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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
This book studies methods to concretely address inverse problems. An inverse problem arises when the causes that produced a given effect must be determined or when one seeks to indirectly estimate the parameters of a physical system.
The author uses practical examples to illustrate inverse problems in physical sciences. He presents the techniques and specific methods chosen to solve inverse problems in a general domain of application, choosing to focus on a small number of methods that can be used in most applications.
This book is aimed at readers with a mathematical and scientific computing background. Despite this, it is a book with a practical perspective. The methods described are applicable, have been applied, and are often illustrated by numerical examples.
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Seitenzahl: 282
Veröffentlichungsjahr: 2016
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Table of Contents
Cover
Dedication
Title
Copyright
Preface
Book layout
Acknowledgments
PART 1: Introduction and Examples
1 Overview of Inverse Problems
1.1. Direct and inverse problems
1.2. Well-posed and ill-posed problems
2 Examples of Inverse Problems
2.1. Inverse problems in heat transfer
2.2. Inverse problems in hydrogeology
2.3. Inverse problems in seismic exploration
2.4. Medical imaging
2.5. Other examples
PART 2: Linear Inverse Problems
3 Integral Operators and Integral Equations
3.1. Definition and first properties
3.2. Discretization of integral equations
3.3. Exercises
4 Linear Least Squares Problems – Singular Value Decomposition
4.1. Mathematical properties of least squares problems
4.2. Singular value decomposition for matrices
4.3. Singular value expansion for compact operators
4.4. Applications of the SVD to least squares problems
4.5. Exercises
5 Regularization of Linear Inverse Problems
5.1. Tikhonov’s method
5.2. Applications of the SVE
5.3. Choice of the regularization parameter
5.4. Iterative methods
5.5. Exercises
PART 3: Nonlinear Inverse Problems
6 Nonlinear Inverse Problems – Generalities
6.1. The three fundamental spaces
6.2. Least squares formulation
6.3. Methods for computing the gradient – the adjoint state method
6.4. Parametrization and general organization
6.5. Exercises
7 Some Parameter Estimation Examples
7.1. Elliptic equation in one dimension
7.2. Stationary diffusion: elliptic equation in two dimensions
7.3. Ordinary differential equations
7.4. Transient diffusion: heat equation
7.5. Exercises
8 Further Information
8.1. Regularization in other norms
8.2. Statistical approach: Bayesian inversion
8.3. Other topics
Appendices
Appendix 1: Numerical Methods for Least Squares Problems
A1.1. Conditioning of the least squares problems
A1.2. Normal equations
A1.3.
QR
factorization
A1.4. SVD and numerical methods
Appendix 2: Optimization Refreshers
A2.1. Local and global algorithms
A2.2. Gradients, Hessians and optimality conditions
A2.3. Quasi-Newton methods
A2.4. Nonlinear least squares and the Gauss–Newton method
Appendix 3: Some Results from Functional Analysis
A3.1. Hilbert spaces
A3.2. Linear operators in Hilbert spaces
A3.3. Spectral decomposition of compact self-adjoint operators
Bibliography
Index
End User License Agreement
List of Tables
3 Integral Operators and Integral Equations
Table 3.1.
Geomagnetic prospection. Condition number of the matrix according to n
6 Nonlinear Inverse Problems – Generalities
Table 6.1.
Comparison of the sensitivity and of the adjoint state method for the computation of the gradients
7 Some Parameter Estimation Examples
Table 7.1.
Directional derivative computation by finite differences
Table 7.2.
Isomerization of α-pinene. Exact and estimated parameters
Guide
Cover
Table of Contents
Begin Reading
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To my wife Elisabeth, to my children David and Jonathan
Series Editor
Nikolaos Limnios
Numerical Methods for Inverse Problems
Michel Kern
First published 2016 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.
Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:
ISTE Ltd27-37 St George’s RoadLondon SW19 4EUUK
www.iste.co.uk
John Wiley & Sons, Inc.111 River StreetHoboken, NJ 07030USA
www.wiley.com
© ISTE Ltd 2016
The rights of Michel Kern to be identified as the author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988.
Library of Congress Control Number: 2016933850
British Library Cataloguing-in-Publication Data
A CIP record for this book is available from the British Library
ISBN 978-1-84821-818-5
Preface
This book studies methods to concretely address (on a computer) inverse problems. But what is an inverse problem? An inverse problem appears whenever the causes that produced a given effect must be determined, or when we seek to indirectly estimate the parameters of a physical system.
The most common example in the everyday life of many of us comes from the medical field: the medical ultrasound that informs if an unborn baby is in good health involves the solution of an inverse problem. A probe, placed on the belly of the patient, emits and receives ultrasounds. These are deflected, and reflected, by the tissues of the fetus. The sensor receives and interprets these echoes to return an image of the contours of these tissues. The image is effectively obtained in an indirect manner. We will see further examples throughout this book.
