Stable Approximate Evaluation of Unbounded Operators

133 Pages·2007·2.195 MB·English

by  Charles W. Groetsch (auth.)

Lecture Notes in Mathematics (cid:49)(cid:56)(cid:57)(cid:52)
Editors:
J.-M.Morel,Cachan
F.Takens,Groningen
B.Teissier,Paris
Charles W. Groetsch
Stable Approximate
Evaluation of
Unbounded Operators
(cid:65)(cid:66)(cid:67)
Author
CharlesW.Groetsch
TheTraubertChair
SchoolofScienceandMathematics
TheCitadel
Charleston,SC(cid:50)(cid:57)(cid:52)(cid:48)(cid:57)
USA
e-mail:[email protected]
LibraryofCongressControlNumber:2006931917
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In Memory of Joaquin Bustoz, Jr.
1939-2003
Preface
This monograph is a study of an aspect of operator approximation theory
that emerges from the theory of linear inverse problems in the mathematical
sciences.Suchinverseproblemsareoftenmodeledbyoperatorequationsofthe
firstkindinvolvingacompactlinearoperatordefinedonaHilbertspace.The
conventional solution operator for the inverse problem is the Moore-Penrose
generalizedinverseofthemodeloperator.Exceptintheunusualcasewhenthe
modeloperatorhasfiniterank,theMoore-Penroseinverseisadenselydefined,
closed,unboundedoperator.Thereforeboundedperturbationsinthedatacan
be amplified without bound by the solution operator. Indeed, it is a common
experience of those who have dealt with such inverse problems to find that
low amplitude noise in the data expresses itself as high amplitude oscillations
in the computed solution. The successful treatment of these inverse problems
therefore requires two ingredients: an approximation of the Moore-Penrose
inversecoupledwithastabilizationtechniquetodampenspuriousoscillations
in the approximate solution.
Hereweconsiderstabilizedevaluationofanunboundedoperatorasaprob-
leminitsownrightinoperatorapproximationtheory.Bystabilizedevaluation
we mean that the value of an unbounded operator at some vector in its do-
mainisapproximatedbyapplyingboundedlinearoperatorstoanapproximate
data vector that is not necessarily in the domain of the original unbounded
operator. Questions of convergence and orders of approximation will be of
foremost concern. A unifying thread for the discussion is a classical theorem
of von Neumann on certain bounded “resolvents” of closed densely defined
unbounded operators. This result is the bridge that allows passage from the
unbounded operator to a class of approximating bounded operators. When
von Neumann’s theorem is combined with the spectral theorem for bounded
self-adjoint operators a general scheme for stabilized evaluation of the un-
bounded operator results. Particular cases of the general scheme, notably the
Tikhonov-Morozov method and its variants, are studied in some detail and
finite-dimensional realizations are dealt with in the final chapter.
VIII Preface
The key idea of von Neumann’s proof involves regarding the graph of the
operator as a subspace embedded in a product Hilbert space (in this sense
the proof is truly Cartesian). We also show that this notion, combined with
von Neumann’s alternating projection theorem, applied in the product space,
can be used to give an alternate non-spectral proof of one of the best known
operator stabilization methods.
Ourintentisforthemonographtobereasonablyself-contained.Webegin
withafairlyinformalintroductorychapterinwhichanumberofmodelinverse
problems leading to the evaluation of unbounded operators are introduced.
The next chapter fills in background material from functional analysis and
operator theory that is helpful in the sequel. We hope that this approach
will make the monograph a useful source of collateral reading for students
in graduate courses in functional analysis and related courses in analysis and
applied mathematics.
Much of the work that is reported here was originally carried out in col-
laboration with my friends Otmar Scherzer of the University of Innsbruck,
Austria and Martin Hanke-Bourgeois of the University of Mainz, Germany.
WithbothOtmarandMartinIhadthehappyexperienceofopenandfriendly
collaborations in which my benefits exceeded my contributions.
While writing these notes I learned of the tragic death of my earliest
colleague and coauthor, Joaquin Bustoz, Jr., of Arizona State University.
Besideshismanyresearchpapersonsummabilitytheoryandspecialfunctions,
Joaquin’s important andlastingcontributions tothemathematical education
of disadvantaged youth made his passing a great loss to the profession, to say
nothing of the personal sense of loss felt by his friends and colleagues. This
monograph is fondly dedicated to Joaquin’s memory.
I am grateful to the Charles Phelps Taft Research Center for providing
support, in the form of a Faculty Fellowship, during the preparation of this
work.
Seabrook Island, S.C. Charles Groetsch
May, 2006
Contents
1 Some Problems Leading to Unbounded Operators ......... 1
1.1 Introduction ............................................ 1
1.2 Some Unbounded Operators .............................. 2
1.2.1 Differentiation and Multiplication ................... 3
1.2.2 Dirichlet-to-Neumann Map ......................... 4
1.2.3 Potential Theory .................................. 5
1.2.4 Moment Theory................................... 8
1.3 Unbounded Operators and the Heat Equation............... 9
1.3.1 Time Reversal .................................... 10
1.3.2 Source Identification ............................... 11
1.3.3 Diffusivity from Temperature History ................ 12
1.3.4 The Steady State.................................. 13
1.3.5 Surface Phenomena................................ 13
1.4 Overview............................................... 16
1.5 Notes .................................................. 17
2 Hilbert Space Background ................................. 19
2.1 Hilbert Space ........................................... 19
2.2 Weak Convergence....................................... 22
2.3 Bounded Linear Operators................................ 24
2.4 Unbounded Operators.................................... 30
2.5 Pseudo-inversion ........................................ 41
