-
UNIVERSITÄT LINZJOHANNES KEPLER JKU
Technisch-NaturwissenschaftlicheFakultät
A Discontinuous Galerkin Method for SolvingTotal Variation
Minimization Problems
MASTERARBEITzur Erlangung des akademischen Grades
Diplomingenieurim Masterstudium
Industriemathematik
Eingereicht von:Stephen Edward Moore
Angefertigt am:Institut für Numerische Mathematik
Beurteilung:O. Univ. Prof. Dipl. Ing. Dr. Ulrich LangerPriv.
Doz. Dr. Johannes Kraus
Mitwirkung:Prof. Dr. Massimo Fornasier
Linz, August, 2011
-
i
To my parentsStephen Moore and Janet Moore.
-
Abstract
The minimization of functionals which are formed by an L2-term
and a Total Variation(TV) term play an important role in
mathematical imaging with many applicationsin engineering, medicine
and art. The TV term is well known to preserve sharp edgesin
images.
More precisely, we are interested in the minimization of a
functional formed by a dis-crepancy term and a TV term. The first
order derivative of the TV term involvesa degenerate term which
could happen in flat areas of an image. Many well knownmethods have
been proposed to solve this problem.
In this thesis, we present a relaxed functional associated with
the TV minimizationproblem. The relaxed functionals are well-posed
and produce a sequence of solutionsminimizing our original
TV-functional. The relaxed functional results in an
IterativelyReweighted Least Squares method that approximates the TV
minimization.
Considering the Euler-Lagrange equation, the minimizer of the
relaxed functional isequivalent to the solution of a second order
elliptic partial differential equation. Wediscretize this partial
differential equation in the framework of Discontinuous
Galerkin(DG) Finite Element Method (FEM) with linear functions on
each element. Specifi-cally, we consider the Symmetric Interior
Penalty Galerkin method. The discretizationleads to a system of
linear equations.
The existence and uniqueness of the solution to the DG
variational form of Discon-tinuous Galerkin and the discrete DG
problem is studied, and a-priori error estimatesare reported.
The Discontinuous Galerkin Finite Element Method in combination
with iterativelyreweighted least squares method is implemented
.
Finally, numerical results are presented that demonstrate the
accuracy of the numeri-cal solution using the proposed methods.
ii
-
Acknowledgments
First of all, I would like to thank my supervisors Prof. Ulrich
Langer for giving me theopportunity to write this thesis and for
organizing financial support; Priv. Doz. Dr.Johannes Kraus for his
patience, timely and friendly discussions and encouragements,the
absence of which would not have made this work possible and lastly
to Prof.Massimo Fornasier for his invaluable advice and discussions
most of which will not beforgotten.
Special thanks go to Dr. Satyendra Tomar, who assisted in my
understanding to theassembling of the Discontinuous Galerkin Finite
Element Method; Dipl. Ing. AndreasLanger for the readiness to
discuss some pertinent aspects of my work and for proof-reading the
Chapter 2. Also to Dr. Veronika Pillwein for her timely discussions
andlecture and Prof. Martin Gander for making time to read and make
suggestionstowards the completion of the thesis.
This work was additionally supported by the Austrian Science
Foundation – Fonds zurFörderung der wissenschaftlichen Forschung
(FWF) – through the DoktoratskollegsComputational Mathematics:
Numerical Analysis and Symbolic Computation.
I am indebted to the European Union for the Erasmus Mundus
Scholarship, TechnicalUniversity of Kaiserslautern, Germany and the
Institute of Computational Mathe-matics at the Johannes Kepler
University, Austria for the environment and technicalsupport.
Last but not least I am grateful to my family for their support
during my stay abroadand also my friends in Germany and
Austria.
Stephen Edward MooreLinz, August 2011
iii
-
Contents
1 Introduction 1
2 Problem Formulation and Analysis 42.1 Functions of Bounded
Variation . . . . . . . . . . . . . . . . . . . . . . 42.2
Existence and Uniqueness of Minimizers . . . . . . . . . . . . . .
. . . 82.3 Euler-Lagrange Equations and a Relaxation Algorithm . .
. . . . . . . 9
3 DG Finite Element Discretization 133.1 Some Basic Function
Spaces . . . . . . . . . . . . . . . . . . . . . . . . 133.2 DG
Variational Formulations . . . . . . . . . . . . . . . . . . . . .
. . 143.3 Some Basic Properties . . . . . . . . . . . . . . . . . .
. . . . . . . . . 183.4 The DG Finite Element Equations . . . . . .
. . . . . . . . . . . . . . 213.5 A Priori Error Estimates . . . .
. . . . . . . . . . . . . . . . . . . . . . 223.6 DG-Version of
IRLS Algorithm . . . . . . . . . . . . . . . . . . . . . . 26
4 Numerical Results 284.1 DG for Diffusion Problems . . . . . .
. . . . . . . . . . . . . . . . . . . 28
4.1.1 Dirichlet Problem . . . . . . . . . . . . . . . . . . . .
. . . . . . 284.1.2 Neumann Problem . . . . . . . . . . . . . . . .
. . . . . . . . . 284.1.3 Poisson Problem with Known Solution . . .
. . . . . . . . . . . 29
4.2 Iteratively Reweighted Least Squares Algorithm . . . . . . .
. . . . . . 314.2.1 Denoising Problem . . . . . . . . . . . . . . .
. . . . . . . . . . 314.2.2 Diffusion Problem, TV-Minimization
Problem and IRLS . . . . 33
5 Conclusion and Outlook 365.1 Conclusion . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . 365.2 Outlook . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36
List of Notations and Function Spaces 38
Bibliography 40
iv
-
Chapter 1
Introduction
In this thesis we will concentrate on the numerical solution of
2D Total Variation(TV) Minimization problems, as they appear in
image processing by a Discontinu-ous Galerkin (DG) Finite Element
Method (FEM) in combination with an IterativelyReweighted Least
Squares (IRLS) method.
TV methods and similar approaches based on regularizations with
L1-norms (andsemi-norms) have become a very popular tool in image
processing and inverse prob-lems due to peculiar features that
cannot be realized with smooth regularizations. TVtechniques had
particular success due to their ability to realize cartoon-type
recon-structions with sharp edges [9]. Within the last decade,
there have been an explosionof new developments in this field. The
TV methods started with an introduction ofa variational denoising
model by Rudin, Osher and Fatemi consisting of minimizingtotal
variation among all functions within a variance bound [40]. It was
shown to beequivalent to an unconstrained minimization problem of
the form:
minu
(2λ∫
Ω|∇u| dx+ ‖Ku− g‖2L2
), (1.1)
where u : Ω → R,Ω ⊂ Rn, n = 1, 2, K : L2(Ω) → L2(Ω), g ∈ L2(Ω)
and λ > 0. TheEuler-Lagrange-equation of the functional in (1.1)
reads as follows :
−λ div(∇u|∇u|
)+K∗(Ku− g) = 0, (1.2)
where K∗ is the adjoint of K. In the literature, several
numerical strategies for effi-ciently solving (1.2) have been
proposed. We only mention the following 3 approaches:
(i) The fixed point iteration method [15, 41, 42, 43]: Once the
coefficients 1/|∇u|are fixed at a previous iteration u, various
iterative solver techniques have beenconsidered. There exist
excellent inner solvers but the outer solver can be slow.Further
improvements are still useful.
1
-
CHAPTER 1. INTRODUCTION 2
(ii) The explicit time marching scheme [33, 38, 40]: It turns
the nonlinear partialdifferential equation (1.2) into a parabolic
equation before using an explicit Eulermethod to march in time to
convergence. The method is quite reliable but oftenslow as
well.
(iii) The primal-dual (PD) method [12, 13, 14]: It solves for
both the primal and thedual variable together in order to achieve
faster convergence with the Newtonmethod (and a constrained
optimisation with the dual variable).
In this thesis, we will consider an Iteratively Reweighted Least
Squares (IRLS) algo-rithm to solve the TV-minimization problem.
Under certain assumptions, the IRLScan be related to the
minimization of the L1-norm of derivatives [24]. By consideringthe
Euler-Lagrange equation, the minimizer of the relaxed functional is
equivalent tothe solution of a second order Partial Differential
Equation(PDE) having the form ofa reaction -diffusion equation with
a specially chosen diffusion coefficient serving asweights.
Introducing suitable boundary conditions, the problem can be
formulated ina variational form.
The application of the IRLS to (1.2) results in a double
minimization algorithm [24],which will be discussed in more detail
later.
The two main ingredient involved in our approach are :
1. An Iteratively Reweighted Least Squares algorithm is used to
reconstruct asequence that converge to the solution of the original
TV minimization problem.It is known to have linear rate of
convergence which can be modified to yield asuper linear rate of
convergence [24, 32].
2. The Discontinuous Galerkin Finite Element Method (DGFEM) is
usedto discretize the continuous, infinite dimensional problem. For
an introductionto DGFEMs, we recommend the book by Rivière [39] or
the survey article byArnoldi, Brezzi, Cockburn and Marrini [4].
We will combine these ingredients to construct a new efficient
numerical method forTV minimization problems in the following way
:
• Firstly, we analyze the conditions that relate the TV
minimization problem tothe iteratively reweighted least squares
algorithm. We will use results from [17].
• Secondly, we present the discontinuous Galerkin finite element
method and astandardized assembling of the elemental matrix
[39].
• Finally, we will present numerical experiments using this
particular combinationof methods and discuss the results.
-
CHAPTER 1. INTRODUCTION 3
The rest of the thesis is organized as follows: In Chapter 2,
starting from the total vari-ation minimization problem, we will
present a link between the TV and the iterativelyreweighted least
squares method, we then derive the Euler-Lagrange equation of
thefunctional. In Chapter 3, after recapitulating the concepts of
Discontinuous GalerkinFinite Element Method (h-version), we will
provide some analysis for existence anduniqueness and present also
some analysis on error estimates, particularly, a-prioriestimates
in L2 and the DG energy norms . In Chapter 4, we present some
numericalresults to illustrate the efficiency of the combination of
DG methods and the IRLSalgorithm. Finally, in Chapter 5, we draw
some conclusions and discuss some futurework.
-
Chapter 2
Problem Formulation and Analysis
In this chapter, we define the space of functions of bounded
variation and give someproperties that form the basis of its
applications in regularization methods followingmostly the
expositions in [2, 21, 22, 26]. In solving the minimization
functional (1.1),we introduce an associated well-posed relaxed
functional following from [25]. Finally,we derive the corresponding
Euler-Lagrange equation of the relaxed functional whichis a second
order elliptic partial differential equation.
