An Introduction to Identification Problems Via Functional Analysis

256 Pages·2001·8.611 MB·English

by  Alfredo Lorenzi

INVERSE AND ILL-POSED PROBLEMS SERIES
An Introduction to Identification Problems
via Functional Analysis
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INVERSE AND ILL-POSED PROBLEMS SERIES
An Introduction to
Identification
Problems via
Functional Analysis
A. Lorenzi
my BPm
UTRECHT · BOSTON · KÖLN · TOKYO
2001
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VSP Tel: [phone]
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©VSP BV 2001
First published in 2001
ISBN 90-6764-349-1
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otherwise, without the prior permission of the copyright owner.
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To my wife Adriana
and my son Luca
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Preface
The present book was originated by two courses dedicated to the Theory
of Inverse Problems that I held in the academic years 1996-97 and 1997-1998
at the Department of Mathematics "F. Enriques" of the Universitä degli
Studi di Milano. The first year the course was delivered to the students
of the Doctorate in Computational Mathematics and Operative Research
(MA. C. R. O.), while in the second year the course was dedicated to fourth-
year students and Ph.D. students in Mathematics and/or MA.C.R.O.
The text is essentially devoted to that aspect of the Theory of Inverse
Problems which is usually related to as the Identification of Parameters
(numbers, vectors, matrices, functions) appearing in differential (or inte-
grodifferential) equations. When the latter describe some physical, chemical,
geological, ... phenomena such parameters are either quite unknown or only
partially known (more often they are only qualitatively known). However,
their knowledge is in most cases essential, since they describe, e.g., the in-
trinsic properties of the material or the substance under consideration. As a
consequence, such characteristics, usually invariant, are the actual aim of in-
vestigation, in contrast to the determination of the so-called "state variable"
(i. e. of the solution to the differential equation). The latter is not often of
interest in itself, as it varies accordingly with the initial and/or boundary
conditions or with the exterior forces, currents, concentrations and so on.
The presentation of the topics contained in this book is essentially math-
ematical, but addressed to people who are not excessively familiar with mod-
ern Functional Analysis. The prerequisites are, in fact, restricted to the first
10 sections of Chapter 5 of the famous book by W. Rudin (1991).
The main aim of the first two chapters consists in showing how it is pos-
sible to solve some identification problems related to linear differential equa-
tions when having initially at our disposal only simple tools from the Theory
of Matrices and Ordinary Differential Equations. Therefore the problems il-
lustrated in Chapters 1 and 2 essentially have an introductory nature (in
some cases we restrict ourselves to setting the problem, in others the problem
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iv A. Lorenzi. An Introduction to identification Problems
is solved only under simplified, though sufficiently meaningful, hypotheses).
Basically we can say that the keystone to solving such problems is the Spec-
tral Theory in finite-dimensional spaces. In turn, this gives a motivation to
introduce the Functional Analysis for Linear Bounded Operators as well as
the corresponding Spectral Theory.
Once we have introduced the basic concepts and have developed a con-
venient theory for the questions under examination, we again analyze the
previous finite-dimensional problems as well as their infinite-dimensional
variants from the point of view of the theory of Linear Bounded Operators
in abstract Banach spaces.
Only subsequently the abstract results are applied to actual cases, after
the formal analogies have played their role of psychological persuasion. Con-
sequently, speaking about, or considering, identification problems related to
particular classes of partial differential or integrodifferential equations will
no longer be as hard as climbing sixth-grade routes.
Finally, the last conceptual leap consists in a limited (and guided) intro-
duction to the theory of Analytic Semigroups of Linear Bounded Operators
related to linear closed sectorial operators. This approach is made possible
by the fact that any such analytic semigroup admits a simple representation
by a Dunford integral. Consequently, for its treatment we will only need a
good introduction to the Theory of the Riemann Integral in Banach spaces
and the basic properties of Banach-valued holomorphic functions.
Using the Theory of Analytic Semigroups, we can solve a number of
identification problems for parabolic equations related to space dimension 1.
However strange it may seem, in the first chapter, which has an in-
troductory character, we have privileged the analysis of the existence and
uniqueness of the solutions to identification problems rather than that of
their continuous dependence on the data. Apart from the fact that this is
more natural for people approaching Inverse Problems when provided with
a mathematical culture, we note that for the majority of problems explic-
itly dealt with in this book, with the exception of those in Chapter 2, the
existence and uniqueness of the solution implies almost everywhere its con-
tinuous dependence on the data in the same spaces where the existence has
been proved. In fact, the basic tool used to solve the identification prob-
lems discussed in this book is the Principle of the Contraction Mapping
after Banach-Caccioppoli. Such a principle, as is well-known, provides, in
addition to existence and uniqueness, also the continuous dependence on
the data. However, Chapter 10 is explicitly devoted to a general enough
treatment of the continuous dependence on the data.
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Preface ν
On the other hand, in Chapter 2, where we exhibit some simple severely
ill-posed identification problems, the question of the continuous dependence
is immediately dealt with.
Summing up, we can say that the spirit pervading the entire book con-
sists in the attempt of transferring to the reader the capability of deter-
mining a suitable system of fixed-point equations, rather than committing
everything to the study of resolution techniques, even though these have
been sufficiently outlined.
I would also like to point out that part of the material contained in
Chapter 1 is an elaboration of some problems dealt with in the fine book by
A.M. Denisov Introduction to the Theory of Inverse Problems (2000).
Finally, I want to thank my students of the 9-th cycle of the Doctorate
MA. C. R. 0. Elisabetta Cordero, Francesca Lunardini, Dario Malchiodi, Si-
mona Perotto and Claudia Salani to have actively attended my course and
to have contributed with patience, but with personality and enthusiasm,
to a large part of the first version of this book. Moreover, I want to thank
Jonathan Hacon, student of the 13-th cycle of the Doctorate in Mathematics
for his help and criticism in preparing the final version of the book.
I hope that the extensive discussions with my students may have given an
original character to the material that was initially only sketched.
Milan, February 2000
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