Table Of Content1792
Lecture Notes in Mathematics
Editors:
J.–M.Morel,Cachan
F.Takens,Groningen
B.Teissier,Paris
3
Berlin
Heidelberg
NewYork
Barcelona
HongKong
London
Milan
Paris
Tokyo
Dang Dinh Ang
Rudolf Gorenflo
Vy Khoi Le
Dang Duc Trong
Moment Theory and
Some Inverse Problems
in Potential Theory
and Heat Conduction
1 3
Authors
DangDinhANG VyKhoiLE
DepartmentofMathematics DepartmentofMathematics
andInformatics andStatistics
HoChiMinhCityNationalUniversity UniversityofMissouri-Rolla
227NguyenVanCu,Q5 Rolla,Missouri65401
HoChiMinhCity USA
VietNam e-mail:[email protected]
e-mail:[email protected]
RudolfGORENFLO DangDucTRONG
DepartmentofMathematics DepartmentofMathematics
andInformatics andInformatics
FreeUniversityofBerlin HoChiMinhCityNationalUniversity
Arnimallee3 227NguyenVanCu,Q5
14195Berlin HoChiMinhCity
Germany VietNam
e-mail:gorenfl[email protected] e-mail:
http://www.fracalmo.org [email protected]
Cataloging-in-PublicationDataappliedfor.
Die Deutsche Bibliothek - CIP-Einheitsaufnahme
Moment theory and some inverse problems in potential theory and heat
conduction / Dang Dinh Ang .... - Berlin ; Heidelberg ; New York ; Barcelona
; Hong Kong ; London ; Milan ; Paris ; Tokyo : Springer, 2002
(Lecture notes in mathematics ; 1792)
ISBN 3-540-44006-2
MathematicsSubjectClassification(2000):
30E05,30E10,31A35,31B20,35R25,35R30,44A60,45Q05,47A52
ISSN0075-8434
ISBN3-540-44006-2Springer-VerlagBerlinHeidelbergNewYork
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Foreword
In recent decades, the theory of inverse and ill-posed problems has im-
pressivelydevelopedintoahighlyrespectablebranchofAppliedMathematics
and has had stimulating effects on Numerical Analysis, Functional Analysis,
Complexity Theory, and other fields. The basic problem is to draw useful
information from noise contaminated physical measurements, where in the
case of ill-posedness, naive methods of evaluation lead to intolerable am-
plification of the noise. Usually, one is looking for a function (defined on a
suitable domain) that is close to the true function assumed to exist as un-
derlying the situation or process the measurements are taken from, and the
above mentioned gross amplification of noise (mathematically often caused
by the attempt to invert an operator whose inverse is unbounded) makes the
numericalresultssoobtaineduseless,these”results”hidingthetruesolution
under large amplitude high frequency oscillations.
There is an ever growing literature on ways out of this dilemma. The
way out is to suppress unwanted noise, thereby avoiding excessive suppres-
sion of relevant information. Various methods of ”regularization” have been
developed for this purpose, all, in principle, using extra information on
the unknown function. This can be in the form of general assumptions on
”smoothness”, an idea underlying, e.g., the method developed by Tikhonov
andPhillips(minimizationofaquadraticfunctionalcontaininghigherderiva-
tives in an attempt to reproduce the measured data) and various modifica-
tionsofthismethod.Anotherefficientmethodistheso-called”regularization
by discretization” method where one has to find a kind of balance between
the fineness of discretization and its tendency to amplify noise. Yet another
method,theso-called”descriptiveregularization”method,consistsinexploit-
ingaprioriknowncharacteristicsoftheunknownfunction,suchasregionsof
nonegativity, or monotonicity, or convexity that can be used in a scheme of
linear or nonlinear fitting to the measured data, fitting optimal with respect
toappropriateconstraints.Manyramificationsandcombinationsoftheseand
othermethodshavebeenanalyzedtheoreticallyandusedinnumericalcalcu-
lations.Ourmonographdealswiththemethodcalledthe”momentmethod”.
The moments considered here are of the form
(cid:1)
µ = u(x)dσ , n=1,2,3,...,
n n
Ω
where Ω is a domain in Rk, dσ is, either a Dirac measure, n ∈ N, or a
n
measure absolutely continuous with respect to the Lebesgue measure, i.e.,
dσ =g (x)dx, n∈N,
n n
g (x) being Lebesgue integrable on Ω. The idea of the moment method is to
n
reconstruct an unknown function u(x) from a given set (µn)n∈I, I ⊂ N, of
the moments of u(x). Then the problem arises as to whether a knowledge of
momentsofu(x)uniquelydeterminesthisfunction.Forthemomentproblems
considered in this monograph, unless stated otherwise, the knowledge of the
vi
complete sequence of moments of u(x) uniquely determines the function.
