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NUMERICAL METHODS FOR NONLINEARELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS
TIAGO SALVADOR
Department of Mathematics and StatisticsFACULTY OF SCIENCE
McGill University, Montreal
MAY 2017
A thesis submitted to McGill Universityin partial fulfillment of the
requirements for the degree ofDoctor of Philosophy
© Tiago Salvador 2017
ABSTRACT
The goal of this thesis is to widen the class of provably convergent schemes for elliptic
partial differential equations (PDEs) and improve their accuracy. We accomplish this by
building on the theory of Barles and Souganidis, and its extension by Froese and Oberman
to build monotone and filtered schemes.
The first problem considered is the widely studied class of first order Hamilton-Jacobi
(HJ) equations. The goal is to construct provably convergent accurate schemes, together
with an efficient solver, by making use of the large number of discretizations and solvers
already available. To this end, we build filtered schemes, whose main idea is to blend a
stable monotone convergent scheme with an accurate scheme while retaining the advan-
tages of both: stability and convergence of the former, and higher accuracy of the latter.
Indeed, we are able to build schemes which are second, third, and fourth order accurate
in one dimension, as well as schemes that are second order accurate in two dimensions.
Moreover, the schemes are explicit, allowing us to use the easy-to-implement fast sweeping
method. Using different accurate schemes (e.g. from standard centred differences, higher
order upwinding and ENO interpolation), the accuracy of the filtered schemes is validated
with computational results for the eikonal equation, as well as other HJ equations (both in
one and two dimensions).
The second problem considered is the 2-Hessian equation, a fully nonlinear PDE related
to the intrinsic curvature for three-dimensional manifolds. The goal is to build numerical
methods to compute its solution on a bounded domain given prescribed boundary data.
We propose two distinct methods. The first is provably convergent to the unique viscosity
solution. The second has higher accuracy and converges in practice, but lacks a formal
proof of convergence. The PDE is elliptic on a restricted set of functions: a convexity-
type constraint is needed for the ellipticity of the PDE operator, which poses additional
difficulties when building the numerical methods. Solutions with both discretizations are
obtained using Newton’s method. Computational results are presented on a number of
exact solutions which range in regularity from smooth to non-differentiable, and in shape
from convex to non-convex.
The third and last problem is to build a provably convergent scheme for the nonlinear
PDE that governs the motion of level sets by affine curvature. It is closely related to mean
curvature but exhibits instabilities not found in the former. These instabilities and the lack
ii
Abstract iii
of regularity of the affine curvature operator posed unexpected and additional difficulties
in building monotone schemes. A standard finite difference scheme is proposed and an
example that illustrates its nonlinear instability is given. We build provably convergent
monotone finite difference schemes. Numerical experiments demonstrate the accuracy and
stability of the discretization, as well as the fact that our approximate solutions capture the
affine invariance and morphological properties of the evolution.
ABRÉGÉ
L’objectif de cette thèse est d’élargir la classe de schémas à convergence prouvable pour les
équations aux dérivées partielles (EDPs) elliptiques, et d’améliorer leur précision. Nous
l’accomplissons en nous appuyant sur la théorie de Barles et Souganidis, et l’extension de
celle-ci par Froese et Oberman pour construire des schémas monotones et filtrés.
Le premier problème considéré est la classe largement étudiée des équations de
Hamilton-Jacobi (HJ) du premier ordre. L’objectif est de construire des schémas pré-
cis et convergents avec un solveur efficace, en utilisant le grand nombre de discrétisations
et de solveurs déjà disponibles. À cette fin, nous construisons des schémas filtrés, dont
l’idée principale est de combiner un schéma monotone, convergent, et stable, et un schéma
précis, tout en conservant les avantages des deux. On préserve la stabilité et la convergence
du premier et la plus grande précision du deuxième. En effet, nous sommes en mesure de
construire des schémas de deuxième, troisième et quatrième ordre précis en une dimension,
ainsi que des schémas de deuxième ordre précis en deux dimensions. En outre, les schémas
sont explicites, ce qui nous permet d’utiliser la méthode dite de “Fast Marching”, qui est
facile à mettre en oeuvre. En utilisant différents schémas précis (par exemple, à partir de
différences centrées standard, de “upwinding” à ordre supérieur, et d’interpolation ENO),
la précision des schémas filtrés est validée avec des résultats informatiques pour l’équation
eikonal et d’autres équations de HJ (aussi bien en une qu’en deux dimensions).
Le deuxième problème considéré est l’équation 2-Hessienne, une EDP entièrement
non linéaire associée à la courbure intrinsèque sur des les variétés tridimensionnelles.
L’objectif est de construire des méthodes numériques pour en calculer la solution sur
un domaine borné, et avec les données à la frontière prescrites. Nous proposons deux
méthodes distinctes. On prouve que la première converge à la solution de viscosité
unique. Le seconde a une plus grande précision et converge en pratique, mais on manque
d’une preuve formelle de convergence. L’EDP est elliptique sur un ensemble restreint de
fonctions: une contrainte de type convexe est nécessaire pour l’ellipticité de l’opérateur
PDE, ce qui pose des difficultés supplémentaires lors de la construction des méthodes
numériques. Les solutions avec les deux discrétisations sont obtenues en utilisant la
méthode de Newton. Des résultats numériques sont présentés sur un certain nombre de
solutions exactes qui varient en régularité, de lisses à non différentiables, et en forme, de
convexes à non convexes.
iv
Abrégé v
Le troisième et dernier problème consiste à construire un schéma convergent pour
l’EDP non linéaire régissant le mouvement des ensembles de niveau par courbure affine.
Il est étroitement lié au problème de courbure moyenne, mais présente des instabilités
non trouvées dans ce-dernier. Ces instabilités, ainsi que le manque de régularité de
l’opérateur de courbure affine, posent des difficultés inattendues et supplémentaires
dans la construction de schémas monotones. Un schéma aux différences finies standard
est proposé, et on présente un exemple qui illustre son instabilité non linéaire. Nous
construisons un schéma aux différences finies monotones, dont on prouve la convergence.
Des essais numériques démontrent la précision et la stabilité de la discrétisation, ainsi
que le fait que nos solutions approximatives captent l’invariance affine et les propriétés
morphologiques de l’évolution.
STATEMENT OF CONTRIBUTION
Here we summarize the contributions of each chapter of this thesis, which we elaborate
more on each chapter.
Chapter 4 was published in the Journal of Computational Physics where my advisor
Prof. Adam Oberman is a coauthor [OS15]. He directed the research and I conducted the
research. The contributions of this chapter are:
• To build provably convergent accurate schemes for first order Hamilton-Jacobi equa-
tions.
Chapter 5 was published in IMA Journal of Numerical Analysis, where my advisor
Prof. Adam Oberman and Prof. Froese are coauthors [FOS16]. They directed the research
and I conducted the research. The contributions of this chapter are:
• To provide an accurate numerical method to compute solutions of the 2-Hessian
equation.
• To build a provably convergent monotone scheme for the same PDE.
Chapter 6 has been submitted for publication and is currently under review. My advisor
Prof. Adam Oberman is a coauthor [OS15]. He directed the research and I conducted the
research. The contributions of this chapter are:
• To study the nonlinearly instabilities exhibit by a standard finite difference scheme
for the PDE that governs the motion of level sets under affine curvature.
• To build a provably convergent monotone scheme for the same PDE.
vi
LIST OF FIGURES
3.1 Examples of filter functions: discontinuous filter (left), continuous filter (right). 36
4.1 Profile of the solutions of the five examples considered in one dimension (at the
top, eikonal equation examples, at the bottom, HJ equations examples). . . . . . 53
4.2 Exact solution and solutions obtained with the monotone scheme and the 2nd
order upwind filtered scheme with 50 grid points for the first example of the
eikonal equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.3 Active stencils in the accurate scheme in the last iteration for the solutions of
the second example considered: −i means that i points to the left were used in
the interpolation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.4 Log-log plot of the errors for the one-dimensional examples of the eikonal
equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.5 Log-log plot of the errors for the one-dimensional examples of HJ equations. . . 59
4.6 Profile and contour plots of the solutions of the three examples considered in
two dimensions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.7 Log-log plot of the errors for the two-dimensional examples in the l∞ norm. . . 64
4.8 Log-log plot of the errors for the two-dimensional examples in the l1 norm. . . 64
4.9 Log-log plot of the errors for the second example in the l∞ norm in regions
(x, y) ∈ R2 : |x + y| > 0.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.10 Log-log plot of the errors for the third example in the l∞ norm in regions
(x, y) ∈ R2 : x2 + y2 ≥ 1, x ≥ 0.1. . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5.1 Failure of the parabolic solver using the naive finite difference scheme: section
z = 0.9 of the initial guess (left) and the solution after 25 iterations (right). . . . 76
5.2 Elements of V1 (solid) and elements of V2 \ V1 (dashed). . . . . . . . . . . . . . . 85
5.3 Surface plots of the level sets of the solution to Example 5.7 on a 30 × 30 × 30
grid with the naive finite differences (left) and the 27-point monotone scheme
(right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.4 Plot of the curves t 7→ u(t, t, t) of the solution of Example 5.7 on a 30 × 30 × 30
grid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
vii
viii LIST OF FIGURES
5.5 Surface plots of the level sets of the solution to Example 5.8 on a 30 × 30 × 30
grid with the naive finite differences (left) and the 27-point monotone scheme
(right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
5.6 Plot of the curves t 7→ u(t, t, t) of the solution of Example 5.8 on a 30 × 30 × 30
grid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
6.1 Solution of the one-dimensional model equation using standard finite differ-
ences at times t ∈ 0, 1, 2, 5. Here dt = h2/2 on a 256-point grid. . . . . . . . . . 114
6.2 Plot of the solution obtained described in Example 6.1 at time t ∈ 0, 1, 5, 20on a 128-point grid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
6.3 Lack of convergence when using a standard finite difference scheme: example
6.6 (d) with the standard finite difference solver: level sets of the solution at
times t = 0 (upper left), t = 15 (upper center), t = 17 (upper right), t = 20
(lower left), t = 40 (lower center) and t = 50 (lower right) with dt = h2/2 on a
32 × 32 grid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
6.4 Neighbour points of the stencil for nθ = 3 (smaller circle) and nθ = 7 (larger
circle). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
6.5 (Top:) Evolution of an ellipse by (left) affine curvature, (right) mean curvature.
(Bottom:) Evolution by affine curvature of (left) a diamond and (right) a flatter
diamond. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
6.6 Evolution of a fan-shape like curve under affine curvature motion (top) and
mean curvature (bottom) at time t ∈ 0, 0.05, 0.1, .2 (see Example 6.5 for more
details). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
6.7 Plot of the zero level sets in Example 6.10 of Φt(u φ) and(
Φt(det φ)2/3(u))
φ
for regular elliptic scheme (left), standard scheme (center) and regular filtered
scheme (right) at time t = 1 with φ given by (a) (top) and (b) (bottom). . . . . . 134
LIST OF TABLES
4.1 Accuracy in the l∞ norm and order of convergence of the schemes for the first
example of the eikonal equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.2 Accuracy in the l∞ norm and order of convergence of the schemes for the
second example of the eikonal equation. . . . . . . . . . . . . . . . . . . . . . . . 56
4.3 Accuracy in the l∞ norm and order of convergence of the schemes for the third
example of the eikonal equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.4 Accuracy in the l∞ norm and order of convergence of the schemes for the fourth
example (H(p) = p2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.5 Accuracy in the l∞ norm and order of convergence of the schemes for the fifth
example (H(p) = cos(p)2 + |p|). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.6 Accuracy and order of convergence of the schemes for the first example in two
dimensions in the l∞ norm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.7 Accuracy and order of convergence of the schemes for the second example in
two dimensions in the l∞ norm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.8 Accuracy and order of convergence of the schemes for the second example in
two dimensions in the l1 norm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.9 Accuracy and order of convergence of the schemes for the third example in two
dimensions in the l∞ norm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5.1 Elements of G1 up to permutations. . . . . . . . . . . . . . . . . . . . . . . . . . . 84
5.2 nS is the number of ν directions available in the stencil, i.e., nS = #Vnθ. . . . . 84
5.3 Accuracy in the l∞ norm of the schemes for the first example using the Newton
solver. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.4 Accuracy in the l∞ norm and order of convergence of the schemes for the
second example using the Newton solver. . . . . . . . . . . . . . . . . . . . . . . 91
5.5 Accuracy in the l∞ norm and order of convergence of the schemes for the third
example using the Newton solver. . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5.6 Accuracy in the l∞ norm and order of convergence of the schemes for the fourth
example using the Newton solver. . . . . . . . . . . . . . . . . . . . . . . . . . . 92
ix
x LIST OF TABLES
5.7 Accuracy in the l∞ norm and order of convergence of the schemes for the fifth
example using the Newton solver. . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.8 Accuracy in the l∞ norm and order of convergence of the schemes for the sixth
example using the Newton solver. . . . . . . . . . . . . . . . . . . . . . . . . . . 94
6.1 Coordinates of the neighbours in the first quadrant of a stencil with width nθ. . 118
6.2 Accuracy in the l∞ norm and order of convergence of the schemes for Example
6.6 with regularized schemes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
6.3 Error in the l∞ norm of the whole computational domain at time t = 0.1 for the
time dependent Example 6.7 and 6.8. . . . . . . . . . . . . . . . . . . . . . . . . . 132
6.4 Difference in the l∞ norm between Φt(gv u0) and gv Φt(u0) for v = 1, 2 for
Example 6.9. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
CONTENTS
Abstract ii
Abrégé iv
Statement of contribution vi
List of Figures vii
List of Tables ix
Acknowledgements xiii
1 Introduction 1
1.1 Literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Organization of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2 Viscosity Solutions 11
2.1 Motivation of viscosity solutions . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Definition of viscosity solution . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3 Comparison principle and uniqueness . . . . . . . . . . . . . . . . . . . . . . 15
2.4 Existence of viscosity solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3 Numerical Schemes 21
3.1 Monotone finite difference schemes . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2 Elliptic finite difference schemes . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.3 Building Elliptic Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.4 Filtered finite difference schemes . . . . . . . . . . . . . . . . . . . . . . . . . 29
4 Filtered Schemes for Hamilton-Jacobi equations 39
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.2 Discretization and solvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.3 Computational Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
xi
xii CONTENTS
5 2-Hessian equation 67
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
5.2 Background on the equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5.3 Discretization and solvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.4 Computational results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
6 Affine curvature 99
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
6.2 The affine curvature PDE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
6.3 Numerical methods for the model equation . . . . . . . . . . . . . . . . . . . 105
6.4 Nonconvergence of standard finite differences . . . . . . . . . . . . . . . . . 113
6.5 Convergent finite difference methods . . . . . . . . . . . . . . . . . . . . . . . 117
6.6 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
6.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
7 Conclusions 137
Bibliography 139
ACKNOWLEDGEMENTS
First and foremost I would like to thank my supervisor Professor Adam Oberman for
his guidance and support during my PhD. Without him this work would not have been
possible. I would also like to thank Professor Brittany Froese who I coauthor a manuscript
with and had many fruitful discussions. The staff in the Mathematics department were
always there to help me with any bureaucratic issues and deserve my many thanks. I
am also grateful for the funding I received that made this work possible, including the
scholarship from FCT, the Portuguese national funding agency for science, research and
technology.
