two steps: prediction , which is based on the equation for the state evolution, using the Markov kernel for the stochastic or deterministic dynamical system that maps \(\mathbb{P}(v_{j}\vert Y _{j})\) into \(\mathbb{P}(v_{j+1}\vert Y _{j})\); and analysis , which incorporates data via Bayes’s formula and maps \(\mathbb{P}(v_{j+1}\vert Y _{j})\) into \(\mathbb{P}(v_{j+1}\vert Y _{j+1})\). All but one of the algorithms we study (the optimal proposal version of the particle filter) will also reflect these two steps.
Kody Law, Andrew Stuart, Konstantinos Zygalakis
Chapter 5. Discrete Time: MATLAB Programs
Abstract
This chapter is dedicated to illustrating the examples, theory, and algorithms, as presented in the previous chapters, through a few short and easy-to-follow MATLAB programs. These programs are provided for two reasons: (i) For some readers, they will form the best route by which to appreciate the details of the examples, theory, and algorithms we describe; (ii) for other readers, they will be a useful starting point to develop their own codes. While ours are not necessarily the optimal implementations of the algorithms discussed in these notes, they have been structured to be simple to understand, to modify, and to extend. In particular, the code may be readily extended to solve problems more complex than those described in Examples 2.​12.​7, which we will use for most of our illustrations.
Kody Law, Andrew Stuart, Konstantinos Zygalakis
Chapter 6. Continuous Time: Formulation
Abstract
In this chapter, and in all subsequent chapters, we consider continuous-time signal dynamics and continuous-time data. This takes us into a part of the subject that is potentially rather technical, a fact that can obscure the structure manifest in the continuous-time formulation. In order to avoid technicalities that can obfuscate the derivations, and in order to create space to highlight the structure present in the continuous-time models, we proceed as follows: we adopt an approach whereby the derivation of many key equations proceeds in a nonrigorous fashion from the discrete-time setup, by formally taking the limit \(\tau \rightarrow 0\), where \(\tau\) is the time increment between observations. We then concentrate on studying the properties of the resulting limiting continuous-time problems, and algorithms for them.
Kody Law, Andrew Stuart, Konstantinos Zygalakis
Chapter 7. Continuous Time: Smoothing Algorithms
Abstract
In this chapter, we describe various algorithms for the smoothing problem in continuous time. We begin, in Section 7.1, by describing the Kalman–Bucy smoother , which applies in the case of linear dynamics when the initial conditions and the observational noise are Gaussian; the explicit Kalman–Bucy formulas are useful for the building of intuition. In Section 7.2, we discuss MCMC methods to sample from the smoothing distributions of interest. However, as in the discrete-time case, sampling the posterior can be prohibitively expensive. For this reason, it is of interest to identify the point that maximizes probability, using techniques from optimization, rather than explore the entire probability distribution—the variational method. This optimization approach is discussed in Section 7.3. Section 7.4 is devoted to numerical illustrations of the methods introduced in the previous sections. The chapter concludes with bibliographic notes in Section 7.5 and exercises in Section 7.6.
Kody Law, Andrew Stuart, Konstantinos Zygalakis
Chapter 8. Continuous Time: Filtering Algorithms
Abstract
In this chapter, we describe various algorithms for determination of the filtering distribution μ t in continuous time. We begin in Section 8.1 with the Kalman–Bucy filter, which provides an exact algorithm for linear problems. Since the filtering distribution is Gaussian in this case, the distribution is entirely characterized by the mean and covariance; the algorithm comprises a system of differential equations for the mean and the covariance. In Section 8.2, we discuss the approximate Gaussian methods introduced in Section 4.​2 in the discrete-time setting. Similarly to the case of the Kalman–Bucy filter, we again obtain a differential equation for the mean; for the extended Kalman (ExKF) filter, we also obtain an equation for the covariance, and for the ensemble Kalman filter (EnKF), we have a system of differential equations coupled through their empirical mean and covariance. In Section 8.3, we discuss how the particle filter methodology introduced in Section 4.​3 extends to the continuous case, while in Section 8.4, we study the long-time behavior of some of the filtering algorithms discussed in the previous sections. Finally, in Section 8.5, we present some numerical illustrations and conclude with bibliographic notes and exercises in Sections 8.6 and 8.7 respectively.
Kody Law, Andrew Stuart, Konstantinos Zygalakis
Chapter 9. Continuous Time: MATLAB Programs
Abstract
This chapter is dedicated to illustrating the examples, theory, and algorithms presented in the preceding three chapters through a few short and easy-to-follow MATLAB programs. We have followed the same principles as in Chapter 9, and again the code may be readily extended to solve problems more complex than those described in Examples 6.​46.​8, which will be used for most of our illustrations.
Kody Law, Andrew Stuart, Konstantinos Zygalakis
Backmatter
Metadaten
Titel
Data Assimilation
verfasst von
Kody Law
Andrew Stuart
Konstantinos Zygalakis
Copyright-Jahr
2015
Electronic ISBN
978-3-319-20325-6
Print ISBN
978-3-319-20324-9
DOI
https://doi.org/10.1007/978-3-319-20325-6