The space of functions of bounded variation BV (Ω) plays an
important role in manyproblems in the calculus of variations. For
instance, BV -spaces are used in treatingthe minimal surface
problem [26, 5] and in the theory and numerics of
hyperbolicconservation laws. It was introduced to image processing
by Rudin, Osher and Fatemi(ROF) [40] and has subsequently found
applications in the related field of inverse prob-lems [20]. The
interesting feature of the minimization problem from the ROF
modelis that they are well suited for problems with discontinuous
solutions.
2.1 Functions of Bounded VariationIn this section, we denote by
Ω a simply connected, bounded, nonempty subset ofRn, n = 1, 2, 3
with Lipschitz continuous boundary Γ = ∂Ω. We use the symbol ∇to
denote the gradient of a smooth function u : Rn → R, i.e., ∇u =
(∂u∂x1, ..., ∂u
∂xn
).
Let C10(Ω;Rn) denote the space of vector-valued functions ϕ =
(ϕ1, ..., ϕn) whosecomponent function ϕi are
continuously-differentiable and compactly supported on Ω,i.e., each
ϕi vanishes outside some compact subset of Ω. The divergence of ϕ
is givenby
divϕ =n∑i=1
∂ϕi∂xi
.
The Euclidean norm is denoted by | · | and given by |ϕ(x)| =
[∑ni=1(ϕi(x))2]1/2 forϕ ∈ C10(Ω;Rn). The Sobolev space W 1,1(Ω)
denotes the closure of C10(Ω) with respectto the norm
4
-
CHAPTER 2. PROBLEM FORMULATION AND ANALYSIS 5
‖ u ‖W 1,1(Ω) =∫
Ω
[|u(x)|+
n∑i=1
∣∣∣∣∣∂u(x)∂xi∣∣∣∣∣]dx,
where | · | denotes the modulos.
Definition 2.1 ([26]). The total variation of a function u ∈
L1(Ω) is defined by∫Ω|Du| = sup
{∫Ωu divϕdx : ϕ ∈ C10(Ω;Rn) : ‖ϕ‖L∞(Ω) ≤ 1
}. (2.1)
Further, we define the space of functions of bounded variation
as
BV (Ω) ={u ∈ L1(Ω) :
∫Ω|Du|
-
CHAPTER 2. PROBLEM FORMULATION AND ANALYSIS 6
Theorem 2.4 (Structure theorem for BV functions). Let u ∈ BV
(Ω). Then thereexist a Radon measure µ on Ω and a µ-measurable
function σ : Ω→ Rn such that
|σ(x)| = 1 µ a.e.,∫Ωu divϕdx = −
∫Ωϕ · σ dµ,
for any ϕ ∈ C10(Ω,Rn).
Theorem 2.4 essentially follows from the Riesz representation
theorem, see [22] fordetails. Here the crucial difference to
Sobolev spaces is that the measure Du need notnecessarily be
represented as a Lebesgue measurable function.
The following properties of BV functions play a central role in
the analysis of totalvariation minimization problems, see [26] for
proofs.
Theorem 2.5 (Lower semicontinuity). Let (uj)j∈N be a sequence of
functions inBV (Ω) which converge in L1loc(Ω) to a function u,
then∫
Ω|Du| ≤ lim inf
j→∞
∫Ω|Duj|,
whereL1loc(U) =
{u : U → R | v ∈ L1(V ) ; for each V ⊂⊂ U
}. (2.5)
Recall that in a real vector space X, a set F ⊂ X is called
convex if, for any u, v ∈ F ,θu+(1−θ)v ∈ F for any θ ∈ [0, 1]. For
functions from such an X to the real numbers,we allow the value +∞,
i.e. we consider functions from X to R̄ = R ∪ {+∞}.
Definition 2.6 (Convexity). Let X be a real vector space, F ⊂ X
convex, ϕ : F → R̄.ϕ is called a convex functional if for any u, v
∈ F ,
ϕ(θu+ (1− θ)v) ≤ λϕ(u) + (1− θ)ϕ(v), (2.6)
for all θ ∈ [0, 1]. ϕ is strictly convex if (2.6) holds strictly
for all θ ∈ (0, 1).
For any ϕ : X → R̄, the set
domϕ = {u ∈ X : ϕ(u) 1 as well.
-
CHAPTER 2. PROBLEM FORMULATION AND ANALYSIS 7
Theorem 2.8 (Compactness). Let in addition Ω ⊂ Rn be a bounded
Lipschitz domain.Then
BV (Ω) ⊂⊂ Lp(Ω) for 1 ≤ p < p∗ = n/(n− 1),and
BV (Ω) ⊂ Lp∗(Ω).These compactness properties are analogous to
those of functions in W 1,1(Ω). There isanother type of compactness
corresponding to a type of convergence that also carriesinformation
on the gradient (see [5]).Definition 2.9 (BV -weak∗ convergence). A
sequence (uj)j∈N ⊂ BV (Ω) converges tou ∈ BV (Ω) in the BV -weak∗
topology, denoted by uj ∗⇀ u, if and only if
uj → u in L1(Ω) and Duj M⇀ Du,
where Duj M⇀ Du denotes a weak convergence of measures, which is
defined as∫ΩϕDuj →
∫ΩϕDu for all ϕ ∈ C0(Ω;Rn).
Theorem 2.10 (BV -weak∗ compactness.). Let (uj)j∈N ∈ BV (Ω) with
‖uj‖BV (Ω) uni-formly bounded, then there exists a subsequence
(uj
k)
k∈N and u ∈ BV (Ω) such that
uj∗⇀ u in BV (Ω).
Theorem 2.11 (Approximation). Let u ∈ BV (Ω), then there exists
a sequence (uj)j∈N ⊂BV (Ω) ∩ C∞(Ω) such that
limj→∞
∫Ω|u− uj| dx = 0,
andlimj→∞
∫Ω|Duj| =
∫Ω|Du|.
The latter result shows a difference to approximation results
for Sobolev spaces: weobtain the approximation but not in the BV
semi-norm, whereas for Sobolev spacesapproximation in the
corresponding norm is possible.
The well-known coarea formula relates the total variation of a
function to the regularityof its level sets [5, 26].Theorem 2.12
(Coarea formula). Let u ∈ BV (Ω) and Lt := {x ∈ Ω : u(x) < t},
thenLt has finite perimeter for L1 a.e. t ∈ R and∫
Ω|Du| =
∫R
(∫Ω|D1
Lt|)dt.
Conversely, u ∈ L1(Ω) and ∫R
(∫Ω|D1
Lt|)dt
-
CHAPTER 2. PROBLEM FORMULATION AND ANALYSIS 8
Since BV (Ω) contains discontinuous functions, the standard DG
finite element spacesare contained in it [36]. This will be
discussed later in Chapter 3 .
2.2 Existence and Uniqueness of MinimizersAs in [21, 40, 22,
26], we consider the minimization in BV (Ω) of the functional
J(u) = ‖Ku− g‖2L2(Ω) + 2λ∫
Ω|Du|, (2.7)
where K : L2(Ω)→ L2(Ω) is a bounded linear operator, g ∈ L2(Ω)
is given, and λ > 0is a fixed regularization parameter. Several
numerical strategies to efficiently performtotal variation
minimization have been proposed in literature [11, 10, 23, 27, 37].
How-ever, in the following we will only discuss how to adapt an
iteratively reweighted leastsquares algorithm to this particular
situation.
In order to guarantee the existence of minimizers for (2.7) we
assume that:
J is coercive in L2(Ω), i.e., there exists C > 0 such that {u
∈ L2(Ω) : J(u) ≤ C}is bounded in L2(Ω).
For smooth u, one can approximate the TV-term in (2.7) by a
smooth, convex func-tional
Eε(u) =∫
Ωϕε(|∇u|) dx, (2.8)
where ϕε ∈ C1(Ω), (i.e, continuously differentiable) and defined
as:
ϕε(z) =
12εz
2 + ε2 if 0 ≤ |z| ≤ ε,|z| if ε ≤ |z| ≤ 1/ε,
ε
2z2 + 12ε if |z| ≥ 1/ε.
Note thatϕε(z) ≥ |z| and lim
ε→0ϕε(z) = |z|, pointwise.
We now consider the following relaxed functional
Jε(u) = ‖Ku− g‖2L2(Ω) + 2λEε(u) = ‖Ku− g‖2L2(Ω) + 2λ∫
Ωϕε(|∇u|) dx, (2.9)
which approximates J pointwise from above, i.e.,
Jε(u) ≥ J(u), (2.10)
andlimε→0
Jε(u) = J(u). (2.11)
-
CHAPTER 2. PROBLEM FORMULATION AND ANALYSIS 9
Since Jε is convex and smooth, by taking the Euler-Lagrange
equations, we have thatuε is a minimizer for Jε if and only if
−λ div(ϕ′ε(|∇u|)|∇u|
∇u)
+K∗ (Ku− g) = 0. (2.12)
2.3 Euler-Lagrange Equations and a Relaxation Al-gorithm
In this section we want to provide an algorithm to compute
efficiently minimizers of theapproximating functionals Jε. First,
we want to derive the Euler-Lagrange equationsassociated to Jε. In
the following we assume that ϕε is continuously differentiable andΩ
is an open, bounded and connected subset of Rn with Lipschitz
boundary ∂Ω (see[25]).
Proposition 2.13. If u is a minimizer in W 1,2(Ω) = H1(Ω) of Jε,
then u solves thefollowing of Euler-Lagrange equations 0 =
−λdiv
(ϕ′ε(|∇u|)|∇u| ∇u
)+K∗ (Ku− g) in Ω,
ϕ′ε(|∇u|)|∇u|
∂u∂~ν
= 0 ∂Ω.(2.13)
The equations (2.13) are the necessary condition for the
computation of minimizersof Jε. The nonlinear div
(ϕ′ε(|∇u|)|∇u| ∇u
)term constitutes the main complication for the
numerical solution of these equations. Sometimes, the second
term is not easy to treat.
In order to compute efficiently solutions of (2.13), we
introduce a new functional givenby:
J̃(u,w) = ‖Ku− g‖2L2(Ω) + 2λ∫
Ω
(w|∇u(x)|2 + 1
w
)dx, (2.14)
where u ∈ W 1,2(Ω) := V and w ∈ L2(Ω) is such that ε ≤ w ≤ 1/ε
almost everywhere.While the variable u is the function to be
reconstructed, the function w is called thegradient weight.
For any given u(0) ∈ V and w ∈ L2(Ω) (for example w(0) := 1 ),
we define the followingiterative double-minimization algorithm
:{
u(k+1) = arg minu∈V J̃(u,w(k)),w(k+1) = arg minε≤w≤ 1
εJ̃(u(k+1), w). (2.15)
We have the following convergence result (see [25]). We will
include the proof for thesake of completeness of this thesis.