I spent five amazing years at McGill and they will definitely be marked by the people I
met and became friends with. I was lucky enough to have amazing officemates: Bilal, Eric,
Chris, Geoff, were always available to discuss work and bounce off ideas or simply take a
break. A special thanks to Alexandra Tcheng for many lengthy and helpful discussions.
Daphna Harel and Sanam Joon thank you for your friendship, support and advice. To
all the friends I made at intramural soccer at McGill, playing with you was a pleasure
and provided the perfect escape from mathematics, in particular 2v2 soccer with Adam
Alcolado. Allison Kolly, thank you for you love and support, this last year and half would
have been much tougher without you.
There are two people who I met only once I arrived to Montreal and to who I am
more than grateful for all they did for me: Adriana Simões and Francisco Salvador. They
welcomed me in Montreal like family and made me feel like I was at home. One of the
reasons moving to Montreal was so easy was definitely because of them and that is why I
will be eternally grateful and forever in their debt.
Last but not least, I would like to thank my parents. Despite being on the other side of
the ocean, their love, support and advice was and will always be invaluable to me.
xiii
CHAPTER 1
INTRODUCTION
In this chapter, we discuss the three distinct elliptic partial differential equations (PDEs)
that are the subject of this thesis in Chapters 4, 5 and 6: Hamilton-Jacobi (HJ) equations
(with special focus on the eikonal equation), 2-Hessian equation and the PDE that governs
the motion of level sets by affine curvature. A literature review on each problem is
presented. We conclude the chapter with a description of the organization of the thesis.
Monotone schemes, introduced by Barles and Souganidis in [BS91], constitute the ideal
framework to build provably convergent schemes for elliptic PDEs. In a nutshell, the
theory says that monotone, stable, consistent schemes converge to the unique viscosity
solution of an elliptic PDE. This result makes monotone schemes very desirable. For
instance, they have been highly successful in building convergent schemes for the Monge-
Ampère equation, including the case of Optimal Transportation [BFO14], which suggest
they could be extended to more general geometric and optimal transportation type PDEs.
Despite the clear requirements of the theory, building monotone schemes is a challenge
since the definition does not provide any insight on how to build such schemes. Moreover,
conditions to ensure monotonicity are different for first and second order equations, and
for explicit and implicit schemes. Using the related notion of elliptic schemes, the work in
[Obe06a] provides a unifying and convenient reformulation of monotone schemes.
Another issue that arises when dealing with monotone schemes is the question of
accuracy. Unfortunately, the accuracy of monotone schemes is less than ideal in some
situations: it is at most first (resp. second) order for first (resp. second) order equations.
To address this, filtered schemes were introduced in [FO13] in the context of the Monge-
Ampère equation, where the goal was to overcome the reduction in accuracy based on the
use of a wide-stencil monotone scheme.
The main goal of this thesis, building on the foundation of monotone schemes, elliptic
and filtered schemes, is thus in widening the class of equations covered and extending the
theory and accuracy of the methods.
1.1 L ITERATURE REVI EW
In this section, we give an overview of some relevant existing literature on the three distinct
problems studied in this thesis.
1
2 1.1. LITERATURE REVIEW
1.1.1 Hamilton-Jacobi equations
In this thesis, we are interested in HJ equations of the form
H(x, ∇u) = f(x), x ∈ Ω,
u(x) = g(x), x ∈ Γ,
where ∇u is the gradient of the function u, Ω is an open set, Γ is the boundary of Ω and the
Hamiltonian H is a nonlinear Lipschitz continuous function. HJ equations appear in many
applications, such as optimal control, differential games, image processing, computer
vision and geometric optics.
We take a particular interest on the special case (take H(p) = |p|) of the eikonal equation
|∇u(x)| = f(x), for x outside Γ,
u(x) = g(x), for x on Γ.
where f > 0 and Γ is here a closed, bounded set. The eikonal equation has a wide range
of applications in geometric optics, computer vision, optimal control, etc. Moreover,
as pointed out in [BLZ10], high order schemes are particularly important in the high
frequency wave propagation where the eikonal equation is coupled to a transport equation
through its gradient [QS99, SVST94].
There are already a large number of discretizations and solvers available for HJ equa-
tions. The simplest approximations are finite difference schemes based on a Cartesian
grid. In this class, monotone schemes are provably convergent [BS91], but only first order
accurate [Obe06b]. In general, higher order finite difference schemes for HJ equations are
neither monotone, nor stable. For example, the centered difference scheme is unstable for
the eikonal equation [Set99b, Section 4.3]. The filtered schemes presented in this thesis are
designed to remain stable while allowing for a wide choice of accurate discretizations.
Higher order accurate schemes have been built, but only by giving up other desirable
properties (e.g. ease of implementation, fast solvers, or the convergence proof). Semi-
Lagrangian schemes [FF02, CF07], are accurate, but they involve solving the characteristic
ordinary differential equations, and are generally more complicated to implement. Central
schemes [LT00] achieve second order accuracy, at the expense of a slightly more compli-
cated, non-explicit formulation. The ENO and WENO schemes [OS91, Shu07, JP00] are
accurate, and while not provably convergent, they are effective in practice. Combinations
of WENO and central schemes have been implemented, achieving higher order accuracy
[BL03]. The ENO based schemes use adaptive stencils, which complicates the use of fast
solvers (however see [ZZQ06] for a sweeping method). Fast marching methods require
Chapter 1. Introduction 3
specialized data structures to implement, are usually first order accurate (however see
[ABM+11] for higher order methods) and only apply to the eikonal equation. A compact
upwind second order scheme for the eikonal equation was proposed in [BLZ10]. In the
same spirit of the filtered schemes presented here, a higher order scheme for HJ equations
was presented by Abgrall in [Abg09]. The convergence of this scheme also follows from
an adaptation of the Barles-Souganidis convergence proof.
1.1.2 2-Hessian equation
The second equation studied in this thesis is the 2-Hessian equation in three dimensions, a
fully nonlinear PDE of the form
S2[u] ≡ uxxuyy + uxxuzz + uyyuzz − u2xy − u2
xz − u2yz = f.
This is a particular instance of a much larger k-Hessian family of PDEs in n-dimensional
space, that include the Laplace equation, ∆u = f , when k = 1, and the Monge-Ampère
equation det D2u = f , when k = n.
Geometric PDEs have been proven to be especially useful in image analysis [Sap06].
In particular, the Monge-Ampère equation in the context of Optimal Transportation has
been used in three dimensional volume based image registration [HZTA04]. While the
2-Hessian equation is unfamiliar outside of Riemannian geometry and elliptic regularity
theory, it is closely related to the scalar curvature operator, which provides an intrinsic
curvature for a three dimensional manifold. Thus, one would expect that scalar curvature
equations would have been used in these contexts, which is not the case. Reasons for this
may include a lack of effective PDE solvers for this operator. Indeed, there are very few
publications devoted to solving the 2-Hessian equation. In the early work of [SG10] a
quadratically constrained eigenvalue minimization problem is solved. In the unpublished
work of [Awa14], an iterative method with quadratic convergence rate is proposed. Gauss-
Seidel and semi-implicit solvers, that relate to the ones we present here, are also discussed.
The 2-Hessian operator also appears in conformal mapping problems. Conformal
surface mappings have been used for two dimensional image registration [AHTK99,
GWC+04], but they do not generalize directly to three dimensions. Quasi-conformal maps
have been used in three dimensions [WWJ+07, ZG11], however these methods are still
being developed.
More generally, k-Hessian equations appear in some problems in differential geometry:
generalizations of the Yamabe problem [Via00b, Via00a], the Calabi-Yau problem [AV10]
and the Christoffel-Minkowski problem [GM03].
4 1.1. LITERATURE REVIEW
Scalar curvature and the 2-Hessian equation
The 2-Hessian equation corresponds to scalar curvature, as we discuss below, and solving
the 2-Hessian PDE (or a related one) allows for the construction of hyper-surfaces of
prescribed curvatures, for example scalar curvature [GG02]. We start by briefly discussing
the Gaussian curvature due to its relation to the Monge-Ampère equation.
The Gaussian curvature of a two-dimensional surface is the product of the principal
curvatures, κ1, κ2 of the surface. It is an intrinsic quantity: it does not depend on the
embedding of the surface in space. Locally, the surface can be defined as the graph of a
function u(x), whose gradient of the function vanishes at x. Then the Gaussian curvature
at x is given by the determinant of the Hessian of u(x), det(D2u) = κ1κ2, which is the two
dimensional Monge-Ampère operator applied to u (if the gradient of u does not vanish at
x, additional first order terms appear).
In higher dimensions, curvature is a tensor rather than a scalar quantity. The curvature
tensor is defined by the sectional curvature, K(p, x), which is given by the Gaussian
curvature of the geodesic surface defined by the tangent plane, p, at x. The scalar curvature
(or the Ricci scalar), which is the trace of the curvature tensor, is the simplest curvature
invariant of a Riemannian manifold. It can be characterized as a multiple of the average
of the sectional curvatures. If we choose coordinates so that a three dimensional surface
is given by the graph of a function u(x) whose gradient vanishes at x, then the scalar
curvature is given by the 2-Hessian operator:
1
2
(
trace(D2u)2 − trace(
(D2u)2))
= κ1κ2 + κ1κ3 + κ2κ3
where κ1, κ2, κ3 are the three principal curvatures. Again, if the gradient of u does not
vanish at x, additional first order terms appear. However the equation above holds in
general if we replace the principal curvatures with the eigenvalues of the Hessian, which
leads to the 2-Hessian equation.
Since the second order terms pose the primary challenge in the solution of nonlinear
elliptic equations, we focus on the 2-Hessian equation in this work. In a similar way, the
Monge-Ampère equation can be related to the equation for Gauss curvature through the
inclusion of appropriate first order terms. In [BFO14] an extension of the Monge-Ampère
equation with first order nonlinear terms was studied and the primary challenge was the
boundary conditions.
Chapter 1. Introduction 5
Related work on curvature equations
The 2-Hessian equation is closely related to a curvature PDE in three dimensions. In two
dimensions there are several works on the evolution of curves using curvature, going
back to the seminal paper of Osher and Sethian [OS88]. In [Obe04], a finite difference
monotone scheme is given for the motion of level sets by mean curvature. The advantage
of monotone discretizations is that they have a convergence proof, and convergent schemes
are more stable and allow for faster solvers [Set96]. The surface evolver [Bra92] is a tool to
evolve two dimensional surfaces by curvature based on the minimization of its energy. In
[Sap06] one can find a relation between geometric PDEs and image analysis. For a review
of the numerical methods for curvature flows see [DDE05a].
Related work on the Monge-Ampère equation
In this thesis we study a fully nonlinear elliptic PDE, while most of the curvature flows
lead to quasilinear parabolic equations. Thus, we also review some of the related work on
the Monge-Ampère equation, a fully nonlinear elliptic PDE. For an extended review on
numerical methods for fully nonlinear elliptic PDEs see [FGN13].
The Monge-Ampère equation has been exhaustively studied. Consistent schemes
using either finite elements [Nei13, BN12] or finite differences [LR05] have been proposed.
However, these schemes are not monotone and therefore do not fall within the convergence
framework of Barles and Souganidis [BS91]. They require instead the PDE solution to be
sufficiently smooth and the numerical solver to be well initialized. Using wide stencil
discretizations, consistent monotone schemes were built [FO11a, FO11b], which are thus
provably convergent but have limited accuracy due to their directional resolution. This
limitation has been overcome recently. By introducing filtered schemes, which blend a
monotone scheme with an accurate (but possibly unstable) scheme, the authors in [FO13]
were able to obtain a provably convergent scheme with improved accuracy. Two other
solutions, specific to particular dimensions, have been proposed as well: in the two
dimensional setting using a mixture of finite differences and ideas from discrete geometry
[BCM14] and in the three dimensional setting using ideas from discretizations of optimal
transport based on power diagrams [Mir15].
The Monge-Ampère problem is related to the problem of prescribed Gauss curvature as
already mentioned. Numerical methods for the problem of prescribed Gauss curvature can
be found in [MO16, Fro16a] . The Gauss curvature flow is also used in image processing
for surface fairing [EE09].
6 1.1. LITERATURE REVIEW
1.1.3 Affine curvature
The planar motion of level sets by affine curvature is governed by the nonlinear PDE
ut = |∇u| (k[u])1/3 . (AC)
Here u = u(x, y) : R2 → R, ∇u = (ux, uy) denotes the gradient of u, and k[u] denotes the
curvature of the level set of u
k[u] = div
(
∇u
|∇u|
)
=uxxu2
y − 2uxuyuxy + uyyu2x
(u2x + u2
y)3/2. (1.1)
The affine curvature PDE is closely related to the well known PDE for motion of level sets
by mean curvature
ut = ∆1u := |∇u| k[u]. (MC)
The affine curvature evolution is one of the most fundamental geometric evolution
equations, after the mean curvature evolution. It was introduced by Sapiro and Tannen-
baum in [ST94] and [AST98] and has applications in mathematical morphology, edge
detection, image smoothing, and image enhancement (see [Sap06]).
The study of the affine curvature PDE is motivated by the recent work of Jeff Calder
[CS16], which provides an application to the statistics of large data sets. The convex
hull peeling algorithm [Cha85] provides an affine invariant notion of the median and the
quantiles of multidimensional probability distributions. While there is more than one
way to measure data depth [Bar76], affine invariance is an important property for such
measures [LPS+99]. The level sets of the solution of the affine curvature PDE, with right
hand side given by the probability density ρ, also give an affine invariant notion of the
depth of ρ. According to [CS16], these two notions of depth are equivalent: the rescaled
data depth layers of N data points sampled from the density, ρ, given by convex hull
peeling algorithm converge, in the limit N → ∞, to the levels given by the solution of
the PDE. Compared to convex hull peeling, the PDE characterization is efficient in terms
of the number of data points N : an efficient density estimation method can be used to
approximate ρ, and afterwards the PDE solver does not depend on N . This kind of limiting
PDE approach has already been shown to be effective for non-dominated sorting [CEH14].
Euclidean and Affine Curvature
We begin with a parametric description of the affine curvature evolution, and make a
comparison with the more familiar mean curvature evolution. We refer to [Sap06, Chapter
2] for more details. Consider a curve described parametrically C(s) : [a, b] ∈ R → R2
Chapter 1. Introduction 7
where s parameterizes the curve. If the curve is parameterized by the Euclidean arc length,∣
∣
∣
dCds
∣
∣
∣ = 1, then the curvature, k, is defined, up to a sign, by |k| := |Css|. It follows then
that circles have constant mean curvature. Letting t and n denote respectively the unit
Euclidean tangent and the Euclidean normal of the curve, we have
dCds
= t andd2Cds2
= kn.
The affine curvature arises from a different parameterization of the curve. Define the
parameter, r by the condition that the vectors Cr and Crr form a parallelogram of area 1,
[Cr, Crr] = 1.