In practice, one has available only a finite set µ1,...,µm of moments, and
furthermoretheseareusuallycontaminatedwithnoise,thereasonbeingthat
theyareresultsofexperimentalmeasurements.Thequestionthenis:Towhat
extent, can the true function u(x) be recovered from the finite set (µi)1≤i≤m
of moments? Note that in the latter situation, the question of existence of a
solutionuplaysaminorrole.Themomentsbeingonlyapproximatelyknown,
the problem is reduced to one of ”regularization”, namely, to the problem of
fitting the function u(x) as closely as possible to the available data, that is,
to the given approximate values of the moments, u(x) being assumed to lie
in a nice function space and to obey a known or stipulated restriction to the
size of an appropriate functional. In our theory of regularization, the index
m, i.e., the number of the given moment values mentioned above, will play
theroleoftheregularizationparameter.Inillustrationofthetheory,weshall
study several concrete cases, discussing inverse problems of function theory,
potentialtheory,heatconductionandgravimetry.Wewillmakeessentialuse
of analyticity or harmonicity of the functions involved, and so the theory
of analytic functions and harmonic functions will play a decisive role in our
investigations.Wehopethatthismonograph,whichisafruitofseveralyears
of joint efforts, will stimulate further research in theoretical as well as in
practical applications.
It is our pleasure to acknowledge with gratitude the valuable assistance
of several researchers with whom we could discuss aspects of the theory of
moments,eitherafterpresentationinconferencesandseminarsorinpersonal
exchangeofknowledgeandopinions.Specialthanksareduetoourcolleagues
JohannBaumeister,BerndHofmann,SergioVessella,LotharvonWolfersdorf
andMasahiroYamamoto.Theyhavestudiedthewholemanuscriptandtheir
detailed constructive-critical remarks have helped us much in improving it.
Our thanks are also due to the anonymous referees for their valuable sugges-
tions. Last not least, we highly appreciate the supports granted by Deutsche
ForschungsgemeinschaftinBonnwhichmadepossibleseveralmutualresearch
visits, furthermore the supports given by the Research Commission of Free
University of Berlin, Ho Chi Minh City Mathematical Society, Ho Chi Minh
City National University, and the Vietnam Program of Basic Research in
the Natural Sciences. Last not least we are grateful to Ms. Julia Loutchko
for her help in the final corrections and preparations of the manuscript for
publishing.
Dang Dinh Ang, Rudolf Gorenflo,
Vy Khoi Le and Dang Duc Trong
Berlin, Ho Chi Minh City, Rolla-Missouri: March
2002
Table of Contents
Introduction.................................................. 1
1 Mathematical preliminaries .............................. 5
1.1 Banach spaces ......................................... 5
1.2 Hilbert spaces.......................................... 6
1.3 Some useful function spaces.............................. 8
1.3.1 Spaces of continuous functions ..................... 8
1.3.2 Spaces of integrable functions ...................... 9
1.3.3 Sobolev spaces ................................... 10
1.4 Analytic functions and harmonic functions................. 12
1.5 Fourier transform and Laplace transform .................. 14
2 Regularization of moment problems by truncated
expansion and by the Tikhonov method .................. 17
2.1 Method of truncated expansion........................... 19
2.1.1 A construction of regularized solutions .............. 19
2.1.2 Convergenceofregularizedsolutionsanderrorestimates 22
2.1.3 Error estimates using eigenvalues of the Laplacian .... 27
2.2 Method of Tikhonov .................................... 30
2.2.1 Case 1: exact solutions in L2(Ω) ................... 30
2.2.2 Case 2: exact solutions in Lα∗(Ω), 1<α∗ <∞ ...... 36
2.2.3 Case 3: exact solutions in H1(Ω) ................... 42
2.3 Notes and remarks...................................... 45
3 Backus-Gilbert regularization of a moment problem ..... 51
3.1 Introduction ........................................... 51
3.2 Backus-Gilbert solutions and their stability ................ 54
3.2.1 Definition of the Backus-Gilbert solutions ........... 54
3.2.2 Stability of the Backus-Gilbert solutions ............ 59
3.3 Regularization via Backus-Gilbert solutions ............... 63
3.3.1 Definitions and notations.......................... 64
3.3.2 Main results .................................... 73
4 The Hausdorff moment problem: regularization and error
estimates ................................................. 83
4.1 Finite moment approximation of (4.1)..................... 84
4.1.1 Proof of Theorem 4.1.............................. 88
viii Table of Contents
4.1.2 Proof of Theorem 4.2.............................. 89
4.2 A moment problem from Laplace transform................ 92
4.3 Notes and remarks...................................... 94
5 Analytic functions: reconstruction and Sinc approximations 99
5.1 Reconstruction of functions in H2(U): approximation by
polynomials............................................ 99
5.2 Reconstruction of an analytic function: a problem of optimal
recovery............................................... 106
5.3 Cardinal series representation and approximation:
reformulation of moment problems........................ 120
5.3.1 Two-dimensional Sinc theory ...................... 120
5.3.2 Approximation theorems .......................... 123
6 Regularization of some inverse problems in potential