Intuitively, the observation of an effect may not be sufficient to determine its cause. If I go inside a room and I note that the temperature is (nearly) uniform, it is difficult for me to know what the distribution of temperature was 2 h earlier. It is said that the inverse problem to determine the temperature in the past is “ill-posed”. This definition contrasts with the question of determining the future evolution of the temperature, which is, in a sense that we will specify, “well-posed”. As Molière’s character Monsieur Jourdain does when he speaks prose, it is so common to solve well-posed problems that we (almost) do it without thinking.
Solving inverse problems thus requires the mastery of techniques and specific methods. This book presents some of those chosen for their very general domain of application. It focuses on a small number of methods that will be used in most applications:
– the reformulation of an inverse problem in the form of
minimization of a square error functional
. The reason for this choice is mainly practical: it makes it possible to carry out calculations at a reasonable cost;
– the
regularization
of ill-posed problems and in particular Tikhonov’s method;
– the use of the
singular value decomposition
to analyze an ill-posed problem;
– the
adjoint state method
to calculate the gradient of the functionals to minimize when these are not quadratic.
These tools will help to address many (but not all!) inverse problems that arise in practice. Two limitations should be however kept in mind. On the one hand, many inverse problems will make use of different techniques (we will mention a few of them). On the other hand, even when the presented tools can be employed, they are rarely sufficient on their own to completely analyze a complex physical application. Most often, it will be necessary to supplement these tools with a fine analysis of the particular situation to make the most of it (redundancy or not of the data, fast or slow variation of the parameters looked for, etc.).
It is common, in this type of preface, to justify the existence of the presented book! It is true that the question is legitimate (many books already exist on the subject as can be seen in the bibliography), and I do not claim any originality about the content. Nonetheless, readers might still be interested to find a book that discusses both linear and nonlinear problems. In addition, this book can be used as an introduction to the more advanced literature.
This book is aimed at readers with a rather substantial mathematical and a scientific computing background, equivalent to a masters in applied mathematics. Nevertheless, it is a book with a practical perspective. The methods described therein are applicable, and have been applied, and are often illustrated by numerical examples.
The prerequisites to approach this book are unfortunately more numerous that I would have wished. This is a consequence of the fact that the study of inverse problems calls upon many others areas of mathematics. A working knowledge of (both theoretical and numerical) linear algebra is assumed, as is a familiarity with the language of integration theory. Functional analysis, which is what linear algebra becomes when it abandons the finite dimensional setting, is ubiquitous, and the Appendices herein serve as reminders of concepts directly useful in this book. An important part of the examples comes from models of partial differential equations. Here again, the reader will benefit from a prior knowledge of analysis methods (weak formulations, Sobolev spaces) and of numerical analysis (finite element method, discretization schemes for differential equations).
Book layout
We start the book with some general remarks on inverse problems. We will introduce the fundamental concept of an ill-posed problem, which is characteristic of inverse problems.
In Chapter 2, we will give several examples of inverse problems, originating from several areas of physics.
An important source of linear inverse problems will be introduced in Chapter 3: the integral equations of the first kind. After outlining the main properties of integral operators, we will show that they lead to ill-posed problems. Finally, we will introduce discretization methods, leading to least squares problems.
The study of these problems is the subject of the subsequent two chapters. In Chapter 4, we will study their mathematical properties in a Hilbertian context: the geometric aspect, and the relationship with normal equations, as well as the questions of existence and uniqueness of the solutions. We will also introduce the fundamental tool, both for theoretical analysis and for numerical approximation, that is the singular value decomposition, first for matrices, then for operators between Hilbert spaces. Reminders regarding the numerical aspects of inverse problems can be found in Appendix 1. Techniques for solving ill-posed problems are the subject of Chapter 5, especially Tikhonov’s regularization method and spectral truncation. Tikhonov’s method will be first addressed from a variational perspective before bringing clarification with singular value decomposition. We will discuss the question of the choice of the regularization parameter and will finish by a short introduction to iterative method.
In the second part, we will discuss nonlinear problems, which are essentially problems of parameters estimation in differential or partial differential equations. In Chapter 6, we will see how to formulate identification problems in terms of minimization and explore the main difficulties that we can expect therefrom. Appendix 2 contains reminders about the basic numerical methods in optimization. Chapter 7 will address the important technique of the adjoint state to compute the functional gradient involved in least squares problems. We will see in several examples how to conduct this computation in an efficient way.
We conclude this second part by briefly introducing issues that could not be discussed in this book, giving some bibliographic hints.
We have compiled reminders regarding the numerical methods of linear algebra for least squares problems, reminders on optimization, as well as some functional analysis results and supplements on linear operators in the appendices.
