模型参数估计的反问题理论与方法(英文影印版)

    书籍
    数学书籍
Prompted by recent developments in inverse theory, Inverse Problem Theory and Methods for Model Parameter Estimation is a completely rewritten version of a 1987 book by the same author. In this version there are many algorithmic details for Monte Carlo methods, leastsquares discrete problems, and least-squares problems involving functions. In addition, some notions are clarified, the role of optimization techniques is underplayed, and Monte Carlo methods are taken much more seriously. The first part of the book deals exclusively with discrete inverse problems with afinite number of parameters, while the second part of the book deals with general inverse problems. ...

Preface
1　 The General Discrete Inverse Problem
　1.1　 Model Space and Data Space
　1.2　 States of Information
　1.3　 Forward Problem
　1.4　 Measurements and A Priori Information
　1.5　 Defining the Solution of the Inverse Problem
　1.6　 Using the Solution of the Inverse Problem
2　 Monte Carlo Methods
　2.1　 Introduction
　2.2　 The Movie Strategy for Inverse Problems
　2.3　 Sampling Methods
　2.4　 Monte Carlo Solution to Inverse Problems
2.5　 Simulated Annealing
3　 The Least-Squares Criterion
3.1　 Preamble: The Mathematics of Linear Spaces
3.2　 The Least-Squares Problem
3.3　 Estimating Posterior Uncertainties
3.4　 Least-Squares Gradient and Hessian
4　 Least-Absolute-Values Criterion and Minimax Criterion

　　Physical theories allow us to make predictions: given a complete description of a physical system, we can predict the outcome of some measurements. This problem of predicting the result of measurements is called the modelization problem, the simulation problem, or the forward problem. The inverse problem consists of using the actual result of some measurements to infer the values of the parameters that characterize the system. .
　　While the forward problem has (in deterministic physics) a unique solution, the inverse problem does not. As an example, consider measurements of the gravity field around a planet: given the distribution of mass inside the planet, we can uniquely predict the values of the gravity field around the planet (forward problem), but there are different distributions of mass that give exactly the same gravity field in the space outside the planet. Therefore, the inverse problem -- of inferring the mass distribution from observations of the gravity field -- has multiple solutions (in fact, an infinite number).
　　Because of this, in the inverse problem, one needs to make explicit any available a priori information on the model parameters. One also needs to be careful in the representation of the data uncertainties.
　　The most general (and simple) theory is obtained when using a probabilistic point of view, where the a priori information on the model parameters is represented by a probability distribution over the 'model space.' The theory developed here explains how this a priori probability distribution is transformed into the a posteriori probability distribution, by incorporating a physical theory (relating the model parameters to some observable parameters) and the actual result of the observations (with their uncertainties).
　　To develop the theory, we shall need to examine the different types of parameters that appear in physics and to be able to understand what a total absence of a priori information on a given parameter may mean.
　　Although the notion of the inverse problem could be based on conditional probabilities and Bayes's theorem, I choose to introduce a more general notion, that of the 'combination of states of information,' that is, in principle, free from the special difficulties appearing in the use of conditional probability densities (like the well-known Borel paradox).
　　The general theory has a simple (probabilistic) formulation and applies to any kind of inverse problem, including linear as well as strongly nonlinear problems. Except for very simple examples, the probabilistic formulation of the inverse problem requires a resolution in terms of 'samples' of the a posteriori probability distribution in the model space. This, in particular, means that the solution of an inverse problem is not a model but a collection of models (that are consistent with both the data and the a priori information). This is why Monte Carlo (i.e., random) techniques are examined in this text. With the increasing availability of computer power, Monte Carlo techniques are being increasingly used. ..
　　Some special problems, where nonlinearities are weak, can be solved using special, very efficient techniques that do not differ essentially from those used, for instance, by Laplace in 1799, who introduced the 'least-absolute-values' and the 'minimax' criteria for obtaining the best solution, or by Legendre in 1801 and Gauss in 1809, who introduced the 'least-squares' criterion.
　　The first part of this book deals exclusively with discrete inverse problems with a finite number of parameters. Some real problems are naturally discrete, while others contain functions of a continuous variable and can be discretized if the functions under consideration are smooth enough compared to the sampling length, or if the functions can conveniently be described by their development on a truncated basis. The advantage of a discretized point of view for problems involving functions is that the mathematics is easier. The disadvantage is that some simplifications arising in a general approach can be hidden when using a discrete formulation. (Discretizing the forward problem and setting a discrete inverse problem is not always equivalent to setting a general inverse problem and discretizing for the practical computations.)
　　The second part of the book deals with general inverse problems, which may contain such functions as data or unknowns. As this general approach contains the discrete case in particular, the separation into two parts corresponds only to a didactical purpose.
　　Although this book contains a lot of mathematics, it is not a mathematical book. It tries to explain how a method of acquisition of information can be applied to the actual world, and many of the arguments are heuristic.
　　This book is an entirely rewritten version of a book I published long ago (Tarantola, 1987). Developments in inverse theory in recent years suggest that a new text be proposed, but that it should be organized in essentially the same way as my previous book. In this new version, I have clarified some notions, have underplayed the role of optimization techniques, and have taken Monte Carlo methods much more seriously.
　　I am very indebted to my colleagues (Bartolome Coil, Georges Jobert, Klaus Mosegaard, Miguel Bosch, Guillaume Evrard, John Scales, Christophe Barnes, Frrdrric Parrenin, and Bernard Valette) for illuminating discussions. I am also grateful to my collaborators at what was the Tomography Group at the Institut de Physique du Globe de Paris. ...
　　Albert Tarantola
　　Paris, June 2004