Multiscale and Adaptivity: Modeling, Numerics and Applications pdf

Lecture Notes in Mathematics 2040
Editors:
J M. Morel, Cachan
B. Teissier, Paris
For further volumes:
/>Fondazione C.I.M.E., Firenze
C.I.M.E. stands for Centro Internazionale Matematico Estivo, that is, International
Mathematical Summer Centre. Conceived in the early fifties, it was born in 1954 in Florence,
Italy, and welcomed by the world mathematical community: it continues successfully, year
for year, to this day.
Many mathematicians from all over the world have been involved in a way or another in
C.I.M.E.’s activities over the years. The main purpose and mode of functioning of the Centre
may be summarised as follows: every year, during the summer, sessions on different themes
from pure and applied mathematics are offered by application to mathematicians from all
countries. A Session is generally based on three or four main courses given by specialists
of international renown, plus a certain number of seminars, and is held in an attractive rural
location in Italy.
The aim of a C.I.M.E. session is to bring to the attention of younger researchers the origins,
development, and perspectives of some very active branch of mathematical research. The
topics of the courses are generally of international resonance. The full immersion atmosphere
of the courses and the daily exchange among participants are thus an initiation to international
collaboration in mathematical research.
C.I.M.E. Director C.I.M.E. Secretary
Pietro ZECCA Elvira MASCOLO
Dipartimento di Energetica “S. Stecco” Dipartimento di Matematica “U. Dini”
Universit
`
a di Firenze Universit
`
a di Firenze

Via S. Marta, 3 viale G.B. Morgagni 67/A
50139 Florence 50134 Florence
Italy Italy
e-mail: zecca@unifi.it e-mail: fi.it
For more information see CIME’s homepage: fi.it
CIME activity is carried out with the collaboration and financial support of:
- INdAM (Istituto Nazionale di Alta Matematica)
- MIUR (Ministero dell’Universita’ e della Ricerca)
Silvia Bertoluzza

Ricardo H. Nochetto
Alfio Quarteroni

Kunibert G. Siebert
Andreas Veeser
Multiscale and Adaptivity:
Modeling, Numerics
and Applications
C.I.M.E. Summer School,
Cetraro, Italy 2009
Editors:
Giovanni Naldi
Giovanni Russo
123
Silvia Bertoluzza
CNR
Istituto di Matematica Applicata
e Tecnologie Informatiche
Pavia
Italy

Ricardo H. Nochetto
University of Maryland
Department of Mathematics
College Park, MD
USA
Alfio Quarteroni
´
Ecole Polytechnique F
´
ed
´
erale
de Lausanne
Chaire de Modelisation
et Calcul Scientifique (CMCS)
Lausanne
Switzerland
Kunibert G. Siebert
Universit
¨
at Stuttgart
Fakult
¨
at f¨ur Mathematik und Physik
Stuttgart
Germany
Andreas Veeser
Universit
`
a degli Studi di Milano

Dipartimento di Matematica
Milano
Italy
ISBN 978-3-642-24078-2 e-ISBN 978-3-642-24079-9
DOI 10.1007/978-3-642-24079-9
Springer Heidelberg Dordrecht London New York
Lecture Notes in Mathematics ISSN print edition: 0075-8434
ISSN electronic edition: 1617-9692
Library of Congress Control Number: 2011943495
Mathematics Subject Classification (2010): 65M50, 65N50, 65M55, 65T60, 65N30, 65M60, 76MXX
c
 Springer-Verlag Berlin Heidelberg 2012
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Preface
The CIME-EMS Summer School in applied mathematics on “Multiscale and
Adaptivity: Modeling, Numerics and Applications” was held in Cetraro (Italy) from
July 6 to 11, 2009. This course has focused on mathematical methods for systems
that involve multiple length/time scales and multiple physics. The complexity of
the structure of these systems requires suitable mathematical and computational
tools. In addition, mathematics provides an effective approach toward devising

computational strategies for handling multiple scales and multiple physics. This
course brought together researchers and students from different areas such as partial
differential equations (PDEs), analysis, mathematical physics, numerical analysis,
and scientific computing to address the challenges present in these issues. Physical,
chemical, and biological processes for many problems in computational physics,
biology, and material science span length and time scales of many orders of
magnitude. Traditionally, scientists and research groups have focused on methods
that are particularly applicable in only one regime, and knowledge of the system at
one scale has been transferred to another scale only indirectly. Microscopic models,
for example, have been often used to find the effective parameters of macroscopic
models, but for obvious computational reasons, microscopic and macroscopic scales
have been treated separately.
The enormous increase in computational power available (due to the improve-
ment both in computer speed and in efficiency of the numerical methods) allows
in some cases the treatment of systems involving scales of different orders of
magnitude, arising, for example, when effective parameters in a macroscopic model
depend on a microscopic model, or when the presence of a singularity in the solution
produces a continuum of length scales. However, the numerical solution of such
problems by classical methods often leads to an inefficient use of the computational
resources, even up to the point that the problem cannot be solved by direct numerical
simulation. The main reasons for this are that the necessary resolution of a fine scale
entails an over-resolution of coarser scales, the position of the singularity is not
known beforehand, the gap between the scales is too big for a treatment in the same
framework. In other cases, the structure of the mathematical models that treat the
system at the different scales varies a lot, and therefore new mathematical techniques
v
vi Preface
are required to treat systems described by different mathematical models. Finally, in
many cases one is interested in the accurate treatment of a small portion of a large
system, and it is too expensive to treat the whole system at the required accuracy. In