-
CHAPTER 2. PROBLEM FORMULATION AND ANALYSIS 10
Theorem 2.14. The sequence (uk)k∈N has subsequences that
converge to a minimizeruε := u(∞) of Jε. If Jε has a unique
minimizer u∗, then u(∞) = u∗ and the full sequence(uk)k∈N converges
to u∗.
Proof. The proof we present here follows from [25]. Observe
that
J̃(u(k), w(k))− J̃(u(k+1), w(k+1)) = (J̃(u(k), w(k))− J̃(u(k+1),
w(k)))︸ ︷︷ ︸Ak
+ (J̃(u(k+1), w(k))− J̃(u(k+1), w(k+1)))︸ ︷︷ ︸Bk
≥ 0.
Therefore J̃(u(k), w(k)) is a non-increasing sequence and
moreover it is bounded frombelow, since
infε≤w≤ 1
ε
∫Ω
(w|∇u(x)|2 + 1
w
)dx ≥ 0.
This implies that J̃(u(k), w(k)) converges. Moreover, we can
write
Bk =∫
Ωc(w(k), |∇u(k+1)(x)|)− c(w(k+1), |∇u(k+1)(x)|),
where c(t, z) := tz2 + 1t. By Taylor’s formula, we have
c(w(k), z) = c(w(k+1), z) + ∂c∂t
(w(k+1), z)(w(k) − w(k+1)) + 12∂2c
∂t2(ξ, z)|w(k) − w(k+1)|2,
for ξ ∈ conv(w(k), w(k+1)) (the segment between w(k) and
w(k+1)). By definition ofw(k+1), and taking into account that ε ≤
w(k+1) ≤ 1
ε, we have
∂c
∂t(w(k+1), |∇u(k+1)(x)|)(w(k) − w(k+1)) ≥ 0,
and ∂2c∂t2
(t, z) = 2t3≥ 2ε3, for any t ≤ 1/ε. This implies that
J̃(u(k), w(k))− J̃(u(k+1), w(k+1)) ≥ Bk ≥ ε3∫
Ω|w(k)(x)− w(k+1)(x)|2dx,
and since J̃(u(k), w(k)) is convergent, we have
‖w(k) − w(k+1)‖L2(Ω)
→ 0, (2.16)
for n→∞. Since u(k+1) is a minimizer of J(u,w(k)), it solves the
following system ofvariational equation∫
Ω
(w(k)∇u(k+1)(x) · ∇ϕ(x) + λ̃(Ku(k+1) − g)(x)Kϕ(x)
)= 0, (2.17)
for all ϕ ∈ V and λ̃ := λ−1. Therefore we can write
-
CHAPTER 2. PROBLEM FORMULATION AND ANALYSIS 11
∫Ω
(w(k+1)∇u(k+1)(x) · ∇ϕ(x) + λ̃(Ku(k+1) − g)(x)Kϕ(x)
)=∫
Ω(w(k+1) − w(k))∇u(k+1)(x) · ∇ϕ(x),
and for 1p
+ 1q
+ 12 = 1, we have∣∣∣∣∫Ω
(w(k+1)∇u(k+1)(x) · ∇ϕ(x) + (Ku(k+1) − g)(x)Kϕ(x)
)∣∣∣∣≤ ‖w(k+1) − w(k)‖Lp‖∇u(k+1)‖Lq‖∇ϕ‖L2 .
By monotonicity of (J(u(k+1), w(k+1)))k, and since w(k+1) =
ϕ′ε(|∇u(k+1)|)|∇u(k+1)| , we have
J̃(u1, w0) ≥ J̃(u(k+1), w(k+1)) = Jε(u(k+1)) ≥ J(u(k+1)) ≥
C|∇u|(Ω)≥ C‖∇u(k+1)‖L2(Ω).
Moreover, since Jε(u(k+1)) ≥ J(u(k+1)) and J is coercive, we
have that ‖u(k+1)‖L2(Ω)and ‖∇u(k+1)‖L2 are uniformly bounded with
respect to k. Therefore, using (2.16), wecan conclude that∫
Ω
(w(k+1)∇u(k+1)(x) · ∇ϕ(x) + λ̃(Ku(k+1) − g)(x)Kϕ(x)
)→ 0,
for k → ∞, and there exists a subsequence (u(kp+1))p∈N that
converges in V to afunction u(∞). Since w(kp+1) = ϕ′ε(|∇u
(kp+1)|)|∇u(kp+1)| , and by taking the limit for p → ∞, we
obtain
−λdiv(ϕ′ε(|∇u(∞)|)|∇u(∞)|
∇u(∞))
+K∗(Ku(∞) − g
)= 0. (2.18)
This is the Euler-Lagranage equation (2.12) associated to the
functional Jε and there-fore u(∞) is a minimizer of Jε.
Assume now that Jε has a unique minimizer u∗. Then necessarily
u(∞) = u∗. Sinceevery subsequence of (uk)k has a subsequence
converging to u∗, the full sequence (uk)kconverges to u∗.
Since both Jε and J̃(·, w) admit minimizers, their uniqueness is
equivalent to theuniqueness of the solutions of the corresponding
Euler-Lagrange equations. If unique-ness of the solution is
satisfied, then the algorithm (2.15) can be equivalently
reformu-lated as the following two-step iterative procedure: Given
w(0) ∈ L∞(Ω), for k = 0, 1, ...define :
-
CHAPTER 2. PROBLEM FORMULATION AND ANALYSIS 12
• Find u(k+1) ∈ V :∫Ω
(w(k+1)∇u(k+1)(x) · ∇ϕ(x) + λ̃(Ku(k+1) − g)(x)Kϕ(x)
)= 0, ∀ϕ ∈ V.
• Compute directly w(k+1) by
w(k+1) = ε ∨ 1|∇u(k+1)|
∧ 1ε
:= min(
max(ε,
1|∇u(k+1)|
),1ε
).
By a standard fixed point argument, the solution of the equation
is unique for λ ∼ ε.The condition λ ∼ ε is acceptable only for
those applications where the constraints onthe data are weak, e.g.,
when the data is affected by a strong noise (see [25]).
The following result establishes the convergence of the
algorithm.
Theorem 2.15. Let us assume that (εj)j∈N is a sequence of
positive numbers mono-tonically converging to zero. The
accumulation points of the sequence (εj)j∈N of theminimizers of Jεj
are minimizers of J.
Remark 2.16. The proof requires the notion of Γ−Convergence. The
minimizers ofa relaxed functional J̄ can be approximated by the
minimum points of functionals thatare defined in W 1,2(Ω) (see [5],
Section 2.1.4).
-
Chapter 3
DG Finite Element Discretization
In [4], Arnold et al. present a uniform analysis of the
Discontinuous Galerkin FiniteElement Methods. In this chapter, we
start with definitions of some function spacesand derive the
standard and the DG variational formulations of our model
problemwhich follows mostly the work of Groosmann, Roos and Stynes
[28] and B. Rivière[39]. Furthermore, we present a short
introduction to the theory of DG finite elementdiscretization
applied to the variational setting of our PDE. Particularly, we
investigatethe discretization errors which we gain from the
discontinuous Galerkin finite elementmethod. Finally, we present
the DG-version of the iteratively re-weighted least
squaresalgorithm for solving our TV minimization problem.
3.1 Some Basic Function SpacesWe start with the definition of
some functions spaces [1].
Definition 3.1. Let Ω ⊂ Rn be a bounded Lipschitz domain. The
Lebesgue spaceLp(Ω) is given by
Lp(Ω) ={v : Ω→ R | ‖v‖Lp(Ω)
-
CHAPTER 3. DG FINITE ELEMENT DISCRETIZATION 14
whereDαv = ∂α1x1 , ..., ∂
αnxn :=
∂|α|v
∂α1x1, ..., ∂αnxn
denotes the weak derivative of the order α, where α = (α1, ...,
αn) ∈ Nn is a multi-index and |α| = ∑ni=1 αi.The Sobolev space
Hs(Ω) is equipped with the norm
‖v‖s := ∑
0≤|α|≤s‖Dαv‖2L2(Ω)
1/2 . (3.1)Correspondingly, a semi-norm on this space is defined
as
|v|s :=∑|α|=s‖Dαv‖2L2(Ω)
1/2 . (3.2)3.2 DG Variational FormulationsIn this section, we
will derive the standard and the DG variational formulations.
Fur-thermore, we will define some special function spaces needed
for the DG formulations.Let us consider the following Neumann
problem as model problem: Find u such that
−∇ · (w∇u) + λ̃u = λ̃g in Ω, (3.3)w∇u · ~ν = 0 on Γ = ∂Ω.
(3.4)
Here ~ν is the outer unit normal, w ∈ L∞(Ω) is assumed to be
unifomly, g ∈ L2(Ω), andΩ ⊂ R2 is a bounded polygonal Lipschitz
domain. We recall from (2.17) that λ̃ = λ−1,where λ is a positive
regularization parameter. The standard variational formulationof
the Neumann problem (3.3) - (3.4) reads as follows: Find u ∈ H1(Ω)
such that∫
Ω(w∇u · ∇v + λ̃uv) dx =
∫Ωλ̃gv dx ∀v ∈ H1(Ω). (3.5)
The existence and uniqueness of a solution of (3.5) immediately
follows from Lax-Milgram’s lemma.
Let us now derive the DG variational formulation. We start with
a decomposition Tof Ω into triangles or rectangles T such that
Ω =⋃T∈T
T and Ti ∩ Tj = ∅ if Ti, Tj ∈ T , Ti 6= Tj.
We assume that T is a regular triangulation, i.e., the
intersection of any two elementsis either empty or a common vertex
or edge. We also assume that the elements areshape regular i.e.,
there exists a constant γ such that
hT
ρT
≤ γ, ∀T ∈ T ,
-
CHAPTER 3. DG FINITE ELEMENT DISCRETIZATION 15
where hT
denotes the diameter of the element T and ρT
is the diameter of the largestball inscribed in T .
To each element T ∈ T we assign a non-negative integer sT and
define the “broken”Sobolev space of the order s = {s
T: T ∈ T } by
Hs(Ω; T ) := {v ∈ L2(Ω) : v|T ∈ HsT (T ), ∀T ∈ T }.
The associated norm and seminorm are
‖v‖s,T =(∑T∈T‖v‖2
Hs
T (T )
) 12
and |v|s,T =(∑T∈T|v|2
HsT (T )
) 12
, respectively.
If sT
= s for all T ∈ T , we write ‖v‖s,T and |v|s,T instead of
‖v‖Hs(Ω;T ) and |v|Hs(Ω;T ) .If v ∈ H1(Ω; T ) then the composite
gradient ∇T v of a function v is defined by(∇T v)|T = ∇(v|T ), T ∈
T .