Here the brackets denote the determinant of the matrix whose columns are given by those
vectors. By differentiating the last equation, we obtain [Cr, Crrr] = 0, which implies that
Crrr = µCr for some constant µ. Using the defining condition again, we obtain
µ = [Crr, Crrr],
which we define to be the affine curvature of the curve. The affine curvature is the simplest
nontrivial affine invariant of the curve [Su83]. Ellipses have constant affine curvature.
Similarly to the Euclidean curvature, we can define the affine tangent vector and the affine
normal vector, which we denote by ta and na, respectively. The following relation then
holds:
na = k1/3n + f(k, kp)t,
where f is a function of the Euclidean curvature and its first derivative.
Under the affine curvature evolution, any convex curve remains convex; any convex
smooth curve evolves to an ellipse until it collapses to a single point; any smooth curve
becomes convex after a certain time. Moreover, the affine curvature evolution is invariant
under the class of special affine transformations, which are defined by matrices with
determinant 1. Compare this to the mean curvature evolution, which shrinks curves to
circles [Gag84] and is invariant under the smaller class of orthogonal transformations.
Level Set PDE formulation
The Level Set Method [Set99c, OF03] for the affine curvature evolution results in the
PDE (AC). It has the following advantages compared to parameterized curve evolution:
(i) it provides a natural generalization of the flows when the curve becomes singular and
notions such as normals are not well defined; (ii) there is no need to track topological
8 1.1. LITERATURE REVIEW
changes since they are discovered when the corresponding level set is computed; (iii) it
can be discretized on a uniform grid, which is convenient for many applications.
In the Level Set Method, a curve is represented implicitly as the level set of the auxiliary
function u(x, y, t) : R2 × R → R, that is,
C(t) =
(x, y) ∈ R2 | u(x, y, t) = c
for some arbitrary constant c ∈ R. If u satisfies ut = |∇u| β[u(·, t)], for some function β
which depends on the level set of u, then all its level sets move in the normal direction with
speed β. For example, choosing β = 1, we obtain the time-dependent eikonal equation,
ut = |∇u|. Taking β = k[u] or (k[u])1/3 we obtain (MC) and (AC), respectively.
Geometric curve evolution formulation
Equivalently, we can also think of the evolution of a single curve. Denoting by C(p, t) :
S1 × [0, T ) → R2 a family of closed curves, they move with β velocity in the normal
direction if the following equation is satisfied:
∂C(p, t)
∂t= β(p, t)n(p, t), (1.2)
with C0 as the initial condition. A more general formulation may allow for a tangential
component of the velocity. However, if β is a geometric intrinsic characteristic of the curve
(meaning it does not depend on its the parameterization), then the tangential component
in the velocity does not influence the geometry of the deformation of the curve, just its
parameterization. Taking β = k(p, t) or (k(p, t))1/3 we recover the evolution of a single
curve by mean curvature and affine curvature, respectively. Notice that both the evolutions
can be written as∂C(p, t)
∂t= Css
where s denotes the Euclidean arc length and the affine arc length, respectively. For the
latter, we have Css = na(p, t). Hence, despite the fact that the affine differential geometry is
not defined for non-convex curves, we can define still define the affine curvature evolution
using the Euclidean curvature, by taking the velocity to be k1/3n [AST98].
Related numerical work on motion by mean curvature
The popularity and ubiquity of level set methods has given rise to numerous numerical
methods for the PDE governing the mean curvature evolution (MC) (see the review papers
[DDE05b] and [CMM11] for an extended review).
Chapter 1. Introduction 9
In the seminal article [OS88], using techniques from the hyperbolic conservation laws,
a scheme is proposed where each level set moves with velocity proportional to the mean
curvature. Crandall and Lions in [CL96] proposed a class of difference schemes for
quasilinear PDE, which include the motion by mean curvature. The schemes, which can be
made monotone by adding a small perturbation as linear diffusion, have some drawbacks,
such as degeneracy and singularities. This makes the scheme complex as several relations
on the parameters must be satisfied in order to ensure its convergence. Finite element
schemes have also been proposed [Wal96, CHR+05, DDE05b], however the theory does
not ensure the uniqueness of solutions.
In [Obe04] a convergent elliptic wide stencil finite difference scheme for ∆1u is pre-
sented based on taking the median of the values of u sampled in a small approximately
circular neighbourhood of x. The motivation follows from observing, using (1.1), that
∆1u = utt, t =(−uy, ux)
(u2x + u2
y)1/2,
where t is the (Euclidean) unit tangent. The median captures an approximation to utt,
the second tangential derivative of u, since the larger values point in the direction of the
gradient and the smaller values point in the opposite directions. Another elliptic scheme
has been proposed by Catté and Dibos [CDK95] which is equivalent to Oberman’s scheme
when the gradient is nonzero, but it lacks consistency otherwise (see [Tak07]).
Numerical schemes have also been proposed for the equivalent geometric curve evolu-
tion. One can interpret it as the singular limit of a semilinear reaction diffusion equation,
which leads to indirect numerical schemes [BG95, ESS92]. Similarly, Bence, Merriman and
Osher [MBO94] proposed a scheme that consists in repeatedly solving the heat equation
for a short time, by using convolution, followed by thresholding. It can be viewed as
well as a singular limit of a reaction diffusion equation. These thresholding methods are
effective for moving a given curve by the evolution, and allow for large time steps to be
taken. Using the equivalent level set representation moves every level set by the evolution,
but requires a much smaller time step.
Related numerical work on motion by affine curvature
Significantly less studied than the mean curvature evolution, there are still some numerical
methods for affine curvature evolution. The recent article [ERT10] gives a Bence-Merriman-
Osher [MBO94] thresholding scheme. It introduced a regularization of the cube root, which
was needed for theoretical purposes, but not in practice.
Alvarez and Guichard proposed a local scheme which lacks the affine invariance
10 1.2. ORGANIZATION OF THE THESIS
property [Gui94]. A morphological scheme which generalized [CDK95] was proposed
by Guichard and Morel for affine curvature [GM97]. This inf-sup scheme, although
morphologically invariant, has some limitations on the speed at which the level set curves
move. In [Moi98], a nonlocal geometric morphological scheme is presented. The author
introduces a geometrical operator, called affine erosion, based on the concept of σ-chords:
region with area σ enclosed by the segment joining two points in the curve and the curve
itself. The affine erosion operator is nonlocal, fully affine invariant and turns out to be
exactly the set of all the middle points of the σ-chords segments of the curve. The author
thus obtains a fast, but difficult to code, algorithm consistent with the curve evolution.
1.2 ORG ANI ZATION OF THE THES IS
The organization of the thesis is as follows: In chapter 2 we review the theory of viscosity
solutions. In chapter 3 we review the theory of monotone, elliptic and filtered schemes.
In chapter 4 we discuss the filtered schemes built for HJ equations, presenting numerical
results in one and two dimensions. In chapter 5, we discuss the numerical methods pro-
posed for the 2-Hessian equation. Computational results are presented on a number of
exact solutions which range in regularity from smooth to nondifferentiable and in shape
from convex to nonconvex. In chapter 6 we build a provably convergent scheme for the
nonlinear PDE that governs the motion of level sets by affine curvature. Numerical experi-
ments are presented to demonstrate the accuracy and stability of the discretization, as well
as the fact that the approximate solutions capture the affine invariance and morphological
properties of the evolution. Finally, in chapter 7 we summarize the results and discuss
future work.
CHAPTER 2
VISCOSITY SOLUTIONS
In this chapter we discuss viscosity solutions, the appropriate notion of solution for
elliptic partial differential equations (PDEs) when working in the framework of Barles and
Souganidis. The standard reference and the one mainly followed here is [CIL92]. It is also
worth mentioning the textbook by Koike [Koi04].
Viscosity solutions were first introduced in a series of papers by Crandall, P. L. Lions
and Evans [Eva80, CL83, CEL84]. The book by Evans [Eva98] also provides a good
introduction to the first order case focusing on Hamilton-Jacobi equations and applications
to control theory.
2.1 MOTIVATION OF VIS C OS ITY S OLUTIONS
In this section, we give a brief example to motivate the use of viscosity solutions in general.
Consider the following Eikonal equation in the one-dimensional setting
|u′(x)| = 1, x ∈ (−1, 1),
u(±1) = 0.
This PDE has no differentiable solution but there are infinitely many solutions that are
differentiable almost everywhere: u(x) = 1 − |x| and u(x) = min1 − |x| , |x| are two
simple examples. However, we are interested in a definition of solution that amongst
other properties satisfies uniqueness. Thus, from the infinite set of almost everywhere
differentiable solutions, we must select a particular one.
Consider then for each ε > 0 the following equation
−εu′′(x) + |u′(x)| = 1, x ∈ (−1, 1),
u(±1) = 0,
where the term εu′′(x) regularizes the equation. Denote its unique differentiable solution
by uε. It is possible to show that
limε→0
uε(x) = 1 − |x| for all x ∈ [−1, 1].
This technique is called vanishing viscosity. As we will see below, this limit captures
11
12 2.2. DEFINITION OF VISCOSITY SOLUTION
exactly the unique viscosity solution of the PDE. The term “viscosity” is inspired by this
method, but in fact has no relation to the actual definition of viscosity solution.
2.2 DEFINITION OF VIS C OS ITY S OLUTION
In this section, we present the definition of viscosity solution. Viscosity solutions can be
defined in (slightly) different ways depending on the context: we can allow for discontinu-
ous viscosity solutions and the boundary conditions can be treated in a viscosity sense or
not. Here, we present the definition of interest for the framework of Barles and Souganidis
[BS91], which we discuss in detail in chapter 3.
We start by considering the PDE
F (x, u(x), ∇u(x), D2u(x)) = 0, x ∈ Ω, (2.1)
where Ω is a bounded open subset of Rn, F ∈ C(R × R × R
n × Sn) and Sn denotes the set of
symmetric n × n matrices. The theory of viscosity solution applies to the PDEs that satisfy
a monotonicity condition.
Definition 2.2.1. The function F : R × R × Rn × Sn → R is degenerate elliptic (in the sense of
[CIL92]) if
F (x, r, p, M) ≤ F (x, r, p, N), for all x ∈ R, r ∈ R, p ∈ Rn, M N,
where N M if dᵀNd ≤ dᵀMd for all d ∈ Rn. If in addition
F (x, r, p, M) ≤ F (x, s, p, M), for all x ∈ R, r ≤ s, p ∈ Rn, M ∈ Sn,
we say that F is proper.
Example 2.1. By this convention, the Laplacian operator is elliptic when written F (M) =
−tr(M). Some authors use the other convention, without the minus sign.
Before presenting the definition of viscosity solutions, we recall the definition of upper
and lower semicontinuous functions as well as the definition of upper and lower semicon-
tinuous envelopes. To be precise, we recall too the definitions of lim sup and lim inf. Given
a function u : U → R, we have
lim supy∈U→x
u(y) := infr>0
sup u(y) | y ∈ U ∩ B(x, r) ,
Chapter 2. Viscosity Solutions 13
and
lim infy∈U→x
u(y) := supr>0
inf u(y) | y ∈ U ∩ B(x, r) ,
where B(x, r) denotes the open ball centred at x with radius r.
Definition 2.2.2. A function u : U → R is upper (resp. lower) semicontinuous at x ∈ U provided
lim supy∈U→x
u(y) ≤ u(x)(
resp. lim infy∈U→x
u(y) ≥ u(x))
.
We denote by USC(U) (resp. LSC(U)) the collection of functions that are upper (resp. lower)
semicontinuous at all points of U .
Definition 2.2.3. The upper (resp. lower) semicontinuous envelopes of a function u : U → R is
defined by
u∗(x) = lim supy∈U→x
u(y)(
resp.u∗(x) = lim infy∈U→x
u(y))
.
We can now give the definition of viscosity solutions.
Definition 2.2.4. Let u denote the function u : Ω → R.
(i) We say that u ∈ USC(Ω) is a viscosity subsolution of (2.1) if for every φ ∈ C2(Ω) such that
u − φ has a local maximum at x ∈ Ω, F (x, u(x), ∇φ(x), D2φ(x)) ≤ 0.
(ii) We say that u ∈ LSC(Ω) is a viscosity supersolution of (2.1) if for every φ ∈ C2(Ω) such
that u − φ has a local minimum at x ∈ Ω, F (x, u(x), ∇φ(x), D2φ(x)) ≥ 0.
(iii) We say that u is a viscosity solution of (2.1) if u is both a viscosity subsolution and a viscosity
supersolution.
Remark 2.1. When checking the definition of a viscosity solution we can limit ourselves to
considering unique, strict, global maxima (minima) of u − φ with a value of zero at the
extremum. Geometrically, this means that φ touches u at x from above (below). See, for
example, [Koi04, Prop 2.2].
Remark 2.2. As it stands, viscosity solutions are continuous. We can allow for discontinuous
viscosity solutions, by saying that a locally bounded function u is a viscosity solution (2.1)
if u∗ is a viscosity subsolution of (2.1) and u∗ is a viscosity supersolution of (2.1).
Despite not including the boundary conditions, the above definition is still of interest
when we discuss comparison principles in the next section.
14 2.2. DEFINITION OF VISCOSITY SOLUTION
In general, we are interested in the boundary value problem (BVP)
F (x, u(x), ∇u(x), D2u(x)) = 0, if x ∈ Ω,
B(x, u(x), ∇u(x)) = 0, if x ∈ ∂Ω.(2.2)
where B ∈ C(∂Ω × R × Rn) is proper.
Definition 2.2.5. Let u denote the function u : Ω → R.
(i) We say that u ∈ USC(Ω) is a viscosity subsolution of (2.2) if for every φ ∈ C2(Ω) such that
u − φ has a local maximum at x ∈ Ω,
F (x, u(x), ∇φ(x), D2φ(x)) ≤ 0, if x ∈ Ω,
min
F (x, u(x), ∇φ(x), D2φ(x)), B(x, u(x), ∇φ(x))
≤ 0, if x ∈ ∂Ω.
(ii) We say that u ∈ LSC(Ω) is a viscosity supersolution of (2.2) if for every φ ∈ C2(Ω) such
that u − φ has a local minimum at x ∈ Ω,
F (x, u(x), ∇φ(x), D2φ(x)) ≥ 0, if x ∈ Ω,
max
F (x, u(x), ∇u(x), D2u), B(x, u(x), ∇φ(x))
≥ 0, if x ∈ ∂Ω.
(iii) We say that u is a viscosity solution of (2.2) if u is both a viscosity subsolution and viscosity
supersolution of (2.2).
Remark 2.3. Alternatively, we can define viscosity solutions in terms of the function
G : Ω × R × Rn × Sn → R given by
G(x, r, p, M) =
F (x, r, p, M), x ∈ Ω,
B(x, r, p), x ∈ ∂Ω.
It is easy to see that
G∗(x, r, p, M) = G∗(x, r, p, M) = F (x, r, p, M), x ∈ Ω,
G∗(x, r, p, M) = minF (x, r, p, M), B(x, r, p), x ∈ ∂Ω,
G∗(x, r, p, M) = maxF (x, r, p, M), B(x, r, p), x ∈ ∂Ω.