such cases, the region of interest is modeled and discretized with great accuracy,
while the remaining parts of the system are described by some reduced model,
which enormously simplifies the calculation, still providing reasonable boundary
conditions for the region of interest, allowing the required level of detail in such
region.
The outstanding and internationally renowned lecturers have themselves con-
tributed in an essential way to the development of the theory and techniques that
constituted the subjects of the courses. The selection of the five topics of the
CIME-EMS Course was not an easy task because of the wide spectrum of recent
developments in multiscale methods and models. The six world leading experts
illustrated several aspects of the multiscale approach.
Silvia Bertoluzza, from IMATI-CNR Pavia, described the concept of nonlinear
sparse wavelet approximation of a given (known) function. Next she showed how
the tools just introduced can be applied in order to write down efficient adaptive
schemes for the solution of PDEs.
Bjorn Engquist, from ICES University of Texas at Austin, gradually guided the
audience toward the realm of “Multiscale Modeling,” by providing mathematical
ground for state-of-the-art analytical and numerical multiscale problems.
Alfio Quarteroni, from EPFL, Lausanne, and Politecnico di Milano, considered
adaptivity in mathematical modeling for the description and simulation of complex
physical phenomena. He showed that the combination of hierarchical mathematical
models can be set up with the aim of reducing the computational complexity in the
real life problems.
Ricardo H. Nochetto, from University of Maryland, and Andreas Veeser, from
Universit
`
a di Milano, in their joint course started with an overview of the a posteriori
error estimation for finite element methods, and then they exposed recent results
about the convergence and complexity of adaptive finite element methods.
Kunibert G. Siebert, from Universit

¨
at Duisburg-Essen, described the implemen-
tation of adaptive finite element methods using toolbox ALBERTA (created by
Alfred Schmidt and Kunibert G. Siebert, which is freely available).
The main “senior” lecturers were complemented by four young speakers, who
gave account of detailed examples or applications during an afternoon session
dedicated to them. Matteo Semplice, Universit
`
a dell’Insubria, has spoken about
“Numerical entropy production and adaptive schemes for conservation laws,”
Tiziano Passerini, from Emory University, about “A 3D/1D geometrical multiscale
model of cerebral vasculature,” Loredana Gaudio, MOX Politecnico di Milano,
about “Spectral element discretization of optimal control problems,” and Carina
Geldhauser, Universit
¨
at Tuebingen, described “A discrete-in-space scheme converg-
ing to an unperturbed Cahn–Hilliard equation.” Both the lectures and the active
interactions with and within the audience contributed to the scientific success of the
course, which was attended by about 60 people of various nationality (14 different
countries), ranging from first year PhD students to full professors. The present
Preface vii
volume collects the expanded version of the lecture notes by Silvia Bertoluzza, Alfio
Quarteroni (with Marco Discacciati and Paola Gervasio as coauthors), Ricardo H.
Nochetto, Andreas Veeser, and Kunibert G. Siebert. We are grateful to them for such
high quality scientific material.
As editors of these Lecture Notes and as scientific directors of the course, we
would like to thank the many persons and Institutions that contributed to the success
of the school. It is our pleasure to thank the members of the Scientific Committee
of CIME for their invitation to organize the School; the Director, Prof. Pietro
Zecca, and the Secretary, Prof. Elvira Mascolo, for their efficient support during the

organization and their generous help during the school. We were particularly pleased
by the fact that the European Mathematical Society (EMS) chose to cosponsor this
CIME course as one of its Summer School in applied mathematics for 2009. Our
special thanks go to the lecturers for their early preparation of the material to be dis-
tributed to the participants, for their excellent performance in teaching the courses
and their stimulating scientific contributions. All the participants contributed to
the creation of an exceptionally friendly atmosphere in the beautiful environment
around the School. We also wish to thank Dipartimento di Matematica of the
Universit
`
a degli Studi di Milano, and Dipartimento di Matematica ed Informatica
of the Universit
`
a degli Studi di Catania for their financial support.
Catania Giovanni Naldi
Milano Giovanni Russo

Contents
Adaptive Wavelet Methods 1
Silvia Bertoluzza
1 Introduction 1
2 Multiresolution Approximation and Wavelets 2
2.1 RieszBases 2
2.2 Multiresolution Analysis 3
2.3 Examples 9
2.4 Beyond L
2
.R/ 17
3 The Fundamental Property of Wavelets 21
3.1 The Case ˝ D R: The Frequency Domain Point of View