In the following, we assume that each element T ∈ T is the
affine image of a rectangularreference element T̂ (unit square),
i.e., T = FT (T̂ ). The finite element space is definedby
Vh(Ω; T ,F) = {v ∈ L2(Ω) : v|T ◦ FT ∈ Q1(T̂ )}, (3.6)where F =
{F
T: T ∈ T } and Q1(T̂ ) is the space of linear polynomials of
degree one
in each space direction on T̂ . Note that the functions in Vh ≡
Vh(Ω; T ,F) may bediscontinuous across element edges.
Let E be the set of all edges of the given triangulation T ,
with Eint ⊂ E the set of allinterior edges e ∈ E in Ω. Set Γint =
{x ∈ Ω : x ∈ e for some e ∈ Eint}. Let theelements of T be numbered
sequentially: T1, T2, .... Then for each e ∈ Eint there
existindices i and j such that i > j and e = Ti ∩ Tj. Set T :=
Ti and T ′ := Tj . Define thejump (which depends on the enumeration
of the triangulation) and average of eachfunction v ∈ H1(Ω, T ) on
e ∈ Eint by
[v]e = (v|∂T∩e − v|∂T ′∩e), {v}e =12(v|∂T∩e + v|∂T
′∩e).
Furthermore, to each edge e ∈ Eint we assign a unit normal
vector ~ν directed from Tto T ′. If instead e ⊂ Γ then we take the
outward-pointing unit normal vector ~ν on Γ.When there is no danger
of misinterpretation we omit the indices in [v]e and {v}e.
For simplicity, we shall assume that the solution u of (3.5)
belongs to H2(Ω) ⊂H2(Ω; T ). Let us mention that, for more general
problems, it is standard to assumethat u ∈ H2(Ω; T ) and that both
u and ∇u ·~ν are continuous across all interior edges,where ~ν is a
normal to the edge. In particular, we have
[u]e = 0, {u}e = u, e ∈ Eint.
-
CHAPTER 3. DG FINITE ELEMENT DISCRETIZATION 16
Multiply the differential equation (3.3) by a (possibly
discontinuous) test functionv ∈ H1(Ω, T ) and integrate over Ω, we
obtain∫
Ω
(−∇ · (w∇u) + λ̃u
)v dx =
∫Ωλ̃gv dx. (3.7)
First, we consider the term −∇ · (w∇u) in (3.7). Let ~νT
denote the outward-pointingunit normal to ∂T for each T ∈ T .
Integration by parts and elementary transformationsgive us
∫Ω
(−∇ · (w∇u))v dx =∑T∈T
∫T
(w∇u) · ∇ v dx−∑T∈T
∫∂T
(w∇u · ~νT) v ds
=∑T∈T
∫T
(w∇u) · ∇v dx−∑
e∈E∩Γ
∫e(w∇u · ~ν) v ds
−∑e∈Eint
∫e
(((w∇u · ~ν
T)v)|∂T∩e + ((w∇u · ~νT ′ )v)|∂T ′∩e
)ds.
Let ~ν be the unit normal vector points from T to T ′, then the
sum of the integralsover e ∈ Eint can be written as∑
e∈Eint
∫e(((w∇u · ~νT )v)|∂T∩e + ((w∇u · ~νT ′)v)|∂T ′∩e) ds
=∑e∈Eint
∫e(((w∇u · ~ν)v)|∂T∩e − ((w∇u · ~ν)v)|∂T ′∩e) ds
=∑e∈Eint
∫e({w∇u · ~ν}e [v]e + [w∇u · ~ν]e{v}e) ds (3.8)
=∑e∈Eint
∫e{w∇u · ~ν}e [v]e ds.
Here the basic relation
(w1 · ~ν1)z1 + (w2 · ~ν2)z2 = {w · ~ν}[z] + [w · ~ν]{z},
(3.9)
is used in (3.8). Applying the boundary condition and using the
abbreviation∑e∈Eint
∫e{w∇u · ~ν}e [v]e ds =
∫Γint{w∇u · ~ν} [v] ds, (3.10)
we get∫Ω
(−∇ · (w∇u))v dx =∑T∈T
∫T
(w∇u) · ∇ v dx−∫
Γint{w∇u · ~ν} [v] ds. (3.11)
Finally, we add the terms∫Γint
σ[u] [v] ds and τ∫
Γint[u]{w∇v · ~ν} ds, (3.12)
-
CHAPTER 3. DG FINITE ELEMENT DISCRETIZATION 17
where the choices of τ = ±1, 0 will define different DG methods
(see below). Theterms (3.12) vanish for the exact solution u ∈
H1(Ω, T ). The penalty parameter σ ispiecewise constant, i.e.,
σ|e = σe := κh−1e , e ∈ E , κ ≥ 0.
Finally, we obtain∫Ω
(−∇ · (w∇u))v dx =∑T∈T
∫T
(w∇u) · ∇ v dx−∫
Γint{w∇u · ~ν} [v] ds.
+ τ∫
Γint[u]{w∇v · ~ν} ds+
∫Γint
σ[u] [v] ds.
We can now give the primal formulation of the discontinuous
Galerkin method withinterior penalties, and its relation to the
standard variational formulation (3.5).
Theorem 3.3. If u ∈ H2(Ω, T ) is a solution to (3.5), then u
also solves
A(u, v) = g(v), ∀v ∈ H2(Ω, T ), (3.13)
where the linear form is given by the relation
g(v) =∑T∈T
∫Tλ̃gv dx, (3.14)
and the bilinear form is given by the expression
A(u, v) =∑T∈T
(∫Tw∇u · ∇v dx+
∫Tλ̃uv dx
)−∫
Γint{w∇u · ~ν} [v] ds
+ τ∫
Γint[u]{w∇v · ~ν} ds+
∫Γint
κh−1e [u][v] ds. (3.15)
The following choices for τ and κ define some of the well known
DG methods :
• τ = −1 and κ ≥ κ0 sufficiently large define the symmetric
interior penalty(SIPG) method [4, 3, 44].
• τ = +1 and κ > 0 define the non-symmetric interior penalty
(NIPG) method[4, 39].
• τ = +1 and κ = 0 define the method of Baumann and Oden [6,
35].
The proof of Theorem 3.3 follows by taking an arbitrary function
v ∈ H2(Ω, T ), mul-tiplying (3.3) by v and integrate by parts over
each element. The desired consistencyis achieved by applying the
algebraic equality (3.9) and the boundary condition from(3.4) (see
[39] for detailed proof ).
-
CHAPTER 3. DG FINITE ELEMENT DISCRETIZATION 18
3.3 Some Basic PropertiesNext, we look at some basic properties
of these methods ( see [39, 28, 19]). Moreprecisely we will
consider the SIPG method. Firstly, we state the trace inequality
asa requirement for the proof of the coercivity property and then
proceed with someother properties.
Theorem 3.4. Let T be a bounded polygonal domain with boundary
∂T and diameterh
T. Let e be an edge and ~ν a unit outward normal vector to e.
Let p > 0 be an integer.
There exists a constant C independent of hT
such that
‖v‖L2(e)≤ C h−1/2
T‖v‖
L2(T ), ∀v ∈ Qp(T ), ∀e ⊂ ∂T, (3.16)
and‖∇v · ~ν‖
L2(e)≤ C h−1/2
T‖∇v‖
L2(T ), ∀v ∈ Qp(T ), ∀e ⊂ ∂T. (3.17)
Theorem 3.5 (Consistency, [39]). All three of the above methods
are consistent. Thatis, if the exact solution u to (3.5) is in
Hs(Ω) for some s > 3/2 then we have
A(u, v) = g(v), ∀v ∈ H2(Ω, T ). (3.18)
We now look at the continuity and coercivity properties of the
bilinear form A(·, ·)with respect to the DG norm:
‖|v|‖ =(∑T∈T
∫T
(w∇v) · ∇v dx+∫
Ωλ̃v2 dx+
∫Γint
κh−1e [v]2 ds
) 12
, ∀v ∈ H1(Ω, T ).
(3.19)We will also show the proof for coercivity for the sake of
completeness of the thesis.
Definition 3.6 (Coercivity). The bilinear form A(·, ·) is
coercive on Vh if there existsa constant C > 0 such that
A(v, v) ≥ C‖|v|‖2 ∀v ∈ Vh. (3.20)
For the SIPG bilinear form, we have
A(v, v) =∑T∈T
(∫Tw(∇v)2 dx+
∫Tλ̃v2 dx
)− 2
∫Γint{w∇v · ~ν}[v] ds
+∫
Γintκh−1e [v]
2 ds. (3.21)
By using the Cauchy-Schwarz inequality1 we obtain an upper bound
for the secondterm ∫
e{w∇v · ~ν}[v] ds ≤ ‖{w∇v · ~ν}‖
L2(e)‖[v]‖
L2(e).
1Cauchy-Schwarz inequality : ∀x, y ∈ L2(Ω), |(x, y)Ω| ≤
‖x‖L2(Ω)‖y‖L2(Ω).
-
CHAPTER 3. DG FINITE ELEMENT DISCRETIZATION 19
Next, we estimate the average of the fluxes for an interior edge
e shared by T and T ′as follows
‖{w∇v · ~ν}‖L2(e)≤ 12‖w∇v|T ‖L2(e) +
12‖w∇v|T ′‖L2(e) .
Using the trace inequality (3.17), we find
‖{w∇v · ~ν}‖L2(e)≤ 12‖w∇v|T ‖L2(e) +
12‖w∇v|T ′‖L2(e)
≤ C∗2 h− 12T ‖w∇v‖L2(T ) +
C∗2 h
− 12T ′ ‖w∇v‖L2(T ′) , (3.22)
where C∗ is independent of v and h. Also we have
|he| ≤ hT ≤ h, ∀e ⊂ ∂T, T ∈ T .
Using (3.22), we obtain∫e{w∇v · ~ν}[v] ≤ C∗2 |he|
12(h−
12
T‖w∇v‖
L2(T )+ h− 12
T ′‖w∇v‖
L2(T ′)
)×(|he|−
12‖[v]‖
L2(e)
)≤ C∗2
(|he|
12h−
12
T+ |he|
12h−
12
T ′
)(‖w∇v‖2
L2(T )+ ‖w∇v‖2
L2(T ′)
) 12
×(|he|−
12‖[v]‖
L2(e)
)≤ C∗
(‖w∇v‖2
L2(T )+ ‖w∇v‖2
L2(T ′)
) 12 (|he|−
12‖[v]‖
L2(e)
).