Then, u ∈ USC(Ω) is a viscosity subsolution of (2.2) if for every φ ∈ C2(Ω) such that u − φ
has a local maximum at x ∈ Ω, G∗(x, u(x), ∇φ(x), D2φ(x)) ≤ 0. Similarly u ∈ LSC(Ω) is a
viscosity supersolution of (2.2) if for every φ ∈ C2(Ω) such that u − φ has a local minimum
at x ∈ Ω, G∗(x, u(x), ∇φ(x), D2φ(x)) ≥ 0.
Chapter 2. Viscosity Solutions 15
Example 2.2. One can check that u(x) = 1 − |x| is the viscosity solution of
|∇u| = 1, x ∈ (−1, 1),
u(−1) = u(1) = 0.
Here we consider F (p) = |p| − 1, which leads to
2.3 COMPARI S ON P RINC IP LE AND UNIQUENES S
In this section, we discuss comparison principles for degenerate elliptic PDEs. In general,
comparison principles tells us that subsolutions lie below supersolutions. The difference
between the different comparison principles presented here lies in the way the behaviour
at the boundary is treated.
We start by discussing what we call a weak comparison principle, where we assume
that the subsolution lies below the supersolution at the boundary. This crucial assumption
makes the result substantially easier to prove.
Definition 2.3.1 (Weak comparison principle). We say that the PDE (2.1) has a weak com-
parison principle if whenever u is a viscosity subsolution and v a viscosity supersolution of (2.1)
in Ω such that u ≤ v on ∂Ω, u ≤ v in Ω.
We start with a result for first order equations. To simplify the presentation, we define
the set of modulus of continuity functions as the set
M =
ω : R+ → R
+ | ω(·) is continuous , limr→0
ω(r) = 0
.
Theorem 2.3.2. Assume that F (x, r, p, M) = H(x, p) − f(x) where
|H(x, p) − H(y, p)| ≤ ωH (|x − y| (1 + |p|)) , for all x, y ∈ Ω, p ∈ Rn,
with ωH ∈ M. Suppose that H has homogeneous degree α > 0 with respect to its second argument,
i.e., there is α > 0 such that
H(x, µp) = µαH(x, p), for all x ∈ Ω, p ∈ Rn, µ > 0.
Suppose as well that f ∈ C(Ω) and that there is σ > 0 such that minx∈Ω
f(x) = σ. Then the PDE
(2.1) satisfies a weak comparison principle.
Proof. See [Koi04] for a proof.
16 2.3. COMPARISON PRINCIPLE AND UNIQUENESS
The general case for second order equations is more subtle and it requires additional
assumptions on the PDE, beyond degenerate ellipticity. We introduce the concept of strictly
proper and structure condition.
Definition 2.3.3. We say that F is strictly proper (in Ω) if there exists γ > 0 such that
γ(s − r) ≤ F (x, s, p, M) − F (x, r, p, M) for all r ≤ s, x ∈ Ω, p ∈ Rn, M ∈ Sn.
Remark 2.4. If F is strictly proper then F is proper.
Definition 2.3.4. We say that F satisfies the weak structure condition if there exists ωF ∈ Msuch that if M, N ∈ Sn and α ∈ R satisfy
−3α
I 0
0 I
≤
M 0
0 −N
≤ 3α
I −I
−I I
,
then
F (y, r, α(x − y), N) − F (x, r, α(x − y), M) ≤ ωF
(
α |x − y|2 + |x − y|)
for all x, y ∈ Ω and r ∈ Rn.
Remark 2.5. If F satisfies the structure condition, then F is degenerate elliptic. Moreover,
the structure condition arises naturally in the proof of the comparison principle. For more
details see [CIL92].
Theorem 2.3.5. Assume that F is degenerate elliptic, strictly proper and satisfies the weak struc-
ture condition. Then the PDE (2.1) satisfies a weak comparison principle.
Remark 2.6. In [IL90], weak comparison principles are obtained for F proper. Instead of
the strictly properness of F , one assumes the existence of δ > 0 such that either u is a
subsolution of
F (x, u(x), ∇u(x), D2u(x)) + δ = 0
or v is a supersolution of
F (x, v(x), ∇v(x), D2v(x)) − δ = 0.
In particular, if F (x, r, p, M) = H(r, p, M) + f(x) is such that there exits α > 0 such that
H(µr, µp, µM) = µαH(r, p, M), for all µ > 0, r ∈ R, p ∈ Rn, M ∈ Sn,
and there exists σ > 0 such that minx∈Ω
f(x) = σ then a subsolution (resp. supersolution) of
(2.1) can be perturbed to produce a strict subsolution (resp. supersolution) as above. Thus,
Chapter 2. Viscosity Solutions 17
in such case, F satisfies a weak comparison principle. This includes the Monge-Ampère
equation and k-Hessian equations.
The weak comparison principle discussed above is a useful result. For instance, it
implies uniqueness.
Proposition 2.3.6. Assume that the PDE (2.1) has a weak comparison principle. Let u and v be
viscosity solutions of (2.1) in Ω with u = v on ∂Ω. Then u = v in Ω.
Proof. Since u (resp. v) and v (resp. u) are viscosity subsolution and supersolution, re-
spectively, and since u ≤ v (resp. v ≤ u) on ∂Ω, the weak comparison principle yields
u ≤ v (resp. v ≤ u) in Ω. Combining the two inequalities together, we get u = v in Ω as
desired.
However, when building numerical schemes for degenerate elliptic PDEs, in order
to prove their convergence using the Barles-Souganidis theorem, one needs a strong
comparison principle where the boundary conditions are treated in the viscosity sense.
Definition 2.3.7 (Strong comparison principle). We say that the BVP (2.2) has a strong
comparison principle if whenever u is a viscosity subsolution and v a viscosity supersolution of
(2.2), u ≤ v on Ω.
Definition 2.3.8. We say that F satisfies the strong structure condition if there exists ωF ∈ Msuch that if M, N ∈ Sn and α ∈ R satisfy
−3α
I 0
0 I
≤
M 0
0 −N
≤ 3α
I −I
−I I
,
then
F (y, r, p, N) − F (x, r, p, X) ≤ ωF
(
α |x − y|2 + |x − y| (|p| + 1))
for all x, y ∈ Ω, r ∈ Rn and p ∈ R
n.
Theorem 2.3.9. Let B : ∂Ω × ×R × Rn be given by B(x, r, p) = 〈n(x), p〉 + f(x, r), where
f ∈ C(∂Ω × R) is nondecreasing in r for each x ∈ ∂Ω and n(x) denotes the outward unit normal
to x ∈ ∂Ω. Assume that Ω is a compact C1 n-submanifold with boundary of Rn that satisfies the
uniform exterior sphere condition: there exists r > 0 such that B(x + rn(x), r) ∩ Ω = ∅ for all
x ∈ ∂Ω. Assume as well that there is a neighbourhood V of ∂Ω and ωV ∈ M such that
|F (x, r, p, M) − F (x, r, q, N)| ≤ ωV (|p − q| + ‖M − N‖)
18 2.3. COMPARISON PRINCIPLE AND UNIQUENESS
for all x ∈ V , p, q ∈ Rn, M, N ∈ Sn. Suppose as well that F is degenerate elliptic, strictly proper
in Ω and satisfies the strong structure condition. Then the BVP (2.2) satisfies a strong comparison
principle.
The above theorem deals with the case of Neumann boundary conditions. The case
of Dirichlet boundary conditions (B(x, r, p) = r − g(x)) is however more subtle. If the
viscosity subsolution and viscosity supersolution are assumed to be continuous at the
boundary, then the strong comparison principle holds (see [CIL92] for details).
We are interested in the application of the strong comparison principle in the Barles
and Souganidis theorem. In such a framework and in some special cases, it is possible to
show that the boundary conditions are satisfied pointwise by the viscosity subsolution
u and viscosity supersolution v (i.e. u(x) = v(x) = g(x) for all x ∈ ∂Ω), in which case the
strong comparison principle follows from the weak comparison principle. This was the
case in [FJ16].
A strong comparison principle is not available in general in the Dirichlet case when
there exists a viscosity solution that does not satisfy the boundary conditions pointwise.
This is illustrated for the eikonal equation in Example 2.3 below. Another example is
provided in [Fro16a] for the prescribed Gauss curvature problem.
Example 2.3. Consider the function
u(x) =
x, x ∈ (0, 1),
2, x = 1.
One can show that u is a viscosity subsolution of
|u′(x)| = 1, x ∈ (0, 1),
u(0) = 0,
u(1) = 2.
One the other hand, the function u(x) = x is a viscosity solution. This simple example
shows that the Eikonal equation does not satisfy a strong comparison principle: u is a
viscosity subsolution that does not lie below the viscosity supersolution u. However, it is
true that u ≤ u in Ω. This was precisely the result proven in [Fro16b] for the prescribed
Gauss curvature problem: viscosity subsolutions lie below viscosity supersolutions in Ω.
Notice that the viscosity solution u does not satisfy the boundary conditions in the strong
sense.
Chapter 2. Viscosity Solutions 19
Proving a strong comparison principle, in particular in the case of the Dirichlet bound-
ary conditions, is still the subject of current research and not the focus of this thesis.
Moreover, we are interested in problems with continuous viscosity solutions. Thus, we
will assume that the problems considered here satisfy a strong comparison principle.
2.4 EXI S TENC E OF VIS C OS ITY S OLUTIONS
In this section, we briefly discuss the existence of viscosity solutions.
In general, existence of viscosity solutions follows from Perron’s method, although in
some specific cases there exists solution formulas. This is the case of Bellman and Isaacs
equations (see [Koi04]) or the Hopf-Lax formula for some Hamilton-Jacaboi equations (see
[Eva98]).
Similarly to the results obtained for the strong comparison principle, we have slightly
different results depending on the type of boundary conditions, although both results
follow from Perron’s method.
For the Neumman problem, existence follows under the same assumptions used to
prove a strong comparison principle.
Theorem 2.4.1. Under the assumptions of Theorem 2.3.9, there exists a viscosity solution (2.2).
As for the Dirichlet problem, we have the following result.
Theorem 2.4.2. Let B(x, r, p) = r − g(x) and suppose that (2.1) has a weak comparison principle.
Suppose also that there is a strong viscosity subsolution u and a strong viscosity supersolution u of
(2.1) that satisfy the boundary condition u∗(x) = u∗(x) = g(x) for x ∈ ∂Ω. Then
W (x) = sup
w(x) | u ≤ w ≤ u in Ω and w is a strong viscosity subsolution of (2.1)
is a viscosity solution of (2.1) with W (x) = g(x) for all x ∈ ∂Ω.
Remark 2.7. Although the existence of viscosity subsolution and viscosity supersolution
(that satisfy the boundary conditions) is not present in Theorem 2.4.1, their existence
follows from the assumptions in the theorem.
CHAPTER 3
NUMERICAL SCHEMES
In this chapter, we discuss the framework developed in [BS91] that provides conditions
under which approximation schemes converge to the unique viscosity solution of an
elliptic PDE. We will focus on finite difference schemes, but the notions of monotone,
elliptic and filtered schemes can be extended to different frameworks (e.g. finite elements).
3.1 MONOTONE FINI TE DIFFER ENCE S C HEMES
Let D be the computational domain such that Ω ⊆ D and consider a set of discretization
points Gh ⊆ D. Here h is a small parameter relating to the grid resolution, which we
assume to be such that
limh→0
supy∈Ω
minx∈Gh
|x − y| = 0. (3.1)
Let C(Gh) denote the set of grid functions u : Gh → R. Define
GhV = Ω ∩ Gh, ∂Gh = Gh \ Gh
V .
For simplicity, we assume that ∂Gh ⊆ ∂Ω.
Example 3.1. Typically, we take Ω = D to be a n-cube and discretize with a uniform
grid spacing h. For instance in the one dimensional setting (n = 1), if D = [0, 1], then
Gh = x ∈ hZ | x ∈ D and ∂Gh = 0, 1.
Definition 3.1.1. A finite difference scheme is a map F h : C(Gh) → C(Gh), such that given
u ∈ C(Gh) we write, for x ∈ Gh,
F h[u](x) = F h(x, u(x), u(·)), (3.2)
where u(·) indicates the values of the grid function u. We say the finite difference scheme F h has a
stencil of width W if F h depends only on values u(y) for ‖y − x‖∞ ≤ Wh. A solution of the finite
difference scheme F h is a grid function u ∈ C(Gh) which satisfies the equation
F h[u](x) = 0 for all x ∈ Gh. (3.3)
Remark 3.1. The explicit dependence on u(x), which we refer to as the reference variable,
will become clear later when we define elliptic finite difference schemes.
21
22 3.1. MONOTONE FINITE DIFFERENCE SCHEMES
Definition 3.1.2. The finite difference scheme (3.2) is monotone if as function F h : Gh × R ×C(Gh) → R we have
u(·) ≥ v(·) =⇒ F h(x, r, u(·)) ≤ F h(x, r, v(·))
for all x ∈ Gh, r ∈ R and u, v ∈ C(Gh).
Definition 3.1.3. The finite difference scheme F h (3.2) is consistent with the BVP (2.2) (in the
sense of [BS91]) if for any smooth function φ and x ∈ Ω,
lim suph→0,y∈Gh→x,ξ→0
F h(y, φ(y) + ξ, φ(·) + ξ) ≤ G∗(x, u(x), ∇φ(x), D2φ(x)),
lim infh→0,y∈Gh→x,ξ→0
F h(y, φ(y) + ξ, φ(·) + ξ) ≥ G∗(x, u(x), ∇φ(x), D2φ(x)).
where
G(x, r, p, M) =
F (x, r, p, M), x ∈ Ω,
B(x, r, p), x ∈ ∂Ω.
Remark 3.2. Most schemes will have only h as a parameter. For those, we say the scheme is
accurate to order k if for any smooth function φ and x ∈ Gh ∩ Ω
F h[φ](x) = F (x, φ(x), ∇φ(x), D2φ(x)) + O(hk).
The order of accuracy of a scheme (and its consistency) are usually verified by a Taylor
series argument.
Stability is a desirable property in numerical methods, but the precise definition de-
pends on the context. Here we follow [BS91].
Definition 3.1.4. The finite difference scheme F h (3.2) is stable if there exists h0 > 0 such that
for all 0 < h < h0 any solution u ∈ C(Gh) of (3.3) is bounded independently of h.
Our finite difference schemes and their solutions are defined only on a finite set of
discretization points Gh and not on the entire domain Ω as in [BS91]. Therefore, we must
modify the original proof of Barles and Souganidis to account for the different framework.
This is accomplished here with a nearest neighbour extension: letting Uh ∈ C(Gh) be a
solution the approximation scheme on the grid, we define the piecewise constant extension
as
uh(x) = max
Uh(y) | y ∈ Gh, |y − x| = minz∈Gh
|z − x|
(3.4)
Chapter 3. Numerical Schemes 23
for all x ∈ Ω. Likewise, the finite difference schemes are also extended to functions defined
on the whole domain Ω:
F h[u](x) = max
F h[u](y) | y ∈ Gh, |y − x| = minz∈Gh
|z − x|
(3.5)
for all x ∈ Ω.