2.6 Optimization ........................................... 43
2.7 Notes .................................................. 51
3 A General Approach to Stabilization ...................... 53
3.1 A General Method....................................... 53
3.2 Some Cases............................................. 58
3.2.1 The Tikhonov-Morozov Method ..................... 58
3.2.2 The Iterated Tikhonov-Morozov Method ............. 63
3.2.3 An Interpolation-Based Method ..................... 65
X Contents
3.2.4 A Method Suggested by Dynamical Systems .......... 70
3.3 Notes .................................................. 73
4 The Tikhonov-Morozov Method ........................... 77
4.1 The Tikhonov-Morozov Method ........................... 77
4.2 The Discrepancy Criterion................................ 81
4.3 Iterated Tikhonov-Morozov Method ....................... 86
4.4 The Nonstationary Method ............................... 89
4.5 Notes .................................................. 97
5 Finite-Dimensional Approximations........................101
5.1 Stabilization by Projection ...............................101
5.2 Finite Elements .........................................111
5.2.1 A Finite Element Discrepancy Criterion..............116
5.3 Notes ..................................................119
References.....................................................121
Index..........................................................125
1
Some Problems Leading to Unbounded
Operators
His majesty then turned to me, and requested me to explain the reason
why such great effects should proceed from so small a cause ...
The Adventures of Hajji Baba of Isfahan
J. Morier
1.1 Introduction
Unbounded operators can transform arbitrarily small vectors into arbitrar-
ily large vectors – a phenomenon known as instability. Stabilization methods
strive to approximate a value of an unbounded operator by applying a family
of bounded operators to rough approximate data that do not necessarily lie
within the domain of the unbounded operator. This monograph is a math-
ematical study of stabilization techniques for the evaluation of unbounded
operatorsactingbetweenHilbertspaces.Thegoalofthestudyistocontribute
to the theoretical basis of a wide-ranging methodology for inverse problems,
datasmoothingandimageanalysis.Myintentisentirelytheoretical.Idonot
propose to delve into the details of these applications, but rather my aim is
to use the application areas as motivators for the study of specific topics in
the theory of operator approximations.
Stabilization problems inevitably arise in the solution of inverse problems
that are phrased in infinite-dimensional function spaces. The direct versions
of these problems typically involve a highly smoothing operator and conse-
quentlytheinversionprocessisusuallyhighlyill-posed.Thisill-posedproblem
can be viewed on a theoretical level (and often on a quite practical level) as
evaluatinganunboundedoperatoronsomedataspace(therangeofthedirect
operator). The main difficulty arises when inaccuracies (arising in the physi-
calsetting frommeasurementerrors)carrytheobservabledata outsideofthe
theoretical data space. The unbounded nature of the solution operator can
thengiverisetospectacularinstabilitiesintheapproximatesolution.General
stabilization techniques, the subject of this monograph, involve insertion of
2 1 Some Problems Leading to Unbounded Operators
the given data into the data space of the solution operator and subsequent
approximate evaluation of the solution operator. Of course these approxima-
tions must be carried out very carefully in order to preserve stability. A key
point in this process is the choice of parameters defining the approximate
evaluation in such a way as to control the instabilities.
It is hoped that this monograph which explores these ideas, and is
expressed in rigorous operator theory, will be interesting to mathematicians
and will also be potentially useful to those in the scientific community who
confront unstable ill-posed problems in their work. Our intent is also to pro-
vide a source for collateral study for students in courses in applied functional
analysis and related courses. With this in mind the next chapter provides
brief details of some of the operator theory that will be used in the sequel.
Before proceeding to this background material we give a semi-formal intro-
duction to some problems from application areas that lead to the evaluation
of unbounded operators.
We remind the reader that a linear operator T : X → Y from a normed
linear space X into a normed linear space Y is called bounded if the number
(cid:2)Tx(cid:2)
(cid:2)T(cid:2)=sup
x(cid:1)=0 (cid:2)x(cid:2)
is finite (unless further specification is necessary, (cid:2)·(cid:2) will always denote the
norminanappropriatespace).Forlinearoperatorstheconceptsofcontinuity
atapoint,uniformcontinuityandboundednesscoincide.IfT isanunbounded
linear operator, then there is a sequence of unit vectors {z } in the domain
n
of T satisfying (cid:2)Tz (cid:2)→∞ as n→∞. For this reason we frequently say that
n
for an unbounded linear operator T the mapping z → Tz is unstable. That
is, for linear operators we treat the terms stable, bounded and continuous as
synonymous.
1.2 Some Unbounded Operators
In elementary calculus the differentiation process is regarded as basic and
benign. But differentiation is Janus-faced and its darker side is all too fa-
miliar to scientists who must differentiate functions that are given empir-
ically. This sinister face of differentiation is instability. In operator terms
this instability means that the linear operator defined by differentiation is
unbounded on func√tion spaces with conventional norms. For example, the
functions φ (t) = 2sinnπt are unit vectors in the space L2[0,1], yet the
n
L2-norm of the derivatives, (cid:2)φ(cid:2) (cid:2)=πn, is unbounded. This means, for exam-
n
ple, that the differentiation operator defined in the subspace of differentiable
functions in L2[0,1], when considered as a operator taking values in L2[0,1],
is unbounded.