vs.theSpaceDomainPointof View 22
4 Adaptive Wavelet Methods for PDE’s: The First Generation 34
4.1 TheAdaptiveWaveletCollocationMethod 37
5 The New Generation of Adaptive Wavelet Methods 40
5.1 A PosterioriErrorEstimates 41
5.2 Nonlinear Wavelet Methods for the Solution of PDE’s 46
5.3 TheCDD2 Algorithm 48
5.4 Operationson InfiniteMatrices andVectors 51
References 54
Heterogeneous Mathematical Models in Fluid Dynamics
and Associated Solution Algorithms 57
Marco Discacciati, Paola Gervasio, and Alfio Quarteroni
1 Introduction and Motivation 57
2 VariationalFormulationApproach 67
2.1 The Advection–Diffusion Problem 67
2.2 Variational Analysis for the Advection–Diffusion
Equation 68
2.3 Domain Decomposition Algorithms for the Solution
of the Reduced Advection–Diffusion Problem 72
2.4 Numerical Results for the Advection–Diffusion Problem 77
ix
x Contents
2.5 Navier–Stokes/Potential Coupled Problem 80
2.6 Asymptotic Analysis of the Coupled Navier–Stokes/
DarcyProblem 82
2.7 Solution Techniques for the Navier–Stokes/Darcy
Coupling 85
2.8 NumericalResultsfortheNavier–Stokes/DarcyProblem 90
3 Virtual ControlApproach 94
3.1 Virtual Control Approach Without Overlap for AD

Problems 95
3.2 Domain Decomposition with Overlap 105
3.3 Virtual Control Approach with Overlap
for the Advection–Diffusion Equation 108
3.4 Virtual Control with Overlap for the Stokes–Darcy
Coupling 114
3.5 CouplingforIncompressibleFlows 119
References 120
Primer of Adaptive Finite Element Methods 125
Ricardo H. Nochetto and Andreas Veeser
1 Piecewise Polynomial Approximation 125
1.1 Classical vsAdaptivePointwiseApproximation 126
1.2 The Sobolev Number: Scaling and Embedding 127
1.3 ConformingMeshes: The Bisection Method 129
1.4 Finite Element Spaces 133
1.5 Polynomial Interpolation in Sobolev Spaces 134
1.6 AdaptiveApproximation 139
1.7 NonconformingMeshes 143
1.8 Notes 145
1.9 Problems 146
2 Error Bounds for Finite Element Solutions 148
2.1 Model Boundary Value Problem 148
2.2 GalerkinSolutions 149
2.3 Finite Element Solutions and A Priori Bound 150
2.4 A Posteriori Upper Bound 151
2.5 Notes 157
2.6 Problems 158
3 Lower A Posteriori Bounds 159
3.1 Local Lower Bounds 160
3.2 Global Lower Bound 166

3.3 Notes 167
3.4 Problems 168
4 ConvergenceofAFEM 170
4.1 A ModelAdaptiveAlgorithm 171
4.2 Convergence 172
4.3 Notes 178
4.4 Problems 179
Contents xi
5 Contraction PropertyofAFEM 180
5.1 Modules of AFEM for the Model Problem 180
5.2 BasicPropertiesof AFEM 182
5.3 ContractionPropertyof AFEM 185
5.4 Example: Discontinuous Coefficients 189
5.5 ExtensionsandRestrictions 191
5.6 Notes 193
5.7 Problems 193
6 Complexityof Refinement 194
6.1 Chains and Labeling for d D 2 195
6.2 Recursive Bisection 197
6.3 ConformingMeshes: ProofofTheorem1 199
6.4 NonconformingMeshes:Proofof Lemma3 204
6.5 Notes 205
6.6 Problems 206
7 ConvergenceRates 206
7.1 TheTotalError 207
7.2 ApproximationClasses 208
7.3 Quasi-Optimal Cardinality: Vanishing Oscillation 212
7.4 Quasi-Optimal Cardinality: General Data 215
7.5 ExtensionsandRestrictions 218
7.6 Notes 221

7.7 Problems 221
References 223
Mathematically Founded Design of Adaptive Finite Element
Software 227
Kunibert G. Siebert
1 Introduction 227
1.1 TheVariational Problem 229
1.2 TheBasicAdaptive Algorithm 230
2 Triangulations and Finite Element Spaces 232
2.1 Triangulations 232
2.2 Finite Element Spaces 234
2.3 Basis Functions and Evaluation of Finite Element
Functions 240
2.4 ALBERTA Realization of Finite Element Spaces 244
3 RefinementByBisection 246
3.1 Basic Thoughts About Local Refinement 246
3.2 Bisection Rule:Bisectionof One SingleSimplex 248
3.3 Triangulations and Refinements 252
3.4 RefinementAlgorithms 255
3.5 Complexityof RefinementByBisection 260
3.6 ALBERTA Refinement 262
3.7 Mesh Traversal Routines 263
xii Contents
4 AssemblageoftheLinearSystem 268
4.1 TheVariational ProblemandtheLinearSystem 269
4.2 Assemblage:TheOuterLoop 272
4.3 Assemblage:ElementIntegrals 276
4.4 RemarksonIterativeSolvers 283
5 TheAdaptiveAlgorithmandConcluding Remarks 285
5.1 TheAdaptiveAlgorithm 286