A similar bound is obtained if e is a boundary edge. Using the
Cauchy-Schwarzinequality and Young’s inequality for δ > 0 2, we
obtain∑e∈Eint
∫e{w∇v · ~ν}[v] ds ≤ C∗
∑e∈Eint
(‖w∇v|
T‖2
L2(T )+ ‖w∇v|
T ′‖2
L2(T ′)
) 12 (|he|−
12‖[v]‖
L2(e)
)
≤ C∗(∑T∈T‖w∇v‖2
L2(T )
) 12 ∑e∈Eint
|he|−1‖[v]‖2L2(e)
12
≤ C∗w12
(∑T∈T‖w
12∇v‖2
L2(T )
) 12 ∑e∈Eint
|he|−1‖[v]‖2L2(e)
12
≤ δ2∑T∈T‖w
12∇v‖2
L2(T )+ C̃2δ
∑e∈Eint
|he|−1‖[v]‖2L2(e)
We obtain a lower bound for A(v, v) :
A(v, v) ≥ (1− δ)∑T∈T‖w
12∇v‖2
L2(T )+ ‖λ̃‖
L∞(Ω)‖v‖2L2(Ω))
+(κ− C̃
δ
) ∑e∈Eint
|he|−1‖[v]‖2L2(e)
2Young’s inequality : ∀a, b ∈ R, ∀δ > 0, ab ≤ δ2a2 + 12δ
b
2
-
CHAPTER 3. DG FINITE ELEMENT DISCRETIZATION 20
where the positive constant C̃ is independent of hT. We achieve
the coercivity result
with C = 1/2 from (3.20) if we choose δ = 1/2 and κ∗ = 1/2 +
C̃/δ.
Definition 3.7 (Continuity, [8]). If κ > 0 for all ∂T , then
the bilinear form A(·, ·) iscontinuous on Vh equipped with the
energy norm ‖| · |‖, if there exists C̃ > 0 such that
A(u, v) ≤ C̃ ‖|u|‖ ‖|v|‖ ∀ u, v ∈ Vh.
Using the Cauchy-Schwarz inequality, we have
|A(u, v)| ≤∣∣∣∣∣∑T∈T
(∫Tw∇u · ∇v dx+
∫Tλ̃uv dx
)∣∣∣∣∣+∣∣∣∣∫
Γint{w∇u · ~ν}[v] ds
∣∣∣∣+∣∣∣∣∫
Γint{w∇v · ~ν}[u] ds
∣∣∣∣+ ∣∣∣∣∫Γint
κh−1e [u][v] ds∣∣∣∣ .
≤∑T∈T‖w
12∇u‖L2(T )‖w
12∇v‖L2(T ) + ‖λ̃
12u‖L2(Ω)‖λ̃
12v‖L2(Ω)
+∑e∈Eint
‖{w∇u · ~ν}‖L2(e) ‖[v]‖L2(e) +∑e∈Eint
‖{w∇v · ~ν}‖L2(e) ‖[u]‖L2(e)
+∑e∈Eint
κh−1e ‖[u]‖L2(e) ‖[v]‖L2(e)
≤ Cw12∑T∈T
(‖w
12∇u‖2L2(T )
) 12(‖w
12∇v‖2L2(T )
) 12 +
(‖λ̃
12u‖2L2(Ω)
) 12(‖λ̃
12v‖2L2(Ω)
) 12
+ Cw 12∑e∈Eint
(‖w
12∇u‖2L2(T )
) 12(h−1e ‖[v]‖2L2(e)
) 12 + Cw 12
∑e∈Eint
(‖w
12∇v‖2L2(T )
) 12
×(h−1e ‖[u]‖2L2(e)
) 12 +
∑e∈Eint
(κh−1e ‖[u]‖2L2(e)
) 12(κh−1e ‖[v]‖2L2(e)
) 12 (3.23)
≤ C̃
∑T∈T‖w
12∇u‖2L2(T ) + ‖λ̃
12u‖2L2(Ω) +
∑e∈Eint
κh−1e ‖[u]‖2L2(e)
12
×
∑T∈T‖w
12∇v‖2L2(T ) + ‖λ̃
12v‖2L2(Ω) +
∑e∈Eint
κh−1e ‖[v]‖2L2(e)
12≤ C̃‖|u|‖ ‖|v|‖ (3.24)
where C̃ = C̃(w). In (3.23), we have used the trace inequality
(3.22) to obtain (3.24).
Remark 3.8. In general, the bilinear form is not continuous on
the “broken” spaceH2(Ω, T ) with respect to the energy norm [39]
and C is a generic constant independentof h and may have different
values in different places.
Combining the coercivity result (3.20) and continuity result
(3.23), we have the fol-lowing theorem that establishes the
existence and uniqueness of the discontinuousGalerkin solution.
-
CHAPTER 3. DG FINITE ELEMENT DISCRETIZATION 21
Theorem 3.9. Let A : Vh × Vh → R be a bilinear form of the SIPG
(3.21). If κis uniformly bounded below by a sufficiently large
positive constant κ∗ for all edges e,then the DG solution exists
and is unique, where the DG space Vh is defined by (3.6).
Proof. We will refer the reader to Section 2.7.4, [39] for the
proof in more detail.
3.4 The DG Finite Element EquationsA comprehensive introduction
to the finite element method can be found in [7, 8, 39].We present
a short outline for the most important features. The discrete
problem isgiven by the Galerkin scheme:
Find uh ∈ Vh : A(uh, vh) = g(vh), ∀vh ∈ Vh (3.25)
The index h stands for the discretization parameter and
indicates that we want toachieve convergence of the discrete
solution for h→ 0.
The space Vh in (3.6) is of finite dimension, thus it must have
a finite basis. For aquadrilateral mesh and bilinear elements, n =
dimVh = 4Nh where Nh is the numberof elements. Let φi denote the
basis functions, i.e., Vh = span{φi : i = 1, ..., n}.Consequently,
the solution uh is of the form
uh(x) =n∑i=1
uihφi(x), for x ∈ Ω, (3.26)
with coefficients uih ∈ R. Using the notation
uh := (uih)ni=1 ∈ Rn, (3.27)gh
:= (g(φi))ni=1 ∈ Rn, (3.28)Ah[i, j] := A(φj, φi), Ah ∈ Rn×n,
(3.29)
we arrive at the linear algebraic system :
Find uh ∈ Rn : Ahuh = gh, (3.30)
which is equivalent to the original discrete problem
(3.25).Since the bilinear formA(·, ·) is positive definite, the
matrixAh is also positive definite.This is because, for uh ∈ Rn, we
can write
uThAhuh = A n∑j=1
ujφj,n∑i=1
uiφi
.So the positive definiteness of A(·, ·) implies uThAhuh > 0
and (uThAhuh = 0⇐⇒ uh =0). Now, since Ah is positive definite, it
is invertible. Therefore, the linear algebraicsystem has exactly
one solution uh ∈ Rn. Thus, (3.25) has exactly one solution uh ∈
Vh.
-
CHAPTER 3. DG FINITE ELEMENT DISCRETIZATION 22
3.5 A Priori Error EstimatesIn this section, we state
approximation results in the space of polynomials of degreeless
than p in each space direction and also recall the trace
inequalities which enablesus to prove discretization error
estimates, see e.g [39].
Theorem 3.10. Let T be a rectangle in 2D. Let v ∈ Hs(T ) for s ≥
1. Let p ≥ 0 be aninteger. There exists a constant C independent of
v and h and a function ṽ ∈ Qp(T )such that
‖ṽ − v‖Hq(T ) ≤ C h
min(p+1,s)−q|v|Hs(T ) ∀0 ≤ q ≤ s, (3.31)
where h = diam(T ).
The next result yields an approximation that conserves the
average of the normal fluxon each edge.
Theorem 3.11. Let T be a rectangle in 2D. Denote by ~ν the
outward normal to T.Let v ∈ Hs(T ) for s ≥ 2 and p > 0. There
exists an approximation ṽ ∈ Qp(T ) of vsatisfying
∫e∇(ṽ − v) · ~ν = 0 ∀e ⊂ ∂T and the optimal error bounds
‖∇i(ṽ − v)‖L2(T )
≤ Chmin(p+1,s)−i|v|Hs(T ) ∀i = 0, 1, 2, (3.32)
We now state and show an a priori energy error estimate in the
DG-norm.
Theorem 3.12 (Energy error estimate, [39]). Let A(·, ·) be the
SIPG bilinear formfrom Theorem (3.3) with κ > 0. If the exact
solution u to the model problem (3.3) isin H2(T ) and uh ∈ Vh
satisfies (3.25) then there exist a constant C̃ independent of
hsuch that
‖|u− uh|‖ ≤ C̃ h ‖u‖
H2(T ). (3.33)
Proof. We will present the main steps of the proof given in
([39], Sect. 2.8). FromProposition 3.5, we know that
A(u, v) = g(v) ∀v ∈ H2(T ).
Since A(·, ·) is bilinear and
A(uh, v) = g(v) ∀v ∈ Vh,
we get the Galerkin orthogonality
A(uh− u, v) = 0 ∀v ∈ Vh.
Let χ ∈ H2(T ) denote the error: χ := uh − u. We have to find an
upper bound forA(χ, χ) which is independent of uh. To achieve this,
let ũ ∈ Vh be a function that
-
CHAPTER 3. DG FINITE ELEMENT DISCRETIZATION 23
approximates the exact solution u. Using Galerkin orthogonality,
we can write
A(χ, χ) = A(χ, uh)−A(χ, u) = −A(χ, u)= A(χ, ũ)−A(χ, u)= A(χ,
ũ− u).
As u, ũ ∈ C0(Ω̄), we see [ũ− u] = 0 on any interior edge and
ũ is a continuousinterpolant. Therefore, most terms vanish when
A(χ, ũ− u) is expanded. We obtain :
A(χ, χ) = A(χ, ũ− u) =∑T∈T
∫T
(w∇χ · ∇(ũ− u) + λ̃(ũ− u)χ
)−
∑e∈Eint
∫e
[χ]{w∇(ũ− u) · ~ν} −∑e∈Eint
∫e{w∇χ · ~ν} [ũ− u]
+∑e∈Eint
∫eκh−1e [χ] [ũ− u].