Remark 3.3. The extension of F h (3.5) is still monotone as a function F h : Ω ×R× C(Ω) → R.
Indeed, let x ∈ Ω, r ∈ R and u, v ∈ C(Ω) with u(·) ≥ v(·). Then
F h(x, r, u(·)) = F h(y, r, u(·)) for some y ∈ Gh,
≤ F h(y, r, v(·)) since F h is monotone and u(·) ≥ v(·),≤ F h(x, r, v(·)) by (3.5).
We are then ready to present the Barles and Souganidis theorem. The proof is similar
to the proof of Theorem 3.4.3.
Theorem 3.1.5. Assume the BVP (2.2) satisfies a strong comparison principle (see Definition
2.3.7). Let F h by any stable, monotone finite difference scheme consistent with the BVP (2.2) and
Uh ∈ C(Gh) any solution of (3.3). Then
limh→0
uh(x) = u(x), for all x ∈ Ω,
where uh is the piecewise constant extension of Uh (3.4) and u is the viscosity solution of (2.2).
3.2 ELLIP TI C FINITE DI FFERENC E S C HEMES
The definition of monotone schemes does not provide any insight on how to build them.
In this section, we discuss elliptic schemes, introduced by Oberman in [Obe06a]. These
not only guarantee monotonicity but are also easy to build as they draw inspiration from
finite difference approximations.
Definition 3.2.1. A finite difference scheme F h is elliptic if it can be written as
F h[u](x) = F h(x, u(x), u(x) − u(·)),
where F h is nondecreasing in its second and third arguments, i.e.,
r ≤ s, u(·) ≤ v(·) =⇒ F h (x, r, u(·)) ≤ F h(x, s, v(·)),
for all x ∈ Gh, r, s ∈ R and u, v ∈ C(Gh).
24 3.2. ELLIPTIC FINITE DIFFERENCE SCHEMES
Example 3.2. The backward approximation
u(x) − u(x − h)
h= ux(x) + O(h)
is an elliptic scheme for ux. Similarly, the forward approximation
−u(x + h) − u(x)
h= −ux(x) + O(h)
is an elliptic scheme for −ux.
Proposition 3.2.2. Elliptic finite difference scheme are monotone.
Proof. Let x ∈ Gh, r ∈ R and u, v ∈ C(Gh). Then
u(·) ≥ v(·) =⇒ r − u(·) ≤ r − v(·) =⇒ F h (x, r, r − u(·)) ≤ F h(x, r, r − v(·))
since F h is elliptic.
As the name suggests, elliptic finite difference schemes are the discrete counterpart of
degenerate elliptic operators. Ideally, we would like that these approximation schemes
inherit the basic structure of the underlying degenerate elliptic PDE. In particular, we
note that under similar assumptions, the elliptic finite difference schemes enjoy a discrete
comparison principle.
Definition 3.2.3. The comparison principle holds for the finite difference operator F h : C(Gh) →C(Gh), if F h[u] ≤ F h[v] implies u ≤ v, more precisely,
F h[u](x) ≤ F h[v](x) for all x ∈ Gh =⇒ u(x) ≤ v(x) for all x ∈ Gh. (3.6)
Remark 3.4. In the discrete comparison principle (3.6), the boundary conditions are encoded
in F h. For Dirichlet boundary conditions, we define
F h[u](x) = u(x) − g(x), for all x ∈ ∂Gh.
Thus, the assumption F h[u] ≤ F h[v] means u ≤ v at boundary points. Uniqueness of
solutions follows from the discrete comparison principle, since if u, v are solutions, then
F h[u] = F h[v] = 0, so u ≤ v and u ≥ v, and thus u = v.
Definition 3.2.4. The finite difference scheme F h is proper if is strictly increasing on its second
argument, i.e., there exists δ > 0 such that
r ≤ s =⇒ F h(x, r, u(·)) − F h(x, s, u(·)) ≤ δ(r − s)
Chapter 3. Numerical Schemes 25
for all x ∈ Gh and u ∈ C(Gh).
Remark 3.5. Without any loss of generality, we can assume the scheme to be proper. Indeed,
if the scheme is not proper, we can consider instead the scheme F h[u] + εu for arbitrarily
small ε (for example, smaller than the discretization error).
Theorem 3.2.5. If F h is a proper elliptic finite difference scheme, then a comparison principle (3.6)
holds for F h.
Proof. See [Obe06a].
The finite difference schemes may be nonlinear and nondifferentiable, but we will
require them to be Lipschitz continuous.
Definition 3.2.6. The finite difference scheme F h : C(Gh) → C(Gh) is Lipschitz continuous with
constant Kh if Kh is the smallest constant such that
∣
∣
∣F h (x, r, u(·)) − F h (x, s, v(·))∣
∣
∣ ≤ Kh max(|r − s|, ‖u − v‖∞), for all x ∈ Gh.
Remark 3.6. In the definition above, the maximum on the right hand side can be replaced
with a maximum over the neighbouring grid values without changing the Lipschitz
constant.
As we saw in the previous section, we require that our schemes are well-posed in the
sense that solutions exist and are stable. Here we will present a result that shows that these
properties follow from the existence of strict classical subsolutions and supersolutions,
whose definition we give below.
Definition 3.2.7. A function u ∈ C2(Ω) is a strict classical subsolution of (2.2) if there exists
some µ > 0 such that
F (x, u(x), ∇u(x), D2u(x)) ≤ −µ, if x ∈ Ω,
min
F (x, u(x), ∇u(x), D2u), B(x, u(x), ∇u(x))
≤ −µ, if x ∈ ∂Ω.
A function u ∈ C2(Ω) is a strict classical supersolution of (2.2) if there exists some µ > 0 such that
F (x, u(x), ∇u(x), D2u(x)) ≥ µ, if x ∈ Ω,
max
F (x, u(x), ∇u(x), D2u), B(x, u(x), ∇u(x))
≥ µ, if x ∈ ∂Ω.
Lemma 3.2.8. Let F h be elliptic Lipchitz finite difference scheme consistent with the BVP (2.2).
Suppose also that there exists functions v, w ∈ C2(Ω) such that v is a strict subsolution and w is a
26 3.2. ELLIPTIC FINITE DIFFERENCE SCHEMES
strict supersolution of (2.2). Then for sufficiently small h > 0, the approximation scheme F h has a
solution and F h is stable.
Proof. See [Fro16b].
Remark 3.7. Alternatively, in [Obe06a] the author shows that if a scheme is proper, Lipschitz
continuous and elliptic, then solutions exist and are stable.
Although the framework is defined in the context of stationary equation, it can be used
to treat time dependent parabolic equations of the form ut + F [u] = 0. Below we give
sufficient conditions to build a monotone scheme for these equations. The time derivative
in the equation is treated with a forward Euler step which leads to definition of the explicit
Euler map.
Definition 3.2.9. For ρ > 0, define the Euler map Sρ : C(Gh) → C(Gh) by
Sρ[u] = u − ρF h[u].
Proposition 3.2.10. Let F h be a Lipschitz continuous, elliptic scheme with Lipschitz constant Kh.
Assume that the CFL condition ρ ≤ 1/Kh is satisfied. Then
u(·) ≤ v(·) =⇒ Sρ[u] ≤ Sρ[v]
for all u, v ∈ C(Gh).
Proof. See [Obe06a].
Remark 3.8. In addition, if we assume that F h is proper and the strict inequality ρ < 1/Kh
holds, F h has a unique solution and the iterates of the Euler map converge to the solution
for arbitrary initial data. In other words, the Euler map Sρ is a convergent solver to the
unique solution of F h[u] = 0.
The approximate solutions uh,dt of ut + F [u] = 0 with initial condition u(x, 0) = u0(x)
are then defined as
uh,dt(x, n + 1) = Sdt[u(·, n)],
uh,dt(x, 0) = u0(x)
for all x ∈ GhV , n = 0, 1, . . ., with the appropriate boundary conditions. The underlying
finite difference scheme is monotone, a direct consequence of Proposition 3.2.10 provided
dt satisfies the CFL condition. In this specific context of parabolic equations, monotonicity
Chapter 3. Numerical Schemes 27
means that the following discrete comparison principle holds: if u(x, n), v(x, n) are solu-
tions of the scheme, u(x, n) ≤ v(x, n) for all x ∈ Gh implies u(x, n + 1) ≤ v(x, n + 1) for all
x ∈ Gh.
3.3 BUILDING ELLIP TI C SC HEMES
In order to build elliptic finite difference schemes, we study the properties of nonde-
creasing maps. This is required since in general elliptic schemes are built composing
nondecreasing maps with elliptic terms. Moreover, for some nonlinear elliptic PDEs the
domain of ellipticity is restricted and thus we need to build nondecreasing extensions
of the underlying functions. This is the particular case of the 2-Hessian equation (see
chapter 5). In this section we define elliptic schemes for |∇u| and − |∇u|. This is done both
as an illustration of the general principle, and because these schemes will be used later in
the thesis.
We start with some definitions, that also serve as examples.
Definition 3.3.1. For u : R2 → R, define the standard finite differences
uhx(x, y) :=
u(x + h, y) − u(x − h, y)
2h,
uhxx(x, y) :=
u(x + h, y) − 2u(x, y) + u(x − h, y)
h2,
uhxy(x, y) :=
u(x + h, y + h) + u(x − h, y − h) − u(x + h, y − h) − u(x − h, y + h)
4h2,
(3.7)
for ux, uxx, uxy, respectively, and, similarly for uhy , and uh
yy in the y coordinate. These are second
order accurate approximations. Only the operator −uhxx is elliptic, the others are not.
Definition 3.3.2. Define the backward and forward first order derivatives
D−x [u](x, y) :=
u(x, y) − u(x − h, y)
h= ux(x, y) + O,
−D+x [u](x, y) :=
u(x, y) − u(x + h, y)
h= −ux(x, y) + O,
and, similarly D−y [u] and D+
y [u]. Both D−x [u] and −D+
x [u] are elliptic.
Definition 3.3.3 (nondecreasing functions). For x, y ∈ RN we say x ≤ y if xi ≤ yi for all
i = 1, . . . , N . We say the function F : RN → R is nondecreasing, and write F ∈ ND(RN), if
x ≤ y =⇒ F (x) ≤ F (y).
28 3.3. BUILDING ELLIPTIC SCHEMES
Write R+ = x ∈ R | x ≥ 0 and R
− = x ∈ R | x ≤ 0. Furthermore, if F ∈ ND(RN) and
F : RN → R
+, (resp. F : RN → R
−) we write F ∈ ND+(RN) (resp. F ∈ ND−(RN)).
Remark 3.9. When f ∈ C1(RN), if f is nondecreasing in each variable, i.e., fxi≥ 0 for all
i = 1, . . . , N , then f ∈ ND(RN).
Remark 3.10. The set of nondecreasing functions is closed under the composition of func-
tions. In particular, sums of nondecreasing functions are nondecreasing.
We give some simple examples in the form of definitions as they are important on their
on: they constitute the building blocks of some of the elliptic schemes proposed in this
thesis.
Definition 3.3.4. The function x+ = max(x, 0) ∈ ND+(R) and x− = min(x, 0) is in ND−(R).
Definition 3.3.5. Write N(x, y) =√
x2 + y2, and define
N+(x, y) := N(x+, y+) and N−(x, y) := −N(x−, y−).
Then N+ ∈ ND+(R2), and N− ∈ ND−(R2). Furthermore, N+ = N on x, y ≥ 0, N− = −N
on x, y ≤ 0.
Example 3.3 (Upwinding). More generally, if we consider the operator a(x)ux, then the
upwind discretization
a+D−x [u] + a−D+
x [u]
is first order accurate and elliptic.
We now present two simple examples of composing nondecreasing maps with elliptic
terms to build elliptic schemes.
Example 3.4. Define
∣
∣
∣uhx
∣
∣
∣
+= max
−D+x [u], D−
x [u], 0
, −∣
∣
∣uhx
∣
∣
∣
−= min
−D+x [u], D−
x [u], 0
that approximate |ux| and −|ux| to first order. The operators∣
∣
∣uhx
∣
∣
∣
+and −
∣
∣
∣uhx
∣
∣
∣
−are elliptic,
the former is nonnegative, and the latter is nonpositive.
Proof. Since D−x and −D+
x are elliptic and max, min ∈ ND(R2), the composed functions∣
∣
∣uhx
∣
∣
∣
+and −
∣
∣
∣uhx
∣
∣
∣
−are elliptic.
Example 3.5. Define
|∇uh|+ = N(
|uhx|+, |uh
y |+)
, −|∇uh|− = −N(
−|uhx|−, −|uh
y |−)
Chapter 3. Numerical Schemes 29
which are elliptic, consistent with |∇u|, and − |∇u|, respectively, and first order accurate.
Proof. Since |uhx|+ and |uh
y |+ are nonnegative,
|∇uh|+ = N(
|uhx|+, |uh
y |+)
= N+(
|uhx|+, |uh
y |+)
.
Thus, since N+ ∈ ND(R2) and |uhx|+ and |uh
y |+ are elliptic, the composed function |∇uh|+is elliptic.
Similarly, −|uhx|− and −|uh
y |− are elliptic and nonpositive and
−|∇uh|− = N−(
−|uhx|−, −|uh
y |−)
with N− ∈ ND(R2). Thus −|∇uh|− is elliptic.
Consistency and first order accuracy follow from the generalized chain rule: the
discretization of each term is consistent and first order accurate, and N(·) is a 1-Lipschitz
function.
3.4 F ILTERED FI NITE DIFFER ENC E S CHEMES
Although monotone schemes are provably convergent, they are only first (resp. second)
order accurate for first (resp. second) order equations [Obe06b]. The filtered schemes
discussed here are built to achieve higher accuracy while retaining the convergence prop-
erty of the monotone schemes. We start the section with motivating example for filtered
schemes. Then we introduce nearly monotone schemes, a general class of approximation
schemes that includes the filtered schemes, which are introduced after. We choose to focus
on nearly monotone schemes since they capture the underlying reason why the original
result of Barles and Souganidis [BS91] can be exploited, while also providing a general
framework to define filtered schemes in different ways. Indeed, this is done in chapter 6
when building filtered schemes for the PDE that governs the motion of level sets by affine
curvature.
Example 3.6. Consider the one-dimensional Eikonal
|u′(x)| = 1, x ∈ (−1, 1),
u(±1) = 0.
The finite difference schemes
∣
∣
∣uhx
∣
∣
∣
M:= max
−D+x [u], D−
x [u], 0
and∣
∣
∣uhx
∣
∣
∣
A:=
|u(x + h) − u(x − h)|2h
30 3.4. FILTERED FINITE DIFFERENCE SCHEMES
are monotone and accurate, respectively. These are, respectively, the schemes∣
∣
∣uhx
∣
∣
∣
+and
∣
∣
∣uhx
∣
∣
∣ introduced above. The different notation used in this example is to better illustrate
the general idea behind the filtered schemes. Theorem 3.1.5 guarantees the convergence
of the solutions of the monotone scheme to the unique viscosity solution of the PDE. The
accurate scheme is unstable, in particular when used to compute singular solutions, which
suggests we should only use it in regions where the solution is smooth.