5.2 ConcludingRemarks 294
6 Supplement: A Nonlinear and a Saddlepoint Problem 297
6.1 The Prescribed Mean Curvature Problem in Graph
Formulation 297
6.2 TheGeneralized StokesProblem 301
References 308
List of Participants 311
Adaptive Wavelet Methods
Silvia Bertoluzza
Abstract Wavelet bases, initially introduced as a tool for signal and image process-
ing, have rapidly obtained recognition in many different application fields. In this
lecture notes we will describe some of the interesting properties that such functions
display and we will illustrate how such properties (and in particular the simultaneous
good localization of the basis functions in both space and frequency) allow to devise
several adaptive solution strategies for partial differential equations. While some of
such strategies are based mostly on heuristic arguments, for some other a complete
rigorous justification and analysis of convergence and computational complexity is
available.
1 Introduction
Wavelet bases were introduced in the late 1980s as a tool for signal and image pro-
cessing. Among the applications considered at the beginning we recall applications
in the analysis of seismic signals, the numerous applications in image processing
– image compression, edge-detection, denoising, applications in statistics, as well
as in physics. Their effectiveness in many of the mentioned fields is nowadays
well established: as an example, wavelets are actually used by the US Federal
Bureau of Investigation (or FBI) in their fingerprint database, and they are one
of the ingredient of the new MPEG media compression standard. Quite soon it
became clear that such bases allowed to represent objects (signals, images, turbulent
fields) with singularities of complex structure with a low number of degrees of
freedom, a property that is particularly promising when thinking of an application

to the numerical solution of partial differential equations: many PDEs have in fact
S. Bertoluzza ()
Istituto di Matematica Applicata e Tecnologie Informatiche del CNR, v. Ferrata 1, Pavia, Italy
e-mail:
S. Bertoluzza et al., Multiscale and Adaptivity: Modeling, Numerics and Applications,
Lecture Notes in Mathematics 2040, DOI 10.1007/978-3-642-24079-9
1,
© Springer-Verlag Berlin Heidelberg 2012
1
2 S. Bertoluzza
solutions which present singularities, and the ability to represent such solution
with as little as possible degrees of freedom is essential in order to be able to
implement effective solvers for such problems. The first attempts to use such bases
in this framework go back to the late 1980s and early 1990s, when the first simple
adaptive wavelet methods [38] appeared. In those years the problems to be faced
were basic ones. The computation of integrals of products of derivative of wavelets –
object which are naturally encountered in the variational approach to the numerical
solution of PDEs – was an open problem (solved later by Dahmen and Micchelli
in [24]). Moreover, wavelets were defined on R and on R
n
. Already solving a simple
boundary value problem on .0; 1/ (the first construction of wavelets on the interval
[19] was published in 1993) posed a challenge.
Many steps forward have been made since those pioneering works. In particular
thinking in terms of wavelets gave birth to some new approaches in the numerical
solution of PDEs. The aim of this course is to show some of these new ideas. In
particular we want to show how one key property of wavelets (the possibility of
writing equivalent norms for the scale of Besov spaces) allows to write down some
new adaptive methods for solving PDE’s.
2 Multiresolution Approximation and Wavelets

2.1 Riesz Bases
Before starting with defining wavelets, let us recall the definition and some prop-
erties of Riesz bases [14], which will play a relevant role in the following. Let H
denote an Hilbert space and let V Â H denote a subspace. A basis B Df'
k
;k 2 I g
(I Â N index set) of V is a Riesz basis if and only if the following norm equivalence
holds:
k
X
k
c
k
e
k
k
2
H
'
X
k
jc
k
j
2
:
Here and in the following we use the notation A ' B to signify that there exist
positive constants c and C , independent of any relevant parameter, such that cB Ä
A Ä CB. Analogously we will use the notation A


<B(resp. A

>B), meaning that
A Ä CB (resp. A  cB).
Letting P W H ! V be any projection operator (P
2
D P ), it is not difficult to
realize that there exist a sequence G Dfg
k
;k 2 I g such that for all f 2 H we
have the identity
Pf D
X
k2I
hf; g
k
i'
k
:
The sequence g
k
is biorthogonal to the basis B,thatiswehavethat
hg
k
;'
i
iDı
i;k
:
Adaptive Wavelet Methods 3

Moreover the sequence G is a Riesz basis for the subspace P

H (P

denoting the
adjoint operator to P ), and P

takes the form
P

f D
X
k2I
hf; '
k
ig
k
:
2.2 Multiresolution Analysis
We start by introducing the general concept of multiresolution analysis in the
univariate case [39].
Definition 1. A Multiresolution Analysis (MRA) of L
2
.R/ is a sequence fV
j
g
j 2Z
of closed subspaces of L
2
.R/ verifying:

(i) The subspaces are nested: V
j
 V
j C1
for all j 2 Z.
(ii) The union of the spaces is dense in L
2
.R/ and the intersection is null:
[
j 2Z
V
j
D L
2
.R/; \
j 2Z
V
j
Df0g: (1)
(iii) There exists a scaling function ' 2 V
0
such that f'.k/; k 2 Zg is a Riesz
basis for V
0
.
(iv) f 2 V
0
implies f.2
j
/ 2 V

j
.
Several properties descend directly from this definition. First of all it is not
difficult to check that the above properties imply that for all j the set f'
j;k
k 2 Zg,
with
'
j;k
D 2
j=2
'.2
j
k/ (2)
is Riesz basis for V
j
, yielding, uniformly in j , a norm equivalence between the L
2
norm of a function in V
j
and the `
2
norm of the sequence of its coefficients.
Moreover, the inclusion V
0
 V
1
implies that the scaling function ' can be
expanded in terms of the basis of V
1

through the following refinement equation:
'.x/ D
X
k2Z
h
k
'.2x  k/; (3)
with .h
k
/
k
2 `
2
.Z/. The function ' is then said to be a refinable function and the
coefficients h
k
are called refinement coefficients.
Let now f 2 L
2
.R/. We can consider approximations f
j
2 V
j
to f at different
levels j .SinceV
j
 V
j C1
it is not difficult to realize that the approximation f
j C1

of a given function f at level j C1 must “contain” more information on f than f
j
.
The idea underlying the construction of wavelets is the one of somehow encoding
the “loss of information” that we have when we go from f
j C1
to f
j
.Letusfor
instance consider f
j
D P
j
f ,whereP
j
W L
2
.R/ ! V
j
denotes the L
2
.R/-
orthogonal projection onto V
j
. Remark that P
j C1
P
j
D P
j

(a direct consequence of
4 S. Bertoluzza
the nestedness of the spaces V
j
). Moreover, we have that P
j
P
j C1
D P
j
: f
j C1
contains in this case all information needed to retrieve f
j
. We can in this case
introduce the orthogonal complement W
j
 V
j C1
(W
j
? V
j
and V
j C1
D V
j
˚W
j
).

A similar construction can be actually carried out in a more general framework,
in which P
j
is not necessarily the orthogonal projection. To be more general, let
us start by choosing a sequence of uniformly bounded (not necessarily orthogonal)
projectors P
j
W L
2
.R/ ! V
j
verifying the following properties:
P
j
P
j C1
D P
j
; (4)
P
j
.f .k2
j
//.x/ D P
j
f.x k2
j
/; (5)
P
j C1

f 2//.x/ D P
j
f.2x/: (6)
Remark again that the inclusion V
j
 V
j C1
guarantees that P
j C1
P
j
D P
j
.On
the contrary, property (4) is not verified by general non-orthogonal projectors and
expresses the fact that the approximation P
j
f can be derived from P
j C1
f without
any further information on f . Equations (5)and(6) require that the projector P
j
respects the translation and dilation invariance properties (iii) and (iv) of the MRA.
Since f'
0;k
g is a Riesz basis for V
0
there exists a biorthogonal sequence fQ'
0;k
g of

L
2
.R/ functions such that
P
0
f D
X
k
hf; Q'
0;k
i'
0;k
:
Property (5) implies that the biorthogonal basis has itself a translation invariant
structure, as stated by the following proposition.
Proposition 1. Letting Q' DQ'
0;0
we have that
Q'
0;k
.x/ DQ'.x  k/: (7)
Proof. We observe that
P
0
.f .Cn//.x/ D
X
k
hf.Cn/; Q'
0;k
i'.x k/ D

X
k
hf./; Q'
0;k
.n/i'.x k/;
P
0
f.x C n/ D
X
k
hf./; Q'
0;k
i'.x  k C n/:
Thanks to (5)wehavethat
X
k
hf./; Q'
0;k
.n/i'.x  k/ D
X
k
hf./; Q'
0;k
i'.x  k C n/;
and, since f'
0;k
g is a Riesz basis for V
0
, implying that the coefficients (and in
particular the coefficient of '

0;0
D '.x/) are uniquely determined, this implies,
for all f 2 L
2
.R/,
Adaptive Wavelet Methods 5
hf; Q'
0;0
.n/iDhf; Q'
0;n
i;
that is, by the arbitrariness of f , Q'
0;n
DQ'
0;0
.n/. ut
In an analogous way, thanks to property (6) it is not difficult to prove the
following Proposition.
Proposition 2. We have
P
j
f D
X
k
hQ'
j;k
;fi'
j;k
with Q'
j;k

.x/ D 2
j=2
Q'.2
j
x  k/: (8)
Moreover the set fQ'
j;k
;k2 Zg forms a Riesz basis for the subspace
Q
V
j
D
P

j
.L
2
.R// (where P

j
denotes the adjoint of P
j
).
Finally, property (4) implies that the sequence
Q
V
j
is nested.
Proposition 3. The sequence f
Q

V
j
g satisfies
Q
V
j

Q
V
j C1
.
Proof. Property (4) implies that P

j C1
P

j
f D P

j
f .Nowwehavef 2
Q
V
j
implies
f D P

j
f D P


j C1
P

j
f 2
Q
V
j C1
. ut
Corollary 1. The function Q' DQ'
0;0
is refinable.
The above construction derives from the a priori choice of a sequence P
j
, j 2 Z,
of (oblique) projectors onto the subspaces V
j
. A trivial choice is to define P
j
as the
L
2
.R/ orthogonal projector. It is easy to see that all the required properties are
satisfied by such a choice. In this case, since the L
2
.R/ orthogonal projector is self
adjoint, we have
Q
V
j