theory .................................................... 131
6.1 Analyticity of harmonic functions......................... 131
6.2 Cauchy’s problem for the Laplace equation ................ 133
6.3 Surface temperature determination from borehole
measurements (steady case).............................. 145
7 Regularization of some inverse problems in heat conduction147
7.1 The backward heat equation ............................ 147
7.2 Surface temperature determination from borehole
measurements: a two-dimensional problem ................. 155
7.3 An inverse two-dimensional Stefan problem: identification of
boundary values........................................ 164
7.4 Notes and remarks...................................... 169
8 Epilogue.................................................. 171
References.................................................... 175
Index......................................................... 181
Introduction
Amomentproblemiseitheraproblemoffindingafunctionuonadomain
Ω of Rd, d≥1, satisfying a sequence of equations of the form
(cid:1)
udσ =µ (0.1)
n n
Ω
where(dσ )isagivensequenceofmeasuresonΩand(µ )isagivensequence
n n
of numbers, or a problem of finding a measure dσ on Ω satisfying a sequence
of equations of the form (cid:1)
g dσ =µ , (0.2)
n n
Ω
for given g and µ , n = 1,2,... . Although this monograph is devoted ex-
n n
clusively to a study of moment problems of the form (0.1), we shall briefly
mentionaclassicalresultonmomentproblemsoftheform(0.2)intheNotes
and Remarks of Chapter 2. Concerning moment problems of the form (0.1),
if dσ is absolutely continuous with respect to the Lebesgue measure, i.e., if
n
dσ =g dx,
n n
where g is Lebesgue integrable, n=1,2,..., then we have the usual moment
n
problem (cid:1)
ug dx=µ . (0.3)
n n
Ω
If dσ is a Dirac measure, i.e., if
n
dσ =δ(x−x ), x ∈Ω, (0.4)
n n n
then the moment problem consists in finding a function u on Ω from its
values at a sequence of points (x ), i.e.,
n
u(x )=µ , n=1,2,... . (0.5)
n n
Before proceeding further, it seems appropriate to explain how each of the
two foregoing variants of the moment problem (0.1) arises in the framework
of this monograph. In fact, many inverse problems can be formulated as an
integral equation of the first kind, namely,
(cid:1)
b
K(x,y)u(y)dy =f(x), x∈(a,b), (0.6)
a
where (a,b) is a bounded or unbounded open interval of R. Here K(x,y)
and f(x) are given functions and u(y) is a solution to be determined. In
practice, f(x) is a result of experimental measurements and hence is given
onlyatafinitesetofpointsthatisconvenientlypatchedupintoacontinuous
functionoranL2-function.Thisisaninterpolationproblem.Interpolationis
adelicateprocess,and,ingeneral,itisdifficulttoknowthenumberofpoints
D.D.Ang,R.Gorenflo,V.K.Le,andD.D.Trong:LNM1791,pp.1–3,2002.
(cid:1)c Springer-VerlagBerlinHeidelberg2002
2 Introduction
neededtoachieveadesireddegreeofapproximationunlessthefunctionf(x)
is sufficiently smooth. The case that the function represented by the integral
in the above equation can be extended to a function complex analytic in a
strip of the complex plane C containing the real interval [a,b] is of special
interest. Indeed, under the analyticity assumption, if the left hand side of
the equation is known on a bounded sequence (x ) in (a,b) with x (cid:5)=x for
n i j
i (cid:5)= j, then by a well-known property of analytic functions, the function is
known in the strip and a fortiori in (a,b). It follows that the above integral
equation is equivalent to the following moment problem
(cid:1)
b
K(x ,y)u(y)dy =f(x ), n=1,2,... . (0.7)
n n
a
Insomeexamplestobegiveninlaterchapters,wealsohavemomentproblems
of the foregoing form with (x ) unbounded and satisfying certain properties.
n
We shall also deal with multidimensional moment problems
(cid:1)
K(x ,y)u(y)dy =f(x ), n=1,2,... (0.8)
n n
Ω
where Ω is a domain in Rd, d ≥ 1 and (x ) is some infinite sequence (not
n
necessarily in Ω).
As mentioned earlier, we can have moment problems of the form (0.1)
above,withthedσ ’sbeingDiracmeasures.Thismomentproblemwillarise
n
in the reconstruction of a function u analytic in the unit disc U of C from
its values at a given sequence of points (z ) of U,
n
u(z )=µ , n=1,2,... . (0.9)
n n
Moment problems are similar to integral equations except that we now
deal with mappings between different spaces. Hence special techniques are
required.
The purpose of this monograph is to present some basic techniques for
treatments of moment problems. We note that classical treatments are con-
cernedprimarilywithquestionsofexistence(anduniqueness).Fortheclassi-
caltheory,thereaderisreferredto,e.g.,themonographofAkhiezer[Ak]and
the article of Landau [La]. From our point of view, however, the given data
are results of experimental measurements and hence are given only at finite
sets of points that are conveniently patched up into functions in appropriate
spaces, and consequently, a solution may not exist. Furthermore, moment
problems are ill-posed in the sense that solutions usually do not exist and
that in the case of existence, there is no continuous dependence on the given
data. The present monograph presents some regularization methods.
Paralleltothetheoryofmoments,weshallconsidervariousinverseprob-
lems in Potential Theory and in Heat Conduction. These inverse problems
provide important examples in illustration of moment theory, however, they
are also investigated for their own sake. In order to convey the full flavor of
the subject, we have tried to explain in detail the physical models.