Notice that, up to a scaling in h, the difference of the schemes is the centered finite
difference approximation for |uxx|:∣
∣
∣
∣
∣
∣
∣uhx
∣
∣
∣
A −∣
∣
∣uhx
∣
∣
∣
M∣
∣
∣
∣
=h
2
|u(x + h) − 2u(x) + u(x − h)|h2
=h
2
∣
∣
∣uhxx(x)
∣
∣
∣ .
We can then use this as a (local) criteria to decide whether or not to use the accurate scheme.
This leads us to consider the following (filtered) scheme
∣
∣
∣uhx
∣
∣
∣
F=
∣
∣
∣uhx
∣
∣
∣
A, if
∣
∣
∣
∣
∣
∣
∣uhx
∣
∣
∣
A −∣
∣
∣uhx
∣
∣
∣
M∣
∣
∣
∣
≤√
h,
∣
∣
∣uhx
∣
∣
∣
M, otherwise.
(The choice of the factor√
h will be addressed later.) The main idea here is that we decide
which scheme to use by looking into the size of the difference between the monotone
scheme and the accurate scheme, instead of an approximate smoothness criteria. By doing
so, we have∣
∣
∣uhx
∣
∣
∣
F=∣
∣
∣uhx
∣
∣
∣
M+ O(
√h),
which is the crucial property for the convergence proof as we will see below.
We now introduce nearly monotone finite difference schemes
Definition 3.4.1. The finite difference scheme F hN is a perturbation if there is w ∈ M such that
supu∈C(Gh)
supx∈Gh
∣
∣
∣F hN [u](x)
∣
∣
∣ ≤ w(h).
Definition 3.4.2 (Nearly monotone finite difference scheme). A finite difference scheme F h is
nearly monotone if it can be written as
F h[u] = F hM [u] + F h
N [u],
where F hM is a monotone scheme and F h
N is a perturbation.
Nearly monotone schemes are still provably convergent. The convergence proof follows
from a simple adaptation of the Barles and Souganidis convergence proof [BS91]: the
Chapter 3. Numerical Schemes 31
small (uniformly bounded) correction to the scheme due to the lack of monotonicity can
be absorbed into the term usually seen as the consistency error. In fact, in [BS91] the
convergence of “almost monotone” schemes is mentioned as a remark, but no definition
or examples are given.
Theorem 3.4.3. Assume the BVP (2.2) satisfies a strong comparison principle (see Definition
2.3.7). Let F h be any stable, nearly monotone finite difference scheme consistent with the BVP (2.2)
and Uh ∈ C(Gh) any solution of (3.3). Then
limh→0
uh(x) = u(x), for all x ∈ Ω,
where uh is the piecewise constant extension of Uh (3.4) and u is the viscosity solution of (2.2).
Proof. We follow the proof in [FO13], which itself is a modification of the original proof
in [BS91] to include the case of nearly monotone schemes. Here, following [Fro16a], a
simple adaptation is done to account for the fact that solutions of (3.3) are only defined on
a grid. This is achieved by extending the solutions to the entire domain using the nearest
neighbour extension (3.4) as well as the finite difference schemes (3.5).
Define
u(x) = lim suph→0,y→x
uh(y) ∈ USC(Ω) and u(x) = lim infh→0,y→x
uh(y) ∈ LSC(Ω)
From the stability of the solutions Uh, it follows that both u and u are bounded. In addition,
we know that u ≤ u by the definition on lim sup and lim inf.
Assume for now that u is a viscosity subsolution and u is a viscosity supersolution.
Then from the strong comparison principle for (2.2) applied to u and u, we conclude that
u ≤ u. We can then conclude that u := u = u and therefore u is the unique solution of
(2.2), again by the comparison principle for (2.2). The convergence then follows from the
definitions of u and u.
It then remains to show the claim that u is a viscosity subsolution and u is a viscosity
supersolution. We proceed to show that u is a viscosity subsolution since the proof for u is
similar.
Given a smooth test function φ, let x0 ∈ Ω be a strict global maximum of u with
φ(x0) = u(x0). By Lemma 3.4.4 below, we can find sequences with
hn → 0
yn → x0
uhn(yn) → u(x0)
32 3.4. FILTERED FINITE DIFFERENCE SCHEMES
where yn is a global maximizer of uhn − φ.
Define
εn = uhn(yn) − φ(yn). (3.8)
Then εn → u(x0) − φ(x0) = 0 and uhn(x) − φ(x) ≤ uhn(yn) − φ(yn) = εn for any x ∈ Ω. In
particular,
uhn(·) ≤ φ(·) + εn. (3.9)
We know that
u(·) ≥ v(·) ⇒ F hM(x, r, u(·)) ≤ F h
M(x, r, v(·))
for all x ∈ Gh, r ∈ R and u, v ∈ C(Gh) due to the monotonicity of the scheme (Definition
3.1.2). Using now the fact that F h is nearly monotone (Definition 3.4.2) we get that
u(·) ≥ v(·) ⇒ F h(x, r, u(·)) − 2w(h) ≤ F h(x, r, v(·)),
for all x ∈ Gh, r, s ∈ R and u, v ∈ C(Gh). Hence from (3.9) we conclude that
F hn(x, r, φ(·) + εn) − 2w(hn) ≤ F hn(x, r, uhn(·)) (3.10)
for all x ∈ Gh and r ∈ R. We then have
0 = F hn [uhn ](yn) since uhn is a solution
= F hn(yn, uhn(yn), uhn(·))= F hn(yn, φ(yn) + εn, uhn(·)) by (3.8)
≥ F hn(yn, φ(yn) + εn, φ(·) + εn) − 2w(hn) by (3.10).
Finally, taking the lim inf we get
0 ≥ lim infn→∞
F hn(yn, φ(yn) + εn, φ(·) + εn) − 2w(hn)
≥ lim infhn→0,y→x0,ε→0
F hn(y, φ(y) + ε, φ(·) + ε)
= F∗(x0, φ(x0), ∇φ(x0))
= F∗(x0, u(x0), ∇φ(x0))
which shows that u is a subsolution.
In the above proof we required the use of the following Lemma whose proof we present
now for completion.
Chapter 3. Numerical Schemes 33
Lemma 3.4.4. Suppose the family of function uh is bounded uniformly in h. Define
u(x) = lim suph→0,y→x
uh(u) ∈ USC(Ω).
Given a smooth function φ, let x0 be a strict global maximum of u − φ with u(x0) = φ(x0). Then
there are sequences
hn → 0
yn → x0
uhn(yn) → u(x0)
where yn is a global maximizer of uhn − φ.
Proof. From the definition of lim sup, there are sequences such that
hn → 0,
zn → x0,
uhn(zn) → u(x0).
Let yn ∈ Ω be the global maximizers of uhn(·) − φ(·). Then we have
uhn(yn) − φ(yn) ≥ uhn(zn) − φ(zn) → u(x0) − φ(x0) = 0.
In addition, for any δ > 0 and large enough n,
uhn(yn) − φ(yn) ≤ u(yn) − φ(yn) + δ ≤ u(x0) − φ(x0) + δ = δ
where we used the fact that x0 is a global maximum of u − φ with u(x0) = φ(x0). Thus we
conclude that
uhn(yn) − φ(yn) → 0.
Now, we show by contradiction that yn → x0. Suppose not. Then, by passing to a
subsequence if needed there is an R > 0 such that |yn − x0| > R. Moreover, since u − φ has
a strict, global and unique maximum at x0 with value zero, there is a K > 0 such that
u(y) − φ(y) < −K
whenever |y − x0| > R. For n large enough we have
uhn(yn) ≤ u(yn) +K
2
34 3.4. FILTERED FINITE DIFFERENCE SCHEMES
and so
uhn(yn) − φ(yn) ≤ u(yn) − φ(yn) +K
2< −K +
K
2= −K
2
which contradicts the fact that uhn(yn) − φ(yn) → 0. We then conclude that yn → x0.
Finally we see that
∣
∣
∣uhn(yn) − u(x0)∣
∣
∣ =∣
∣
∣uhn(yn) − φ(x0)∣
∣
∣
≤∣
∣
∣uhn(yn) − φ(yn)∣
∣
∣+ |φ(yn) − φ(x0)|→ 0
and therefore uhn(yn) → u(x0) as desired.
The stability of nearly monotone finite difference schemes is one of the assumptions in
Theorem 3.4.3. The next result addresses precisely that. It tells us that not only solutions
for the nearly monotone finite difference scheme exist but that also that these are stable,
provided the perturbation is continuous and the well-posedness of the underlying mono-
tone schemes. The latter assumption was shown to be true in [Obe06a] for finite difference
schemes that are elliptic, proper and Lipschitz continuous.
Proposition 3.4.5. Let S denote the continuous solution operator for the inhomogeneous problem
for the monotone scheme
F hM [u] + g = 0,
for which we suppose that solutions exist. Suppose as well that there exists h0 > 0 such that for all
0 < h < h0 and g ∈ C(Gh) any u ∈ C(Gh) such that F hM [u] + g = 0 is bounded independently of
h. Let F hN be a continuous perturbation and denote by F h[u] = F h
M [u]+F hN [u] the nearly monotone
scheme. Then solutions of F h[u] = 0 exist and F h is stable.
Remark 3.11. Notice that we do not prove that the solutions of the nearly monotone scheme
are unique: only stability is required to to apply Theorem 3.4.3. Moreover, we emphasize
that this result can only be applied to the filtered schemes defined below, when the filter
function is continuous.
Proof. We follow [FO13].
F hN is a perturbation. Therefore there exists C > 0, independent of h, such that
∥
∥
∥F hN [u]
∥
∥
∥ ≤ C
for all u ∈ C(Gh). By assumption S is continuous and so
∥
∥
∥S(F hN)∥
∥
∥ ≤ R
Chapter 3. Numerical Schemes 35
for some R > 0, also independent of h. Since u ∈ C(Gh) is a solution of F h[u] = 0 if and
only if it is a fixed point of S F hN , the above inequality allows us to conclude that F h is
stable. Moreover, it follows that
S(F hN [BR]) ⊆ BR,
where here BR denotes the ball u ∈ C(Gh) : ‖u‖ ≤ R. Hence, by the Brouwer’s fixed
point theorem and due to the continuity of S F hN , we conclude that there exists a fixed
point u∗ of S F hN , i.e., u∗ = S(F h
N [u∗]) which means that
F hM [u∗] + F h
N [u∗] = 0.
Thus u∗ is a solution of F h[u] = 0.
The filtered schemes we define here fit very naturally into the framework of nearly
monotone schemes, while also being general enough to allow for a variety of schemes. The
main idea is to provide a systematic method to blend a monotone scheme with an accurate
scheme and retain the advantages of both: convergence of the former and higher accuracy
of the latter. This is achieved by requiring that the difference between the filtered scheme
and the monotone scheme is uniformly bounded, in other words, that the schemes are
nearly monotone. Filtered schemes were introduced in [FO13] in the context of the Monge-
Ampère equation. There they were used to overcome the reduction in accuracy based
on the use of a wide-stencil monotone scheme that introduces a directional resolution
error. The idea of blending a monotone scheme with an accurate scheme was first seen in
[Abg09] in the context of Hamilton-Jacobi equations.
Definition 3.4.6 (Filter function). We say that S : R → R is a filter function if it is a bounded
function that is equal to the identity in a neighbourhood of the origin and zero away from the origin.
Example 3.7. The following functions, depicted in Figure 3.1,
S(x) =
x, |x| ≤ 1,
0, |x| > 1,and S(x) =
x, |x| ≤ 1,
0, |x| ≥ 2,
−x + 2, 1 ≤ x ≤ 2,
−x − 2, −2 ≤ x ≤ −1.
are examples of filter functions. The former is discontinuous, while the latter is continuous.
36 3.4. FILTERED FINITE DIFFERENCE SCHEMES
-2 -1 1 2
-1.0
-0.5
0.5
1.0
-3 -2 -1 1 2 3
-1.0
-0.5
0.5
1.0
Figure 3.1: Examples of filter functions: discontinuous filter (left), continuous filter (right).
Definition 3.4.7. Let S be a filter function and ε(h) : R+ → R
+ be a nonnegative modulus
function with ε(h) → 0 as h → 0. Let F hM denote a monotone scheme and F h
A an accurate scheme.
The filtered scheme is defined as
F h[u] = F hM [u] + ε(h)S
(
F hA[u] − F h
M [u]
ε(h)
)
.
We start by showing that the filtered schemes are nearly monotone.
Proposition 3.4.8. A filtered scheme is nearly monotone.
Proof. We can write the filtered scheme as
F h[u] = F hM [u] + F h
N [u]
where
F hN [u] = ε(h)S
(
F hA[u] − F h
M [u]
ε(h)
)
.
Since the filter function S is bounded, F hN is a perturbation and therefore F h is a nearly
monotone scheme.
Stability of the filtered schemes follows from the stability of the underlying monotone
scheme, the continuity of the filter function and Proposition 3.4.5. Consistency is a con-
sequence of the nearly monotonicity and the consistency of the underlying monotone
scheme. The convergence of the filtered schemes then follows from Theorem 3.4.3.
Remark 3.12. It is important to notice that there are no requirements on the accurate scheme:
it can simply be constructed with standard higher order finite differences, or it can be
designed to take advantage of known properties of the solutions to the equation under
consideration.
Finally, we explain how filtered schemes can achieve higher accuracy: the parameter
ε(h) must be chosen carefully. Heuristically, it should be large enough to allow the accurate
Chapter 3. Numerical Schemes 37
scheme to be active where the solution is smooth, and small enough to force the monotone
scheme to be active when the solution is singular (in order to guarantee convergence).
Proposition 3.4.9. Suppose that the formal discretization errors of the schemes F hM , F h
A are
O(hβM ) and O(hβA), respectively. Choose α such that βA > βM > α > 0. Then if φ smooth and
ε(h) = O(hα), then F h[φ] = F hA[φ] for sufficiently small h.
Proof. If φ is smooth, then
F hA[φ] − F h
M [φ]
ε(h)=
O(hβA) + O(hβM )
O(hα)= O(hβM −α) ≤ O(1)
for sufficiently small h. Hence,
F h[φ] = F hM [φ] + ε(h)S
(
F hA[φ] − F h
M [φ]
ε(h)
)
= F hM [φ] + F h
A[φ] − F hM [φ] = F h
A[φ]
since S is equal to the identity in a neighbourhood of the origin by definition of filter
function.
In summary, filtered schemes combine a stable, monotone, consistent scheme with
an accurate (but possibly unstable) scheme. The accurate scheme is not required to be
stable on its own. However, independently of the choice made, the combination of the two
schemes is both provably convergent, and (potentially) higher order accurate.
CHAPTER 4
FILTERED SCHEMES FOR HAMILTON-JACOBI EQUATIONS
4.1 INTR ODUC TI ON
In this chapter, we build filtered schemes for first order Hamilton-Jacobi (HJ) partial
differential equations, with special focus on the Eikonal equation. These form a simple and
general class of finite difference schemes that combine a stable, monotone scheme with
an accurate (but possibly unstable) scheme. By construction they are nearly monotone
(see Definition 3.4.2) and thus provably convergent to the unique viscosity solution of the
underlying equation (see Theorem 3.4.3). Moreover, they have the potential to achieve
higher accuracy (when compared to monotone schemes) by making a careful choice of the
filtering mechanism (see Proposition 3.4.9).