D V
j
, and the biorthogonal function Q' belongs itself to V
0
.
Clearly, in the case that f'
0;k
;k 2 Zg is an orthonormal basis for V
0
(as in the
Haar basis case of the forthcoming Example I, or as for Daubechies MRA’s) we
have that Q' D '. Another possibility would be to choose P
j
to be the Lagrangian
interpolation operator. This choice, which we will describe later on, does however
fall outside of the framework considered here, since interpolation is not an L
2
.R/
bounded operator.
Infinitely many other choices are possible in theory but quite difficult to construct
in practice. The solution is then to go the other way round, constructing the function
Q' directly and defining the projectors P
j
by (8)[18]. We then introduce the
following definition:
Definition 2. A refinable function
Q' D
X
k
Q

h
k
Q'.2 k/ 2 L
2
.R/ (9)
is dual to ' if
h'.k/; Q'.l/iDı
k;l
k; l 2 Z:
It is possible to prove that the translates of the dual refinable function are a Riesz
basis for the subspace that they span.
6 S. Bertoluzza
Assuming then that we have a refinable function Q' dual to ', we can define the
projector P
j
using (8).
P
j
f D
X
k2Z
hf; Q'
j;k
i'
j;k
:
The operator P
j
is bounded and it is indeed a projector: it is not difficult to check
that f 2 V

j
) P
j
f D f .
Remark 1. As it happened for the projector P
j
, the dual refinable function Q' is not
uniquely determined, once ' is given. Different projectors correspond to different
dual functions. It is worth noting that P.G. Lemari
´
e[37] proved that if ' is compactly
supported then there exists a dual function Q' 2 L
2
.R/ which is itself compactly
supported.
The dual of P
j
P

j
f D
X
k2Z
hf; '
j;k
iQ'
j;k
is also an oblique projector onto the space Im.P

j

/ D
Q
V
j
,where
Q
V
j
D span < Q'
j;k
;k2 Z >:
It is not difficult to see that since Q' is refinable then the
Q
V
j
’s are nested.
Remark 2. The two different ways of defining the dual MRA are equivalent. A third
approach yields also an equivalent structure. In fact, assume that we have a sequence
Q
V
j
of spaces such that the following inf-sup conditions hold uniformly in j :
inf
v
j
2V
j
sup
w
j

2
Q
V
j
hv
j
; w
j
i
kv
j
k
L
2
.R/
kw
j
k
L
2
.R/

>1; inf
w
j
2
Q
V
j
sup

v
j
2V
j
hv
j
; w
j
i
kv
j
k
L
2
.R/
kw
j
k
L
2
.R/

>1:
(10)
Then it is possible to define a bounded projector P
j
W L
2
.R/ ! V
j

as P
j
v D v
j
,
v
j
being the unique element of V
j
such that
hv
j
; w
j
iDhv; w
j
i8w
j
2
Q
V
j
:
It is not difficult to see that if the sequence
Q
V
j
is a multiresolution analysis (that is, if
it satisfies the requirements of Definition 1) then the projector P
j

satisfies properties
(4)–(6). Conversely the uniform boundedness of the projector P
j
and of its adjoint
Q
P
j
actually implies the validity of the two inf-sup conditions (10).
2.2.1 Wavelets
Whatever the way chosen to introduce the dual multiresolution analysis, we have
now a natural way to define a space W
j
which complements V
j
in V
j C1
.More
precisely we set
Adaptive Wavelet Methods 7
V
j C1
D V
j
˚ W
j
;W
j
D Q
j
.V

j C1
/; Q
j
D P
j C1
 P
j
: (11)
Remark that Q
2
j
D Q
j
,thatisQ
j
is indeed a projector on W
j
. W
j
can also be
defined as the kernel of P
j
in V
j C1
. Iterating for j decreasing the splitting (11)we
obtain a multiscale decomposition of V
j C1
as
V
j C1

D V
0
˚ W
0
˚˚W
j
:
By construction we also have, for all f 2 L
2
.R/, the decomposition
P
j C1
f D P
0
f C
j
X
mD0
Q
m
f:
In other words the approximation P
j C1
f is decomposed as a coarse approximation
at scale 0 plus a sequence of fluctuations at intermediate scales 2
m
;mD 0;:::;j.
If we are to express the above identity in terms of a Fourier expansion, we need
bases for the spaces W
j

. Depending on the nature of the spaces considered such
bases might be readily available (see for instance the construction of interpolating
wavelets). However this is not, in general, the case. A general procedure to construct
a suitable basis for W
j
is the following [30]: define two sets of coefficients:
g
k
D .1/
k
Q
h
1k
; Qg
k
D .1/
k
h
1k
;k2 Z
and introduce a pair of dual wavelets 2 V
1
and
Q
2
Q
V
1
.x/ D
X

k
g
k
'.2x  k/
Q
.x/ D
X
k
Qg
k
Q'.2x k/: (12)
The following theorem holds [18]:
Theorem 1. The integer translates of the wavelet functions and
Q
are orthog-
onal to Q' and ', respectively, and they form a couple of biorthogonal sequences.
More precisely, they satisfy
h ;
Q
.k/iDı
0;k
h .k/; Q'iDh
Q
.k/; 'iD0: (13)
The projection operator Q
j
can be expanded as
Q
j
f D