Using the Cauchy Schwarz inequality, Young’s inequality and the
approximation theo-rems Theorem 3.10 and Theorem 3.11, we obtain a
bound for the first term as follows:
∣∣∣∣∣∑T∈T
∫Tw∇χ · ∇(ũ− u)
∣∣∣∣∣ ≤ w 12(∑T∈T‖∇(ũ− u)‖2
L2(T )
) 12(∑T∈T‖w
12∇χ‖2
L2(T )
) 12
≤ C2δ‖∇(ũ− u)‖2L2(T )
+ δ2‖w12∇χ‖2
L2(T ).∣∣∣∣∣∑
T∈T
∫Tλ̃(ũ− u)χ
∣∣∣∣∣ ≤ ‖λ̃‖L∞(T )(∑T∈T‖ũ− u‖2
L2(T )
) 12(∑T∈T‖χ‖2
L2(T )
) 12
≤ C2δ‖λ̃‖L∞(Ω)‖ũ− u‖2L2(T )
+ δ2‖χ‖2L2(T )
.
By the triangle inequality, we obtain∣∣∣∣∣∑T∈T
∫T
(w∇χ · ∇(ũ− u) + λ̃(ũ− u)χ
)∣∣∣∣∣ ≤ C2δ (1 + ‖λ̃‖L∞(Ω) )‖ũ− u‖2H1(T ) + δ2‖|χ|‖2≤
C̃h2‖u‖2
H2(T )+ δ2‖|χ|‖
2.
The second term can be bounded by applying the Cauchy Schwarz
inequality and
-
CHAPTER 3. DG FINITE ELEMENT DISCRETIZATION 24
Young’s inequality to obtain:∣∣∣∣∣∣∑e∈Eint
∫e[χ]{w∇(ũ− u) · ~ν}
∣∣∣∣∣∣ ≤∑e∈Eint
‖[χ]‖L2(e)‖{w∇(ũ− u) · ~ν}‖
L2(e)
=∑e∈Eint
h−12‖[χ]‖
L2(e)h
12‖{w∇(ũ− u) · ~ν}‖
L2(e)
≤
∑e∈Eint
h−1‖[χ]‖2L2(e)
12 ∑e∈Eint
h‖{w∇(ũ− u) · ~ν}‖2L2(e)
12
≤ C
∑e∈Eint
h−1‖[χ]‖2L2(e)
12 (∑T∈T
hh−1T ‖∇(ũ− u)‖2L2(T )
) 12
≤ δ2‖|χ|‖2 + C2δ‖∇(ũ− u)‖
2L2(T )
≤ δ2‖|χ|‖2 + C̃h2‖u‖2
H2(T ).
Using the approximation properties Theorem 3.10 and Theorem 3.11
and the traceinequalities from Theorem 3.4, we obtain the
bound∣∣∣∣∣∣
∑e∈Eint
∫e[χ]{w∇(ũ− u) · ~ν}
∣∣∣∣∣∣ ≤ C̃ h2‖u‖2H2(T ) + δ2‖|χ|‖2.In general, when ũ is not
continuous, we proceed similarly for the third and fourthterms with
(ũ−u) unknown on the interfaces. Using jump norm, the trace
inequalitiestogetherer with definitions of the trace operators
(average and jump), and using thecoercivity result (3.20), we
obtain
‖|χ|‖2 ≤ A(χ, χ) = A(χ, ũ− u) ≤ C̃h2‖u‖2H2(T ),
where C̃ = C̃(λ̃, w).
Next, we prove an error estimate in the L2 norm.
Theorem 3.13 (L2 error estimate). Assume that Theorem 3.12 holds
and that ourmodel problem is H2-coercive then there exists a
constant C̃ independent of h such that
‖u− uh‖L2(Ω) ≤ C̃h2‖u‖
H2(T ).
Proof. The proof we present follows [39]. Consider the dual
problem
−∇ · (w∇v) + λ̃v = u− uh in Ω,w∇v · ~ν = 0 on Γ = ∂Ω.
We assume that v ∈ H2(Ω) and that there is a constant C that
depends on Ω suchthat
‖v‖H2(Ω)
≤ C‖u− uh‖L2(Ω) . (3.34)
-
CHAPTER 3. DG FINITE ELEMENT DISCRETIZATION 25
Denote χ = u− uh,
‖χ‖2L2(Ω)
=∑T∈T
∫T
(−∇ · (w∇v) + λ̃v)χ.
Integrating by part on each element and applying the basic
inequality (3.9) yields
‖χ‖2L2(Ω)
=∑T∈T
(∫Tw∇v · ∇χdx+
∫Tλ̃vχ dx
)−
∑e∈Eint
∫e{w∇v · ~ν}[χ]
−∑e∈Eint
∫e{w∇χ · ~ν}[v]
=∑T∈T
(∫Tw∇v · ∇χdx+
∫Tλ̃vχ dx
)−
∑e∈Eint
∫e{w∇v · ~ν}[χ],
since v is continuous. By subtracting the Galerkin orthogonality
equation, we obtainfor ṽ ∈ Vh
‖χ‖2L2(Ω)
=∑T∈T
∫T
(w∇χ · ∇(ṽ − v) + λ̃(ṽ − v)χ
)−
∑e∈Eint
∫e{w∇v · ~ν}[χ] +
∑e∈Eint
∫e{w∇ṽ · ~ν}[χ]
−∑e∈Eint
∫e{w∇χ · ~ν}[ṽ]−
∑e∈Eint
∫eκh−1e [χ][ṽ].
By noting that v ∈ H2(Ω) and choosing ṽ ∈ C0(Ω), we are left
with
‖χ‖2L2(Ω)
=∑T∈T
∫T
(w∇χ · ∇(ṽ − v) + λ̃(ṽ − v)χ
)−
∑e∈Eint
∫e{w∇v · ~ν}[χ] +
∑e∈Eint
∫e{w∇ṽ · ~ν}[χ]
=∑T∈T
∫T
(w∇χ · ∇(ṽ − v) + λ̃(ṽ − v)χ
)+
∑e∈Eint
∫e{w∇(ṽ − v) · ~ν}[χ].
(3.35)The first term is bounded in the following way :∣∣∣∣∣∑
T∈T
∫Tw∇(ṽ − v) · ∇χ
∣∣∣∣∣ ≤ C ∑T∈T‖w∇(ṽ − v)‖
L2(T )‖∇χ‖
L2(T )
≤ C̃∑T∈T‖∇(ṽ − v)‖
L2(T )‖∇χ‖
L2(T )
≤ C̃ ‖∇(ṽ − v)‖L2(Ω)‖∇χ‖
L2(Ω).∣∣∣∣∣∑
T∈T
∫Tλ̃(ṽ − v)χ
∣∣∣∣∣ ≤ C‖λ̃‖L∞(Ω)‖ṽ − v‖L2(Ω)‖χ‖L2(Ω)≤ C̃ ‖ṽ − v‖
L2(Ω)‖χ‖
L2(Ω).
-
CHAPTER 3. DG FINITE ELEMENT DISCRETIZATION 26
By the triangle inequality, we obtain∣∣∣∣∣∑T∈T
∫T
(w∇(ṽ − v) · ∇χ+ λ̃(ṽ − v)χ
)∣∣∣∣∣ ≤ C̃‖ṽ − v‖H1(Ω)‖χ‖H1(Ω)≤ C̃ h‖v‖H2(Ω)‖|χ|‖. (3.36)
The last term is bounded by using the Cauchy-Schwarz inequality
and by takingadvantage of the definition of the penalty parameter
:∣∣∣∣∣∣∑e∈Eint
∫e{w∇(ṽ − v) · ~ν}[χ]
∣∣∣∣∣∣ ≤∑e∈Eint
h12e ‖{w∇(ṽ − v) · ~ν}‖
L2(e)h− 12e ‖[χ]‖L2(e)
≤
∑e∈Eint
he‖{w∇(ṽ − v) · ~ν}‖2L2(e)
12
×
∑e∈Eint
h−1e ‖[χ]‖2L2(e)
12
≤ C(∑T∈T
heh−1T ‖∇(ṽ − v)‖2
L2(T )
) 12 ∑e∈Eint
h−1e ‖[χ]‖2L2(e)
12
≤ C̃(∑T∈T
h2T‖v‖2L2(T )
) 12 ∑e∈Eint
h−1e ‖[χ]‖2L2(e)
12≤ C̃ h‖v‖H2(Ω)‖|χ|‖. (3.37)
We substitute (3.36) and (3.37) into (3.35). Therefore, using
the bound (3.34) weobtain
‖χ‖2L2(Ω) ≤ C̃h‖v‖H2(Ω)‖|χ|‖≤ C̃h‖χ‖L2(Ω)‖|χ|‖.
With (3.12), this implies
‖χ‖L2(Ω)
≤ C̃h‖|χ|‖ ≤ C̃h2‖u‖H2(T )
.
3.6 DG-Version of IRLS AlgorithmIn this section, we present the
iteratively reweighted least squares algorithm in com-bination with
the DG discretization. The 1D case has already been presented in
[34].Now we are in the position to proceed with the 2D case.
Indeed, we have all theingredients to assemble our numerical scheme
into the following algorithm.
-
CHAPTER 3. DG FINITE ELEMENT DISCRETIZATION 27
Algorithm 3.14 (Double Minimization, [24]).Input: Data vector g,
ε > 0, initial gradient weight w(0) with ε ≤ w(0) ≤ 1/ε,
number nmax of outer iterations.Parameters: λ̃, κ > 0.Output:
Approximation u∗ of the minimizer of Jε.u
(0)h = 0
for n = 0 to nmax doAssemble the matrix A(n+1)h from Theorem
3.3Compute u(n+1)h such that the matrix A
(n+1)h u
(n+1)h = gh;
Compute the gradient ∇u(n+1)h =∑k∈N u
(n+1)h,k ∇φk;
w(n+1) = min(
max(ε,
1|∇u(n+1)h |
),1ε
).
endforu∗h := u
(n+1)h .
-
Chapter 4
Numerical Results
In this chapter, we present some simulation results obtained
with the DG-FEM. Inparticular, we will show results of our model
problem with Dirichlet boundary condi-tion and with Neumann
boundary conditions. We present numerical convergence ratesfor the
L2-norm of the error for a smooth exact solution. We continue with
results ofthe IRLS, particulary a denoising problem, by presenting
results for the convergenceof the algorithm with varying
regularization parameter and mesh size.
4.1 DG for Diffusion Problems
4.1.1 Dirichlet ProblemLet us consider the Dirichlet boundary
value problem{
−∇ · (w∇u) + λ̃u = λ̃g Ω,u = 0 ∂Ω. (4.1)
with Ω = (−1, 1)2, g(x, y) = −1 if x, y < 0, 1 if x, y >
0, and 0 otherwise. Discretiz-ing problem (4.1) by means of the
SIPG method with penalty parameter κ = 10 asdescribed in Section
3.1, we obtain the results displayed in Figure 4.1. More
precisely,we set w = 1. The plots show that increasing the
regularization parameter λ̃ yieldssolutions that approaches g.