We take a particular interest on the eikonal equation
|∇u(x)| = f(x), for x outside Γ,
u(x) = g(x), for x on Γ.(4.1)
where f > 0 and Γ is here a closed, bounded set. We consider as well HJ equations of the
form
H(x, ∇u) = f(x), x ∈ Ω,
u(x) = g(x), x ∈ Γ,(4.2)
where ∇u is the gradient of the function u, Ω is an open set, Γ is the boundary of Ω and
the Hamiltonian H is a nonlinear Lipschitz continuous function. We always refer to the
eikonal equation specifically, even though it is in fact an HJ equation (take H(p) = |p|).When we refer to HJ equations we always have more general equations in mind.
In general, solutions are not smooth (or even differentiable) and so we consider viscosity
solutions (see chapter 2). The viscosity solutions can be piecewise smooth with a singularity
in the gradient. It therefore makes sense to design high order schemes that provide higher
order accuracy (at least) away from these singularities.
4.1.1 Contribution of this work
We build filtered schemes for first order HJ equations. As discussed in section 3.4, filtered
schemes allow for a wide choice of accurate schemes, which are not required to be stable
39
40 4.2. DISCRETIZATION AND SOLVERS
on their own. Here we exploit this and choose accurate schemes that are designed to take
advantage of known properties of the solutions of HJ equations. The main contribution
is in the choice of the filter function together with a judicious choice of the accurate
scheme. We show that using one-sided higher order finite differences for the accurate
scheme, combined with an upwind monotone scheme results in a very simple, explicit and
accurate scheme for the eikonal equation. We treat as well the general case of first order HJ
equations. In summary, the schemes we introduce have the following properties.
1. They are simple and easy to implement on Cartesian grids. For example, for the
eikonal equation the filtered scheme using the centered difference scheme, is conver-
gent and examples show that second order accuracy is obtained, which results in the
simplest second order accurate finite difference scheme.
2. Higher order explicit schemes are obtained using higher order upwind interpolation.
These higher order schemes can be solved using fast sweeping.
3. Other choices of accurate schemes can be used instead: we implement ENO schemes
for comparison. Any choice of discretization (e.g. the popular discontinuous Galerkin
method) can be used, provided a monotone scheme can also be constructed in the
same setting.
4. For the eikonal equation in one dimension, higher order convergence for the numeri-
cal solution is proved, even for non-smooth solutions.
5. For HJ equations (in general), higher order convergence is obtained locally, in regions
where the solution is smooth.
4.2 D IS C RETIZATION AND S OLVER S
In this section we will discuss the discretization of the monotone and filtered schemes
for both HJ and eikonal equations for different choices of the accurate schemes (centered,
upwind and ENO). We do this both in one and two dimensions. We should point out
that all discretizations for HJ equations can be applied to the eikonal equation, although
we choose to present and use specific discretizations for the eikonal equation given its
importance in the literature.
We consider only the case of regular Cartesian grids since the discretization is simpler
and the idea is clear. It is certainly possible to build filtered schemes using higher order
methods on triangulated grids for example.
Chapter 4. Filtered Schemes for Hamilton-Jacobi equations 41
4.2.1 Monotone schemes
For the eikonal equation, in the one-dimensional case, the monotone scheme is given by
∣
∣
∣uhx
∣
∣
∣
M= max
−u(x + h) − u(x)
h,u(x) − u(x − h)
h, 0
. (4.3)
Since we are working on a Cartesian grid, extending it to the two dimensional case simply
requires the use of the standard Euclidean 2-norm function N : R2 → R given by
N(x, y) =√
x2 + y2. (4.4)
We then define∣
∣
∣∇uh∣
∣
∣
M= N
(
∣
∣
∣uhx
∣
∣
∣
M,∣
∣
∣uhy
∣
∣
∣
M)
(4.5)
which is monotone as desired (see Examples 3.4 and 3.5).
There are several monotone numerical Hamiltonians we could use to discretize HJ equa-
tions. Here we choose to use the Lax-Friedrichs numerical Hamiltonian [KOQ04], because
it has a simple form and it can be used for both convex and nonconvex Hamiltonians:
HhLF [u](x) = Hh
LF (x, p+, p−) = H
(
x,p+ + p−
2
)
− σxp+ − p−
2(4.6)
where σx is the artificial viscosity satisfying σx = max∣
∣
∣
∂H∂p
∣
∣
∣, p = ux and p± are the corre-
sponding forward and backward differences approximations of ux.
The scheme easily generalizes into higher dimensions: in the two-dimensional case we
have
HhLF [u](x, y) = Hh
LF (x, y, p+, p−, q+, q−)
= H
(
x, y,p+ + p−
2,q+ + q−
2
)
− σxp+ − p−
2− σy
q+ − q−
2
(4.7)
where σy = max∣
∣
∣
∂H∂q
∣
∣
∣, q = uy and q± are the corresponding forward and backward differ-
ences approximations of uy.
4.2.2 Accurate schemes
We know that the filtered scheme will converge independently of the choice of the accurate
scheme. Its purpose is to provide additional accuracy in the regions where the solution is
smooth and where the accurate scheme is active. Thus the resulting accuracy of the solution
comes from a judicious choice of the accurate scheme. In addition to the accuracy, the
42 4.2. DISCRETIZATION AND SOLVERS
choice of accurate scheme determines the type of solver we can use (iterative or sweeping),
based on whether an explicit solution formula is available (see subsection 4.2.5).
We first consider the one-dimensional case and then show how, as in the previous
section, the schemes can be generalized for the two-dimensional case.
Centered Schemes: The second order accurate centered scheme are obtained by simply
replacing ux by its second order centered approximation:
∣
∣
∣uhx
∣
∣
∣
C,2=
|u(x + h) − u(x − h)|2h
,
HhC,2[u](x) = H
(
x,u(x + h) − u(x − h)
2h
)
.
Upwind Schemes: The upwind schemes proposed here were first thought for the
eikonal equation, although they can be generalized to HJ equations. In the eikonal equation
case, they are designed to choose the finite difference stencil in terms of the direction of
the characteristics of the solution. This means using the left (right) biased stencil if the
characteristics are being propagated from the left (right). The higher order upwind schemes
generalize the monotone scheme above. They are defined as follows.
Set P ±,n[u] to be the interpolating polynomial of degree n of u at the nodes xj = x ± jh
for j = 0, 1, . . . , n. (The sign in the superscript indicates interpolation to the left or to
the right.) These interpolating polynomials are standard and given in several convenient
explicit forms (see [Ise09]). We give a specific example below. We then set
∣
∣
∣uhx
∣
∣
∣
U,n= max
− d
dxP +,n[u](x),
d
dxP −,n[u](x), 0
,
HhU,n[u](x) = Hh
LF
(
x,d
dxP +,n[u](x),
d
dxP −,n[u](x)
)
.
ENO Schemes: High order essentially non-oscillatory (ENO) are another option for
the accurate discretization. (A refinement of ENO is WENO [JP00], which we choose not
to implement, since the main idea is clear from the ENO examples.) The idea underlying
the ENO schemes is to do a standard interpolation using an adaptive stencil, i.e., the
stencil used depends on the function being interpolated. Starting with two nodes, the ENO
interpolation of order n selects the remaining n − 1 interpolation nodes by successively
adding nodes to the stencil with the smallest Newton divided difference. This way, the rth
node is chosen by comparing two approximations of the derivative of order r + 1, with r
taking successively the values 1, . . . , n − 1.
Let En,± 12 [u] denote the ENO interpolation as explained above, and as defined in [OS91].
Chapter 4. Filtered Schemes for Hamilton-Jacobi equations 43
Then we define the nth-order accurate ENO scheme to be
∣
∣
∣uhx
∣
∣
∣
E,n= max
− d
dxEn, 1
2 [u](x),d
dxEn,− 1
2 [u](x), 0
,
HhE,n[u](x) = Hh
LF
(
x,d
dxEn, 1
2 [u](x),d
dxEn,− 1
2 [u](x)
)
.
Two dimensional schemes. In the case of the eikonal equation we use (4.4) as we did
in subsection 4.2.1. The second order centered scheme becomes
∣
∣
∣∇uh∣
∣
∣
C,2= N
(
∣
∣
∣uhx
∣
∣
∣
C,2,∣
∣
∣uhy
∣
∣
∣
C,2)
, (4.8)
the upwind schemes become
∣
∣
∣∇uh∣
∣
∣
U,n= N
(
∣
∣
∣uhx
∣
∣
∣
U,n,∣
∣
∣uhy
∣
∣
∣
U,n)
, (4.9)
and, finally, the ENO schemes are defined as
∣
∣
∣∇uh∣
∣
∣
E,n= N
(
∣
∣
∣uhx
∣
∣
∣
E,n,∣
∣
∣uhy
∣
∣
∣
E,n)
. (4.10)
The upwind schemes defined here for the eikonal equation recover the 2nd and 3rd order
upwind schemes from ([Set99a], [Cho01] and [ABM+11]). These schemes have been solved
in the literature using Fast Marching algorithms.
As for HJ equations, the extension to two dimensions follows from using the two-
dimensional expression of HhLF as we did with the monotone scheme.
4.2.3 Boundary conditions
In this section we discuss the treatment of boundary conditions for the filtered scheme.
First we discuss the one dimensional case. Note that we solved the internal problem
and so the Dirichlet data is prescribed on the boundary of the computational domain. For
the monotone difference method this leads to a standard application of Dirichlet boundary
conditions.
For higher order accurate methods, the situation is similar to the case of multistep
methods for ordinary differential equations: more information is needed to achieve the
higher accuracy. This information can take the form of additional function values at adja-
cent grid points, or higher derivative information [HNW93]. For practical considerations,
in order to test the accuracy of the solution without introducing errors from the boundary,
we extend the Dirichlet data to more grid points. More precisely, we set the exact solution
(in fact, an nth order approximation of the exact solution is enough) at the n grid points
44 4.2. DISCRETIZATION AND SOLVERS
adjacent to the boundary when using the nth order upwind and ENO filtered schemes.
If the additional information is not available we may lose the higher accuracy. Using
just the first order accurate monotone scheme reduces the order of the global accuracy.
Similarly, using only the available one sided higher order approximations may decrease
the accuracy since the available direction is not the one we are interested in: as we will see
below in the proof of Theorem 4.2.4 in subsection 4.2.7 for the eikonal equation case, we
want to interpolate towards the boundary and not away from it.
In the two-dimensional case, for the eikonal equation, we solved the external problem
and so the boundary of the computational domain did not include the Dirichlet boundary.
This poses an additional difficulty since the schemes need to be carefully defined near the
boundary of the computational domain to prevent computational errors that propagate into
the computational domain. Here we dealt with this issue as is usually done for monotone
schemes: we consider only the one sided differences available. Since the characteristics
go inward, the lack of external information is not a problem. For the (internal) Dirichlet
boundary, we proceed in the same way we did in the one-dimensional case: we set the
exact solution at as many adjacent grid points of the boundary as needed depending on
the order of accuracy of the scheme used.
For general HJ equations, the computational boundary can cause problems, depending
on the discretization used. For the Godunov scheme, which reduces to (4.5) in the case
of the eikonal equation, there are no problems, so this is what we used for the eikonal
equation. However, for general HJ equations in two dimensions using the Lax-Friedrichs
schemes (4.7) with high order interpolation is more complicated [ZZQ06], and can lead to
errors at the computational boundary.
4.2.4 Filtered schemes
We can now define the filtered schemes, which were discussed in detail in section 3.4. Let
F hM denote the monotone discretization of the operator on the grid with spacing h, given
above in subsection 4.2.1. Let F hA denote an accurate discretization of the same operator,
with several possible choices being given above in subsection 4.2.2.
Here we define the filtered scheme using the discontinuous filter function of Exam-
ple 3.7 and taking ε(h) =√
h. This leads to the filtered scheme, F h, given by the following
simple formula:
F h[u] =
F hA[u], if
∣
∣
∣F hA[u] − F h
M [u]∣
∣
∣ ≤√
h,
F hM [u], otherwise.
(4.11)
Chapter 4. Filtered Schemes for Hamilton-Jacobi equations 45
Remark 4.1. The choice of the factor√
h in (4.11) is designed to fit between two rates: large
enough to permit the accurate scheme to be active where the solution is smooth (see
Proposition 3.4.9) , and small enough to force the monotone scheme to be active when the
solution is singular. In the case of the eikonal equation, the monotone scheme is accurate
to O(h) and the accurate scheme is O(h2) or better, thus our choice of the factor√
h.
Below, at the end of subsection subsection 4.3.1, we consider an example where the
Hamiltonian is non-convex, and the observed convergence rate is O(√
h) for the monotone
scheme, and so we take the factor to be smaller than√
h.
Theorem 4.2.1 (Convergence of the filtered schemes for HJ equations). Consider the Dirichlet
problem (4.2) on a bounded domain Ω and assume that it has a strong comparison principle (see
Definition 2.3.7). Let F h be the filtered scheme (4.11) and Uh ∈ C(Gh) be any of its solutions. If
F h is stable then
limh→0
uh(x) = u(x), for all x ∈ Ω
where uh is the piecewise constant extension of Uh (3.4) and u is the viscosity solution of (4.2).
Proof. The filtered scheme F h is consistent since both underlying schemes, F hM and F h
A, are
consistent and is nearly monotone as a consequence of Proposition 3.4.8. Thus, since F h is
stable by assumption, the convergence follows from Theorem 3.4.3.
To apply the theorem, we need to show that the filtered scheme is stable. Stability will
follow from the well-posedness of the monotone schemes and the continuity of the filter
function (see Proposition 3.4.5 and Remark 3.11).
In our setting, although discontinuous, (4.11) has a simple form which allows for
explicit solution formulas as we will see in subsection 4.2.5. These explicit solution
formulas allow us to build fast sweeping solvers, which are appropriate for HJ equations.
In practice the computational results are as good as could be expected. For the purpose of
the proof, a continuous filter is needed but the practical advantages of the discontinuous
one outweigh the lack of rigor.
Remark 4.2. In [FO13], a continuous interpolation between the monotone and accurate
scheme was used. There, this was required to show the stability of the filtered schemes,
but it was also of practical use for a Newton solver.
Theorem 4.2.1 does not provide any information regarding the convergence rate. Prov-
ing higher order convergence requires additional efforts and is possible in specific settings.
For the one-dimensional eikonal equation, we prove higher order convergence in subsec-
tion 4.2.7. For the two-dimensional eikonal equation, second and third order convergence
is proven for smooth solutions in [ABM+11]. We are more interested in demonstrating
46 4.2. DISCRETIZATION AND SOLVERS
the higher order convergence in practice, which is done using numerical simulations. In
particular, in the case of piecewise smooth solutions in two dimensions, we achieve second
order convergence rates in the smooth region, and first order convergence overall in the l∞
norm.