X
k
hf;
Q

j;k
i
j;k
and the functions
j;k
constitute a Riesz basis of W
j
.
8 S. Bertoluzza
For any function f 2 L
2
.R/, P
j
f in V
j
can be expressed as
P
j
f D
X
k
c
j;k
'
j;k

D
X
k
c
0;k
'
0;k
C
j 1
X
mD0
X
k
d
m;k

m;k
; (14)
with d
m;k
Dhf;
Q

m;k
i and c
m;k
Dhf; Q'
m;k
i.
Both f'

j;k
;k 2 Zg and f'
0;k
;k 2 Zg[
0Äm<j 1
f
m;k
;k 2 Zg are bases for V
j
and
(14) expresses a change of basis. Thanks to the density property (1)forj !C1
P
j
f converges to f in L
2
.R/. Then, taking the limit for j !C1in (14), we
obtain
f D
X
k
c
0;k
'
0;k
C
C1
X
mD0
X
k

d
m;k

m;k
:
We will see in the following that, under quite mild assumptions, the convergence is
unconditional.
2.2.2 The Fast Wavelet Transform
The idea is now to design an algorithm allowing to compute efficiently the
coefficients c
j 1;k
.f / and d
j 1;k
.f / directly from the coefficients c
j;k
.f /,which
uniquely identify P
j
f . The key is the refinement equation (9), which gives us a
“fine to coarse” discrete projection algorithm:
c
j 1;k
Dhf; Q'
j 1;k
iD2
.j 1/=2
hf; Q'.2
j 1
k/i
D 2

.j 1/=2
hf;
X
n
Q
h
n
Q'.2
j
2k  n/iD
1
p
2
X
n
Q
h
n
c
j;2kCn
:
An analogous relation holds for d
j 1;k
. On the other hand, thanks to (3), given the
projection P
j 1
f D
P
k
c

j 1;k
'
j 1;k
we are able to express it in terms of basis
functions at the finer scale
P
j 1
f D 2
.j 1/=2
X
k
c
j 1;k
'.2
j 1
k/
D 2
.j 1/=2
X
k
c
j 1;k
X
n
h
n
'.2
j
2k  n/
D

1
p
2
X
k
h
X
n
h
k2n
c
j 1;n
i
'
j;k
:
Adaptive Wavelet Methods 9
Analogously we have
Q
j 1
f D
1
p
2
X
k
h
X
n
g

k2n
d
j 1;n
i
'
j;k
:
Since P
j
f D P
j 1
f CQ
j 1
f we immediately get
P
j
f D
X
k
1
p
2
"
X
n
h
k2n
c
j 1;n
C

X
n
g
k2n
d
j 1;n
#
'
j;k
:
In summary, the one level decomposition algorithm reads
c
j;n
D
1
p
2
X
k
Q
h
k2n
c
j C1;k
d
j;n
D
1
p
2

X
k
Qg
k2n
c
j C1;k
while its inverse, the one level reconstruction algorithm can be written as
c
j C1;k
D
1
p
2
h
X
n
h
k2n
c
j;n
C
X
n
g
k2n
d
j;n
i
:
Once the one level decomposition algorithm is given, giving the coefficient vectors

.c
j;k
/
k
and .d
j;k
/
k
in terms of the coefficient vector .c
j C1;k
/
k
, we can iterate it to
obtain .c
j 1;k
/
k
and .d
j 1;k
/
k
and so on until we get all the coefficients for the
decomposition (14).
2.3 Examples
2.3.1 Example I: Daubechies Wavelets
The Haar basis: The first, simplest, example of a wavelet basis is the Haar basis,
which was introduced in 1909 by Alfred Haar as an example of a countable
orthonormal system for L
2
.R/. In the Haar wavelet case V

j
is defined to be the
space of piecewise constant functions with uniform mesh size h D 2
j
:
V
j
Dfw 2 L
2
.R/ such that wj
I
j;k
is constantg;
where we denote by I
j;k
the dyadic interval I
j;k
WD .k2
j
;.k C 1/2
j
/. It is not
difficult to see that the sequence fV
j
;j2 Zg is indeed a multiresolution analysis.
In particular an orthonormal basis for V
j
is given by the family
'
j;k

WD 2
j=2
'.2
j
k/ with ' D j
.0;1/
: (15)
10 S. Bertoluzza
Letting P
j
W L
2
.R/ ! V
j
be the L
2
.R/-orthogonal projection onto V
j
, clearly
we have
P
j
f D
X
k
c
j;k
.f /'
j;k
;c

j;k
.f / Dhf; '
j;k
i;
and the dual multiresolution analysis f
Q
V
j
g coincides with fV
j
g. The space W
j
is
then the orthogonal complement of V
j
in V
j C1
:
V
j C1
D W
j
˚ V
j
;W
j
?V
j
;
and the L

2
.R/-orthogonal projection Q
j
WD P
j C1
 P
j
onto W
j
verifies
Q
j
f j
I
j C1;2k
D P
j C1
f j
I
j C1;2k
 .P
j C1
f j
I
j C1;2k
C P
j C1
f j
I
j C1;2kC1

/=2
D P
j C1
f j
I
j C1;2k
=2 P
j C1
f j
I
j C1;2kC1
=2; (16)
Q
j
f j
I
j C1;2kC1
D P
j C1
f j
I
j C1;2kC1
 .P
j C1
f j
I
j C1;2k
C P
j C1
f j