4.1.2 Neumann ProblemLet us consider the Neumann boundary value
problem{
−∇ · (w∇u) + λ̃u = λ̃g Ω,∇u · ~ν = 0 ∂Ω, (4.2)
with Ω = (−1, 1)2, g(x, y) = −1 if x, y < 0, 1 if x, y >
0, and 0 otherwise. Discretiz-ing problem (4.2) by means of the
SIPG method with penalty parameter κ = 10 asdescribed in Section
3.1, we obtain the results displayed in Figure 4.2. More
precisely,
28
-
CHAPTER 4. NUMERICAL RESULTS 29
Figure 4.1: DG solution of Dirichlet problem with penalty
parameter κ = 10; λ̃ = 10(left) and λ̃ = 1000 (right).
we set w = 1. The plots show that increasing the regularization
parameter λ̃ yieldssolutions that approaches g.
Figure 4.2: DG solution of Neumann problem with penalty
parameter κ = 10 ; λ̃ = 10(left) and λ̃ = 1000 (right).
4.1.3 Poisson Problem with Known SolutionLet us consider the
Poisson problem{
−4u = g Ω,u = 0 ∂Ω, (4.3)
on the unit square (−1, 1)2. We choose g such that the
analytical solution of the prob-lem is given by u(x, y) = sin(πx)
sin(πy).
-
CHAPTER 4. NUMERICAL RESULTS 30
Figure 4.3: DG solution of Poisson problem with penalty
parameter κ = 10 (left) anderror for the L2-norm (right).
n h en = ‖u− uh‖ β1 1/2 1.4358 ×10−12 1/4 5.3246 ×10−2 2.6963
1/8 1.5578 ×10−2 3.4184 1/16 4.1438 ×10−3 3.7595 1/32 1.0628 ×10−3
3.8996 1/64 2.6871 ×10−4 3.9557 1/128 6.7532 ×10−5 3.9798 1/256
1.6928 ×10−5 3.9899 1/512 4.2367 ×10−6 3.996
Table 4.1: Numerical error estimates for a smooth function.
Table 4.1 shows the numerical error in the L2-norm for equation
(4.3). The convergencerate given by
β = enen+1
as predicted by the theory is O(h2). This implies halving the
mesh size leads toan increse in the error estimate. Table 4.1
demonstrates that as n increases, theconvergance rate β increases
to a factor 4. The error in the L2 norm plotted againstthe mesh
size is shown in Figure 4.3 using a penalty parameter of κ =
10.
-
CHAPTER 4. NUMERICAL RESULTS 31
4.2 Iteratively Reweighted Least Squares Algorithm
4.2.1 Denoising ProblemFor studying the numerical results of the
iteratively reweighted least squares (IRLS)Algorithm 3.14, we use a
regularization functional
J̃(u,w) = λ̃2‖u−G‖2L2(Ω) +
∫Ω
(w|∇u|2 + 1
w
)dx. (4.4)
The functional (4.4) is called a denoising functional where G =
g + η , λ̃ > 0 and η isa uniformly distributed noise. The
corresponding Euler-Lagrange equation is given by{
−∇ · (w∇u) + λ̃u = λ̃G Ω,w∇u · ~ν = 0 ∂Ω. (4.5)
Concerning the numerical work involved in the IRLS algorithm,
each iteration requiressolving the Euler-Lagrange equation and
using the solution to compute new gradientweights w. Figure 4.4
shows the result for 20 iterations of the IRLS algorithm. Weset the
regularization parameter λ̃ = 102, the penalty parameter κ = 10 and
ε = 10−2as in Algorithm 3.14. As the results indicate, there is a
sharp improvement in thesolution within the first 10 iteration
steps. The solution becomes steady till the totalnumber of
iteration is reached. This means an approximate solution is reached
afterfew number of iterations.
J(u)n h = 1/8 h = 1/16 h = 1/32 h = 1/64 h = 1/1281 45.3461
44.2652 51.2620 63.7826 71.74432 43.3240 38.5002 49.8223 58.2257
67.11173 38.6976 41.6775 47.8067 55.4609 63.62225 41.0890 40.0938
45.4033 52.1332 58.610510 42.6990 40.4148 42.9642 47.8849 53.042520
39.7064 38.1429 40.7648 45.4784 48.569030 39.9341 37.2609 41.3725
43.1490 46.502440 38.1215 38.7159 40.3228 43.0523 44.886250 42.8382
39.6733 40.8136 43.2327 44.2599
Table 4.2: Results for the functional J(u) from (4.6) for 50
iterations and decreasingmesh size .
The Table 4.2 shows the values of the convergence of the
iteratively reweighted leastsquares algorithm. For a fixed λ̃ = 104
and penalty parameter κ = 10, the denoisingproblem (4.5) is solved
for 50 iterations with the iteratively reweighted least
squaresalgorithm. The solution is used in computing the
functional
J(u(n)) = λ̃2‖u(n) −G‖2L2(Ω) +
∫Ω|∇u(n)| (4.6)
-
CHAPTER 4. NUMERICAL RESULTS 32
Figure 4.4: Results of the iteratively reweighted least squares
algorithm at iterations1 and 2 (above) and iterations 3 and 5
(middle) and 10 and 20 (below).
for varying mesh sizes h.
In the Table 4.3, the results for different regularization
parameter λ̃ are computed.We set ε = 10−1.
-
CHAPTER 4. NUMERICAL RESULTS 33
J(u)n λ̃ = 102 λ̃ = 103 λ̃ = 1041 35.7395 46.7507 63.11282
35.5685 44.2403 56.87163 34.9391 42.6369 54.23165 34.7120 41.4692
51.473010 35.1925 39.2793 47.555620 34.6311 38.8595 44.664230
34.5625 38.1292 43.676840 34.5749 37.9654 42.958150 34.8817 38.2415
42.1900
Table 4.3: Convergence results for a fixed mesh h = 2−6 computed
for 50 iterationswith different λ̃.
Figure 4.5 shows the value of J(u(n)) for the first 50
iterations for λ̃ = 102 (above),λ̃ = 103 (middle) and λ̃ = 104
(below). It can be observed that the functional J(u(n))converges to
a minimum with increasing iteration. The minimum is achieved
andremains almost the same with regularization parameters 103 and
104. As the regular-ization parameter increases, the solution
converges whiles a decrease in ε (figures fromleft to right) does
not make any significant difference. When λ̃ = 102, the
functionalJ(u(n)) has an irregular behaviour for both ε = 10−2 and
ε = 10−1. This means theregularization parameter λ̃ must be larger
than 102 to achieve good results.
4.2.2 Diffusion Problem, TV-Minimization Problem and IRLSWe
consider an example of our model problem (4.5). In this example, we
set the noiseη = 0, ε = 10−2, κ = 10 and the regularization
parameter λ̃ = 100. In Figure 4.6,the results for some iterations
are plotted. The results show the efficiency of the IRLSalgorithm
to yield sharp edges for problems without noise.
-
CHAPTER 4. NUMERICAL RESULTS 34
Figure 4.5: Convergence for the regularization functional
plotted against the numberof iterations. The results on the left
correspond to ε = 10−1 and on the right ε = 10−3.The plot shows the
values of J(u(n)) for the first 50 iterations for λ̃ = 102
(above),λ̃ = 103 (middle) and λ̃ = 104 (below).
-
CHAPTER 4. NUMERICAL RESULTS 35
Figure 4.6: Results of the IRLS for the varying diffusion
coefficients at iterations 1and 3 (above) and iterations 10 and 20
(below).
-
Chapter 5
Conclusion and Outlook
5.1 ConclusionIn this thesis, we solved a total variation
minimization problem by using a discontin-uous Galerkin finite
element method. In Chapter 1, we considered the Rudin, Osherand
Fattemi (ROF) model. In general, the first order optimality
conditition of themodel yields an ill-posed non-linear second order
partial differential equation. Consid-ering the space of Bounded
Variation of functions and some of its relevant properties,we
showed that the existence and uniqueness of minimizers of the
ill-posed functionalcan be achieved by a relaxation algorithm
called iteratively reweighted least squaresalgorithm which is also
referred to as double minimization algorithm in some booksand
articles.
The well-posed second order partial differential equation
equation obtained from theiteratively reweighted least squares
algorithm is discretized by a standard discontinuousGalerkin (DG)
finite element method. This is particularly useful since the space
ofbounded variation functions contain discontinuous functions. We
presented some the-ory on the primal formulation of a discontinuous
Galerkin finite element method. Weshowed the existence and
uniqueness of the DG solution and also presented a-priorierror
estimates in the energy norm and L2-norm.
Finally, we have discussed some numerical results obtained by DG
discretizationmethod. In Chapter 4, two examples are discussed and
the L2 error are computedfor a known problem. Results from the
iteratively reweighted least squares algorithmare also presented
with an application to a denoising problem. The rate of
convergenceof the iteratively reweighted least squares algorithm is
also presented.
5.2 OutlookThis thesis provides a starting point for preparing
theoritical results for a discontinuousGalerkin finite element
method for solving total variation minimization problems. The
36
-
CHAPTER 5. CONCLUSION AND OUTLOOK 37
DG discretization has high degrees of freedom and requires
efficient solvers. The workcan be continued in the following
way
1. Robust multilevel solver.At each step of the IRLS, we have to
solve a diffusion problem with varyingdiffusion coefficient w
between ε and 1/ε. Here we need robust iterative solversand an
appropriate stopping criteria in order to achieve optimal results.
Onecandidate for such a robust solver is the algebraic multilevel
solver proposed byKraus and Tomar [31] and Kraus [29].
2. Harmonic mean for DG formulation.In the formulation of the
DG, the harmonic mean can be used for averaging ofthe flux. This
approach is particularly useful since the diffusion coefficients
canbe discontinuous [18].
3. Kraus–Tomar Assembling.The assembling process can also be
improved by reducing the total number ofdegrees of freedom. The
Kraus-Tomar assembling of the discontinuous Galerkinfinite element
method reduces the total degree of freedom compared to the
stan-dard assembling. It is useful since the coefficients (gradient
weights) can bediscontinuous [31].
4. Analysis and Simluation of 3D Problems.The work can be
carried on to higher dimension.
-
List of Notations and FunctionSpaces
Notation
N,R,Rn natural numbers, real numbers, n-dimensional Euclidean
spaceL(X, Y ) space of bounded linear operator between Banach
spaces X and YX∗ dual space of the Banach space XV ⊂ U V is a
subset of U ; for Banach spaces U and V. It also denotes
continuous
embedding, i.e. the identity mapping I : V → U is continuousV ⊂⊂
U V is compactly contained in U , i.e. V ⊂ U and V is compact;
for Banach spaces U and V. It also denotes compact embedding,
i.e.I : V → U is compact.