Acknowledgments
My thanks go first to Professor Limnios, who suggested I write this book, from a first version of course notes that I had published on the Internet. I am grateful to him for giving me the opportunity to publish this course by providing more visibility thereto.
The contents of this book owe a lot, and this is a euphemism, to Guy Chavent. This book grew out of lecture notes that I had written for a course that had originally been taught by G. Chavent, and for which he trusted me enough to let me replace him. Guy was also my thesis supervisor and was the leader of the Inria team where I did all my career. He has been, and remains, a source of inspiration with regard to how to address a scientific problem.
I had the chance to work in the particularly stimulating environment of Inria and to meet colleagues who added great scientific qualities to endearing personalities. I am thinking especially of Jérôme Jaffré and Jean Roberts. A special mention for my colleagues in the Serena team: Hend Benameur, Nathalie Bonte, François Clément, Caroline Japhet, Vincent Martin, Martin Vohralík and Pierre Weis. Thank you for your friendship, and thank you for making our work environment a pleasant and an intellectually stimulating one.
I would like to thank all the colleagues who have told me of errors they found in previous versions of the book, the students of the Pôle Universitaire Léonard de Vinci, of Mines–ParisTech and of the École Nationale d’Ingénieurs of Tunis for listening to me and for their questions, as well as the staff of ISTE publishing for their help in seeing the book through to completion.
Michel KERN
February 2016
PART 1Introduction and Examples
2Examples of Inverse Problems
In this chapter, we present a few “concrete” examples of inverse problems, as they occur in the sciences or in engineering. This list is far from exhaustive (see the references at the end of this chapter for other applications).
Among the areas in which inverse problems play an important role, we can mention the following:
– medical imaging (ultrasound, scanners, X-rays, etc.);
– petroleum engineering (seismic prospection, magnetic methods, identification of the permeabilities in a reservoir etc.);
– hydrogeology (identification of the hydraulic permeabilities);
– chemistry (determination of reaction constants);
– radars (determination of the shape of an obstacle);
– underwater acoustics (same objective);
– quantum mechanics (determination of the potential);
– image processing (restoration of blurred images).
From a mathematical point of view, these problems are divided into two major groups:
– linear problems (echography, image processing, etc.), which amount to solving an integral equation of the first kind;
– nonlinear problems, which are mostly questions of parameter estimation in differential or partial differential equations.
2.1. Inverse problems in heat transfer
In order to determine the temperature distribution in an inhomogeneous material occupying a domain (open connected subset) , the conservation of energy is first written as
where T is the temperature, ρ is the density of the fluid, c is the specific heat, represents a heat flux and f is a volume source.
Fourier’s law then connects the heat flux density to the temperature gradient:
where K is the thermal conductivity (which may be a tensor, and depends on the position).
By eliminating we obtain the equation for the temperature, known as the heat equation, in a heterogeneous medium:
This equation must be complemented by boundary conditions on the boundary of the domain Ω and an initial condition.
The direct problem is to determine T knowing the physical coefficients ρ, c and K as well as the source of heat f. This problem is well known, both from the theoretical point of view (existence and uniqueness of the solution) and the numerical point of view. Several inverse problems can be established:
– given a measurement of the temperature at an instant determine the initial temperature. We will discuss it in example 2.1;
– given a (partial) temperature measurement, determine some of the coefficients of the equation.
Note that the first of these problems is linear, while the second is nonlinear: in fact, the application is nonlinear.
EXAMPLE 2.1 (Backward heat equation).– We consider the ideal case of a homogeneous and infinite material (in one spatial dimension to simplify). The temperature is a solution of the heat equation:
(there is no source). It is assumed that the temperature is known at some time tf, or and that the objective is to find the initial temperature .
The problem of determining Tf knowing T0 is the Cauchy problem for the heat equation. It has a unique solution, which continuously depends on the initial data. As we shall see, this is not true for the inverse problem that we consider here. Physically, this is due to the irreversible character of the thermal diffusion. It is well known that the temperature tends to become homogenized over time, and this implies that it is not possible to go back, that is to recover the previous state that can be more heterogeneous than the current state.
Because of the very simplified situation that we have chosen, we can calculate by hand the solution of the heat equation [2.4]. Using the spatial Fourier transform of equation [2.4] (we note the Fourier transform of T(x, t) keeping t as fixed), we obtain an ordinary differential equation (where this time it is k that is used as a parameter) whose solution is
Using the inverse Fourier transform, we can see that the solution at the instant tf is related to the initial condition by a convolution with the elementary solution of the heat equation:
It is well known [CAN 84] that, for any “reasonable” function T0 (continuous, bounded), the function Tf is infinitely differentiable, which mathematically expresses the irreversibility mentioned earlier.
While remaining in the Fourier domain, we can pointwise invert equation [2.5]