I
j C1;2kC1
/=2
DP
j C1
f j
I
j C1;2k
=2 C P
j C1
f j
I
j C1;2kC1
=2: (17)
It is then not difficult to realize that we can expand Q
j
f as
Q
j
f D
X
k
d
j;k
.f /
j;k
with
j;k
D 2
j=2

.2
j
k/;
where
WD 
.0;1=2/
 
.1=2;1/
:
Since the functions
j;k
at fixed j are an orthonormal system, they do constitute
an orthonormal basis for W
j
.Wehavethen
d
j;k
.f / Dhf;
j;k
i:
Daubechies’ compactly supported orthonormal wavelets: In her 1988 ground-
breaking paper [29] Ingrid Daubechies managed to generalize the Haar basis and
construct a class of MRA’s such that both ' and have arbitrarily high regularity R,
are supported in .0; L/ and they generate by translations and dilations orthonormal
bases for the spaces V
j
and W
j
. The projectors P
j

are, as in the Haar case, L
2
orthogonal projectors. Also in this case the scaling function and the dual function
coincide and we have
Q
V
j
D V
j
,
Q
W
j
D W
j
, Q' D ' and
Q
D (see Fig.1 for an
example of scaling and wavelet functions in this framework).
A characteristic of Daubechies’ wavelets is that, unlike the Haar basis, the spaces
V
j
and the function ' are not given directly. By giving an algorithm to construct
them, Daubechies characterizes all the sequences h D .h
k
/
k
for which a unique
solution ' to the refinement equation (3) exists and is orthogonal to its integer
Adaptive Wavelet Methods 11

translates and smooth. Once .h
k
/
k
is built, the spaces V
j
are then defined as the
span of f'
j;k
;k 2 Zg with '
j;k
defined by (2). The function ' is, by construction,
refinable, and the sequence fV
j
;j2 Zgis a multiresolution analysis. The algorithm
to construct the refinement coefficients and the proof that for a given sequence .h
k
/
k
satisfying suitable conditions (3) has indeed a solution with the required properties
is quite technical and it is beyond our scope here to give more details about such
a construction. We refer the interested reader to [30]. The coefficient sequences
themselves are available, already computed, in table form at different resource sites
over the web (see www.wavelet.org).
It is worth noting that the refinement equation is quite powerful and that it is
possible to derive from it a lot of information on the function '. For instance, if we
need to plot the function ' we will need access to point values of such a function.
Since the ' is supported in .0; L/ we have that '.n/ D 0 for all n 2 Z, n 62 .0; L/.
For the remaining integers we can write
'.n/ D

X
k
h
k
'.2n  k/ D
X
`
h
2n`
'.`/:
If we consider the L  1  L  1 matrix H D .h
n;`
/ with h
n;`
D h
2n`
, clearly the
vector .'.1/; : : : ; '.L1// is an eigenvector of H for the eigenvalue 1,whichturns
out to be unique. Once the values in the integers are computed by any eigenvector
computation algorithm, the values in dyadic points are computed recursively thanks
again to the refinement equation, which gives us
'

n
2
j
Á
D
X
k

h
k
'
Â
n  2
j 1
k
2
j 1
Ã
:
Analogous algorithms are available for computing many quantities which are
needed for the application in the numerical solution of PDEs, like for instance point
values of derivatives, integrals and integrals of product of derivatives.
2.3.2 Example II: B-Splines
Many applications of wavelets to PDEs are based on the multiresolution analysis
generated by the spaces V
j
:
V
j
Dff 2 L
2
\ C
N 1
W f j
I
j;k
2 P
N

g:
AbasisforV
j
whose elements are compactly supported can be constructed by
defining the B-spline B
N
of degree N recursively by
B
0
WD 
Œ0;1
;B
N
WD B
0
 B
N 1
D ./
N C1

Œ0;1
;
12 S. Bertoluzza
01234
−0.4
−0.2
0
0.2
0.4
0.6

0.8
1
1.2
1.4
db2 : phi
01234
−1.5
−1
−0.5
0
0.5
1
1.5
2
db2 : psi
Fig. 1 The scaling and wavelet functions ' and generating a Daubechies’ orthonormal wavelet
basis
where  denotes the convolution product. The function B
N
is supported in
Œ0; N C1, it is refinable and the corresponding scaling coefficients are defined by
h
k
D
8
ˆ
ˆ
<
ˆ
ˆ

:
2
N

N C 1
k
!
;0Ä k Ä N C 1;
0 otherwise:
The integer translates of the function ' D B
N
form a Riesz basis for V
j
.ForN
given it is possible to construct an infinite class of compactly supported refinable
functions dual to B
N
. More precisely, analogously to what is done for Daubechies’
wavelets, it is possible [18] to characterize and construct a family of sequences
.
Q
h
k
/
k
for which the solution to the refinement equation (9) exists, is dual to the
B-spline B
N
, has compact support and arbitrarily high smoothness
Q

R. Figures 2
and 3 show the functions ', Q', and
Q
for N D 1,
Q
R D 0 and N D 1,
Q
R D 1,
respectively.