1V characteristic function of the set V, i.e. 1V (x) = 1 if x ∈
V, 0 otherwiseDu distributional derivative of udiv g divergence of
the vector-valued function g∇u classical or weak gradient of the
scalar functional uXV characteristic function of the set V in the
sense of convex analysis∂V boundary of the set VK∗ adjoint in L(Y
∗, X∗) of K ∈ L(X, Y )|x|`p `p norm of x ∈ Rn, 1 ≤ p ≤ ∞|α|, |x|
modulos of α ∈ R or Euclidean norm of x ∈ RnId identity matrix, Id
∈ Rm×n‖v‖V norm of v in the Banach space VO Landau symbol4
Laplacian operator, 4u =
n∑i=1
∂2u∂x2i, for u : Rn → R
Function Spaces
Let Ω ⊂ Rn be open. See [21] as general reference.
38
-
LIST OF NOTATIONS AND FUNCTION SPACES 39
C(Ω,Rn) continuous functions on Ω with values in Rn, denoted by
C(Ω) for n = 1C0(Ω,Rn) functions in C(Ω,Rn) with copmact support in
ΩCk(Ω,Rn) k times continuously differentiable functions on Ω with
values in Rn
denoted by Ck(Ω) for n = 1Ck0 (Ω,Rn) functions in Ck(Ω,Rn) with
compact support in ΩC∞(Ω,Rn) infinitely differentiable functions in
Ω with values in Rn, denoted by
C∞(Ω) for n = 1C∞0 (Ω,Rn) functions in C∞(Ω) for n = 1Lp(Ω,Rn)
Lebesgue space, 1 ≤ p ≤ ∞, on Ω with values in Rn, denoted by
Lp(Ω)
for n = 1W k,p(Ω) Sobolev space of functions with k−th order
weak derivatives in Lp(Ω)W k,p0 (Ω) closure of C∞0 (Ω) in W
k,p(Ω)Hk(Ω), Hk0 (Ω) abbreviations for the Hilbert spces W k,2(Ω),
W
k,20 (Ω)
BV (Ω) space of functions of bounded variation on Ω
-
Bibliography
[1] R. Adams, Sobolev Spaces. Academic Press, New York,
1975.
[2] L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded
Variation andFree Discontinuity Problems. Oxford Mathematical
Monographs,Oxford Univer-sity Press,2000.
[3] D. Arnold, An interior penalty finite element method with
discontinuous elements,SIAM J. Numer. Anal., 19 pp. 742–760,
1982.
[4] D. N. Arnold, F. Brezzi, B. Cockburn and L. D. Marini,
Unified Analysis of Dis-continuous Galerkin Methods for Elliptic
Problems. SIAM Journal on NumericalAnalysis, Vol. 39/5, pp. 1749
–1779, 2002.
[5] G. Aubert and P. Kornprobst, Mathematical Problems in Image
Processing: Par-tial Differential Equations and the Calculus of
Variation. Springer, 2002.
[6] C. E. Baumann and J. T. Oden, A discontinuous hp finite
element method forconvection-diffusion problems Comput. Methods
Appl. Mech. Engrg., 175 pp. 311–341, 1999.
[7] D. Braess, Finite Elemente: Theorie schneller Löser und
Anwendungen in derElastizitätstheorie. 2. Auflage, Springer,
1996.
[8] S. C. Brenner and L. R. Scott, The Mathematical Theory of
Finite Element Meth-ods Springer 2008.
[9] M. Burger and S. Osher, A Guide to the TV Zoo I:Models and
Analysis. In LevelSet and PDE-Based Reconstruction Methods,
Springer, 2009.
[10] A. Chambolle, An algorithm for total variation minimization
and applications. J.Math. Imaging Vision 20:1-2 , pp. 89–97,
2004.
[11] A. Chambolle and P. -L. Lions, Image recovery via Total
Variation Minimizationand Related Problems. J. Math. Anal. Appl.,
Vol 276/2, pp. 845–876, 2002.
[12] T. F. Chan, G. H. Golub and P. Mulet, A nonlinear primal
dual method for totalvariation based image restoration. SIAM J.
Sci. Comput., 20, (6):1964-1977, 1999.
40
-
BIBLIOGRAPHY 41
[13] T. F. Chan and P. Mulet, Iterative methods for total
vari-ation restoration. CAM report 96-38, UCLA, USA;
seehttp://www.math.ucla.edu/applied/cam/index.html, 1996.
[14] T. F. Chan and J. Shen, Image Processing And Analysis:
Variational, PDE,Wavelet and Stochastic Methods. SIAM 2005.
[15] K. Chen and X-C. Tai, A Nonlinear Multigrid Method for
Total Variation Mini-mization from Image Restoration J. Sci.
Comput., Vol 33/2, pp. 115–138 , 2007.
[16] P. G. Ciarlet, The Finite Element Method for Elliptic
Problems. Classics in Ap-plied Mathematics, Vol. 40, SIAM,
Philadelphia, 2002.
[17] I. Daubechies, R. DeVore, M. Fornasier and C. S. Güntürk,
Iteratively Re-weightedLeast Squares Minimization for Sparse
Recovery. pp. 1-38 Communications inPure and Applied Mathematics,
Vol.63, 2010.
[18] B. A. Dios, M. Holst, Y. Zhu and L. Zikatanov, Multilevel
Preconditioners for Dis-continuous Galerkin Approximations of
Elliptic Problems with Jump Coefficient.arXiv:1012.1287v1 [math.NA]
2010.
[19] V. A. Doberov, Preconditioning of Discontinuous Galerkin
Method for Second Or-der Elliptic Problems PhD Thesis , Texas A.
& M. University, 2007.
[20] H. W. Engl, M. Hanke and A. Neubauer, Regularization of
Inverse Problems.Mathematics and Its Applications, vol. 375,
Kluwer, 2000.
[21] L. C. Evans, Partial Differential Equations. Graduate
Studies in Mathematics,American Mathematical Society, 1998.
[22] L. C. Evans and R. F. Gariepy, Measure Theory and Fine
Properties of Functions.CRC Press, Boca-Raton–Ann Arbor– London,
1992.
[23] M. Fornasier, Domain Decomposition methods for linear
inverse problems withsparsity constraints. Inverse Problems 23, pp.
2505–2526, 2007.
[24] M. Fornassier (ed)., Numerical methods for Sparse Recovery.
Radon Series Appliedand Computational Mathematics, de Gruyter,
2010.
[25] M. Fornasier and R. March, Restoration of Color Images by
Vector Valued BVFunctions and Variational Calculus SIAM J. Appl.
Math., Vol. 68 No. 2, 437–460,2007
[26] E. Giusti, Minimal Surfaces and Functions of Bounded
Variation. BirkhäuserBoston, 1984.
[27] T. Goldstein and S. Osher, The split Bregmann method for L1
regularized prob-lems. SIAM Journal on Imaging Sciences Vol.2/2,
pp. 323–343, 2009.
-
BIBLIOGRAPHY 42
[28] C. Groosmann, H-G. Roos and M. Stynes, Numerical Treatment
of Partial Dif-ferential Equations Springer-Verlag, Berlin
Heidelberg 2007.
[29] J. Kraus , Additive Schur complement approximation and
application to multilevelpreconditioning. RICAM: Linz, Bericht-Nr.
2011-22.
[30] J. Kraus and S. Margenov, Robust Algebraic Multilevel
Methods and AlgorithmsRadon Series Comp. Appl. Math., vol. 5,
Walter de Gruyter, Berlin-New York,2009.
[31] J. Kraus and S. Tomar, Multilevel Preconditioning of
Elliptic Problems Discretizedby a Class of Discontinuous Galerkin
Methods. RICAM: Linz, Bericht-Nr. 2006-36.
[32] C. L. Lawson, Contributions to the Theory of Linear Least
Maximum Approxima-tion. Ph.D.thesis, University of California, Los
Angeles, 1961.
[33] Y. Meyer, Oscillating patterns in image processing and
nonlinear evolution equa-tions. American Mathematical Society,
2001.
[34] S. E. Moore, Discontinuous Galerkin for Total Variation
Minimization. ProjectSeminar, Summer Semester, Johannes Kepler
Universität, Linz, 2010.
[35] J. T. Oden, I. Babuška and C. E. Baumann, A discontinuous
hp finites elementmethod for diffusion problems J. Comput. Phys.
Vol. 146, pp. 491–519, 1998.
[36] C. Ortner, A Non-conforming Finite Element Method for
Convex OptimizationProblems. http://www2.maths.ox.ac.uk/oxmos/,
OxMOS: New Frontiers in theMathematics of Solids, 2008.
[37] S. Osher, M. Burger, D. Goldfarb, J. Xu and W. Yin, An
iterative regulariza-tion method for total variation-based image
restoration. Multiscale Model. Simul,Vol.4/2, pp. 460–489,
2005.
[38] S. Osher and A. Marquina, Explicit algorithms for a new
time dependent modelbased on level set motion for nonlinear
deblurring and noise removal SIAMJ. Sci. Comput., Vol. 22/2, pp.
387–405, 2000.
[39] B. Rivière, Discontinuous Galerkin Methods for Solving
Elliptic and ParabolicEquations: Theory and Implementation. SIAM
2008.
[40] L. I. Rudin, S. J. Osher and E. Fatemi, Nonlinear total
variation based noise re-moval algorithms Physica D. 60, pp.
259–268, 1992.
[41] C. R. Vogel, Computational methods for inverse problems
SIAM publications,USA, 2002.
-
BIBLIOGRAPHY 43
[42] C. R. Vogel, A multigrid method for total variation-based
image denoising Bowers,K., Lund, J.(eds.) Computation and Control
IV. Progress in Systems and ControlTheory, Vol.20, Birkhäuser,
Boston, 1995
[43] C. R. Vogel and M. E. Oman, Iterative methods for total
variation denoising.Special issue on iterative methods in numerical
linear algebra (Breckenridge, CO,1994). SIAM J. Sci. Comput. Vol
17/1, pp 227–238, 1996.
[44] M. F. Wheeler, An elliptic collocation-finite element
method with interior penaltiesSIAM J. Numer. Anal. 15, pp. 152–161,
1978.
-
Eidesstattliche Erklärung
Ich, Stephen Edward Moore, erkläre an Eides statt, dass ich die
vorliegende Master-arbeit selbständig und ohne fremde Hilfe
verfasst, andere als die angegebenen Quellenund Hilfsmittel nicht
benutzt bzw. die wörtlich oder sinngemäß entnommenen Stellenals
solche kenntlich gemacht habe.
Linz, August 2011
————————————————Stephen Edward Moore