Remark 4.3. In addition to stationary equations, we can build filtered schemes for time
dependent equations. This can be accomplished by using the filtered scheme on the spatial
part of the operator, and a standard time discretization (forward Euler or strong stability
preserving time discretizations [GKS11]) for the time derivative. As needed, the filter could
also be applied to the time derivative term as well. In this case, with minor modifications,
the proof of convergence for the filtered scheme goes through, since, as it is standard for
viscosity solution, the time derivative can be considered as an additional spatial variable.
Following ideas similar to these, filtered schemes for time dependent equations were built
for first order HJ equations [BFS16], second order Hamilton-Jacobi-Bellman equations
[BPR16] and front propagation [Sah16].
4.2.5 Explicit methods
For upwind schemes, the interpolation is fixed, so we can solve for the reference variable
and build explicit schemes. In contrast, it is difficult to directly build explicit methods for
many of the other schemes. Rather than present the general method for solving for the
reference variable and in order to be concrete (and save space), we give a specific example
below. The general method should then be clear.
Eikonal equations
Example 4.1 (one-dimensional case). Consider first the monotone scheme in the one-
dimensional case (4.3). Solving the equation∣
∣
∣uhx
∣
∣
∣
M= f for the reference variable, u(x),
leads to
u(x) = minu(x + h), u(x − h) + hf(x) (4.12)
since f > 0. Consider now the second order upwind scheme, again in one dimension. The
upwind scheme takes the form
∣
∣
∣uhx
∣
∣
∣
U,2 ≡ 1
2hmax 3u(x) − 4u(x ± h) + u(x ± 2h), 0 = f.
Solving the preceding equation for the reference variable, u(x), leads to
u(x) =1
3min4u(x + h) − u(x + 2h), 4u(x − h) − u(x − 2h) +
2
3hf(x). (4.13)
Chapter 4. Filtered Schemes for Hamilton-Jacobi equations 47
Finally, consider the correspondent filtered scheme. Combining (4.12) and (4.13) and
using the definition of the filtered scheme (4.11) we obtain the following explicit represen-
tation of the solution of the filtered scheme at a reference point in terms of the neighboring
values
u(x) =
1
3min4u(x ± h) − u(x ± 2h) +
2
3hf(x) if
∣
∣
∣
∣
∣
∣
∣uhx
∣
∣
∣
A −∣
∣
∣uhx
∣
∣
∣
M∣
∣
∣
∣
≤√
h,
minu(x + h), u(x − h) + hf(x) otherwise.
Example 4.2 (two-dimensional case). We can also obtain an explicit solution for the filtered
schemes using the upwind scheme in the two-dimensional case as above. In this case
solving for the reference variable u(x, y) requires solving a nonlinear equation of the form
[
(z − a)+]2
+[
(z − b)+]2
= c2
for the unknown z where a, b and c > 0 are constants and (z)+ := maxz, 0. This equation
combines piecewise linear functions with a quadratic function. The unique solution of the
equation is given by
z =
mina, b + c |a − b| ≥ c,
a + b +√
2c2 − (a − b)2
2|a − b| < c,
(4.14)
(see e.g. [Zha05] for a derivation).
In the case of the monotone scheme we get
a = minu(x + h, y), u(x − h, y),
b = minu(x, y + h), u(x, y − h),
c = hf(x).
As for the second order upwind scheme we have
a =1
3min4u(x ± h, y) − u(x ± 2h, y),
b =1
3min4u(x, y ± h) − u(x, y ± 2h),
c =2
3hf(x).
The explicit formula of the filtered scheme can then be obtained as in the one-dimensional
case using the definition of filtered scheme (4.11) and (4.14).
48 4.2. DISCRETIZATION AND SOLVERS
Hamilton-Jacobi equations
Example 4.3 (one-dimensional case). Consider first the monotone scheme (4.6). We know
that
p+ =u(x + h) − u(x)
h, p− =
u(x) − u(x − h)
h.
Thus, solving HhLF [u] = f for the reference variable, u(x), leads to
u(x) =h
σx
[
f(x) − H
(
x,u(x + h) − u(x − h)
2h
)
+ σxu(x + h) + u(x − h)
2h
]
.
Consider now the second order upwind scheme. We have
d
dxP +,2[u](x) =
−3u(x) + 4u(x + h) − u(x + 2h)
2h,
d
dxP −,2[u](x) =
3u(x) − 4u(x − h) + u(x − 2h)
2h.
Thus, solving HhU,2[u] = f for the reference variable, u(x), leads to
u(x) =2h
3σx
[
f(x) − H
(
x,−u(x + 2h) + 4u(x + h) − 4u(x − h) + u(x − 2h)
4h
)
+σx−u(x + 2h) + 4u(x + h) + 4u(x − h) − u(x − 2h)
4h
]
.
The explicit formula of the filtered scheme can then be obtained as in the eikonal
equation case using the definition of filtered scheme (4.11).
For the ENO schemes, we can’t get an explicit formula. However, it is possible to get a
fixed point iteration which was used successfully with a fast sweeping solver in [ZZQ06].
4.2.6 Solution methods
The simplest solver is to use the fixed point iteration
un+1 = un − dt(F h[u] − f) (4.15)
which corresponds to the discrete version of the parabolic equation ut + (F [u] − f) = 0
using a forward Euler step, where F [u] = |∇u| or F [u](x) = H(x, ∇u). The fixed point
iteration will be a contraction in the l∞ norm provided that we choose dt small enough
as dictated by the nonlinear CFL condition [Obe06b], which in the eikonal equation case
means dt = O(h) (see section 3.2 for more details). This will however make the solver
relatively slow.
Chapter 4. Filtered Schemes for Hamilton-Jacobi equations 49
As seen in the previous section, we have explicit formulas for the upwind filtered
schemes. This allows us to use the fast sweeping method [TCOZ03, Zha05], which is a
fast iterative solution method. Each node is updated using Gauss-Seidel iterations with
alternating sweeping ordering of the domain. This allows information to propagate from Γ
along characteristics to the rest of the computational domain. In the case of the eikonal
equations, an alternative would be the Fast Marching Method [Set99a, Tsi95]: the solution
is constructed by using characteristic information to select the next node where the solution
can be obtained. However this requires a complicated data structure which makes it more
difficult to implement. In one dimension, the whole domain is swept with two alternating
ordering of the nodes
• (i = 1, . . . , N) and (i = N, . . . , 1)
which correspond to the two possible directions for the propagation of the characteristics.
In two dimensions we sweep the whole domain with eight alternating ordering of the
nodes
• (i = 1, . . . , N, j = 1, . . . , N),
• (i = 1, . . . , N, j = N, . . . , 1),
• . . .
• (j = N, . . . , 1, i = N, . . . , 1).
corresponding respectively to up-right, up-left, down-left, down-right, right-up, left-up,
left-down and right-down. Here, the first (last) four orderings speed up the convergence
when the characteristics are aligned with the x-axis (y-axis).
For the filtered centered and ENO schemes, we implemented the fixed point solver
(4.15). For the upwind filtered schemes we implemented the fast sweeping solver described
above.
4.2.7 Error estimates in one dimension
In this section, we focus on the eikonal equation in one dimension, with Dirichlet boundary
conditions on the endpoints of an interval. Despite the fact that the solution is Lipchitz
continuous, we are able to prove, when the data f is smooth enough, that the upwind
schemes converge to higher order. This is a consequence of the fact that (i) the solution
is piecewise smooth, and we can express it as a minimum of the two ODE solutions (ii)
50 4.2. DISCRETIZATION AND SOLVERS
the numerical solution is also expressed as the minimum of the left and right branches. A
similar idea was used to obtain higher accuracy for conservation laws in [EFT13].
Here we prove the higher order convergence of a particular scheme: the (unfiltered)
high order upwind schemes. In this case we do not prove convergence of the filtered
scheme which combined the high order upwind scheme with the monotone upwind
scheme. However, we implement the filtered scheme, and we found, in practice, for the
computed solution, the higher order scheme is always active.
Remark 4.4. The reason for using the filtered scheme is that it provides global stability:
intermediate numerical solutions are stable, even though in the final computed solution
the accurate scheme is always active. To use a simile, the filtered scheme acts like training
wheels on a bicycle, maintaining stability even though, ultimately the training wheels do
not touch the ground.
We consider u to be the viscosity solution of the one-dimensional eikonal equation
|u′(x)| = f(x), x ∈ (a, b),
u(x) = g(x), x ∈ Γ = a, b.(4.16)
To start we need first to recall the known Dynamic Programming Principle (DPP).
Proposition 4.2.2. Consider the dynamics
y(t) = α(t) t ∈ (0, +∞),
y(0) = x,
and cost functional
Jx(α(·)) =∫ tx(α)
0f(yx(s; α)ds + g(yx(tx(α), α)),
where A = α(·) : [0, +∞) → −1, 1 ⊂ R, measurable and tx denotes the entry time in Γ.
Hence u is the value function of a minimum cost problem, being given by
u(x) = infα∈A
Jx(α(·)). (4.17)
Proof. See [BCD97, Chapter IV].
We are now able to express u as the minimum of two ODE solutions.
Proposition 4.2.3. The viscosity solution u of (4.16) is given by
u(x) := minua(x), ub(x), (4.18)
Chapter 4. Filtered Schemes for Hamilton-Jacobi equations 51
where ua and ub are respectively the solution of
u′(x) = f(x),
u(a) = g(a),and
−u′(x) = f(x),
u(b) = g(b).(4.19)
Proof. Since f > 0, the only trajectories to be considered in the minimum of (4.17) are
the ones that travel straight to the endpoints a and b. These trajectories are given by the
controls α1 ≡ −1 and α2 ≡ 1, respectively. Hence
u(x) = min Jx(α1(·)), Jx(α2(·)) .
It is now easy to see that ua(x) = Jx(α1(·)) and ub(x) = Jx(α2(·)) and so we are done.
We can now prove our result.
Theorem 4.2.4. For n ≤ 6 and if f ∈ C(n+1)[a, b] the upwind schemes are convergent. Moreover,
if the solution is denoted by uh,n, we have the following error estimate
∣
∣
∣uh,n(a + jh) − u(a + jh)∣
∣
∣ ≤ ChnMn+1 (4.20)
for j = 0, . . . , b−ah
, where C is a constant depending on n, the Lipschitz constant of f , a and b and
Mn = maxx∈[a,b]
∣
∣
∣f (n)(x)∣
∣
∣.
Proof. The idea of the proof consists in solving (4.19) with backward difference schemes
and realize using (4.18) that we recover uh,n, more precisely, the explicit formulas for
upwind schemes discussed in subsection 4.2.5. The assumption n ≤ 6 is needed as the
backward schemes are only stable (and therefore convergent) when n ≤ 6.
Let uh,na and uh,n
b denote respectively the solutions obtained using backward schemes to
solve (4.19). Hence they are the solution of
U−,n[u](x) = f(x)
u(a + jh) given for j = 0, . . . , n − 1,
−U+,n[u](x) = f(x)
u(b − jh) given for j = 0, . . . , n − 1.
Set uh,n(x) := minuh,na , uh,n
b . Under our assumptions we know that uh,na and uh,n
b converge
respectively to ua and ub (see [QSS07] on multistep methods). Therefore the proof is done
if we show that uh,n(x) = uh,n(x).
Rather than prove this for all n, we give a particular example (n = 2) and the general
case should then follow easily. We will use the second order backward differentiation
schemes and will therefore recover the second order upwind schemes. We have that uh,2a is
52 4.3. COMPUTATIONAL RESULTS
the solution of3u(x) − 4u(x − h) + u(x − 2h)
2h= f(x)
and can therefore be written as
uh,2a (x) =
1
3(4u(x − h) − u(x − 2h)) +
2h
3f(x).
Likewise, uh,2b is the solution of
−−3u(x) + 4u(x + h) − u(x + 2h)
2h= f(x)
and so
uh,2b (x) =
1
3(4u(x + h) − u(x + 2h)) +
2h
3f(x)
Using now (4.18), we recover (4.13) as desired.
Thus the accuracy of the numerical solution of (4.16) is determined by the accuracy of
the numerical solution of each of the two linear odes (4.19).
The error estimates result naturally from the error estimates for backward difference
schemes for ODEs which can be found in [QSS07].
Remark 4.5. The requirement f ∈ C(n+1)[a, b] is needed to obtain the order of convergence.
This requirement can be relaxed to f being piecewise C(n+1) in the same regions as the
solution u. The idea is that we only need uh,na and uh,n
b to be high order convergent when
they are active in the minimum of (4.18).
Remark 4.6. Here we assume the exact solution is known near the boundary, but this
assumption can be relaxed. The same order of accuracy can be obtained provided the
boundary conditions are known to sufficient precision near the boundary, i.e., with the
same of order of accuracy. Furthermore, these can be computed from the boundary data
using standard methods [QSS07].
4.3 COMP UTATIONAL RES ULTS
4.3.1 Example solutions in one dimension
In this subsection we discuss the examples considered in one dimension. In all of them
the solution is piecewise smooth with a single singularity. Their purpose is confirm the
improved accuracy of the filtered schemes, as well as the high order convergence of the
upwind schemes for the eikonal equation. All examples are displayed in Figure 4.1.
The first example is the eikonal equation with f(x) = 1 + cos(x) with the Dirichlet
boundary conditions being prescribed at x = ±2. The computational domain is [−2, 2].
Chapter 4. Filtered Schemes for Hamilton-Jacobi equations 53
−2 −1.5 −1 −0.5 0 0.5 1 1.5 20
0.5
1
1.5
2
2.5
3
−2 −1.5 −1 −0.5 0 0.5 1 1.5 20
1
2
3
4
5
6
7
8
9
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.2
0.4
0.6
0.8
1
1.2
1.4
−2 −1.5 −1 −0.5 0 0.5 1 1.5 214.5
15
15.5
16
16.5
17
17.5
18
−2 −1.5 −1 −0.5 0 0.5 1 1.5 20.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Figure 4.1: Profile of the solutions of the five examples considered in one dimension (at the top, eikonalequation examples, at the bottom, HJ equations examples).
The exact solution is given by u(x) = 3 − |x + sin(x)| and it is therefore piecewise smooth
with a singularity at x = 0 (see Figure 4.1). We represent the solution obtained with the
monotone scheme and the 2nd order upwind filtered scheme for 50 mesh points near the
singularity in [−0.4, 0.4] on Figure 4.2.
The second example is again the eikonal equation with f(x) = 1 + e|x|, where the
Dirichlet boundary conditions are once again prescribed at x = ±2 and the computational
domain is [−2, 2]. The exact solution is given by u(x) = 10−|x|−e|x| and as in the previous
example, it is piecewise smooth with a singularity at x = 0 (see Figure 4.1).
The third example, also a solution of the eikonal equation, is given by
u(x) =
x3 + ax x ∈ [0, x0],
1 + a − ax − x3 x ∈ [x0, 1],
with a =1−2x3
0
2x0−1, x0 =
3√2+2
4 3√2and therefore f(x) = 3x2 + a. This example was chosen for two
main reasons: there is no symmetry in the relationship between the singularity and the
grid points, as opposed to the two previous examples where the singularity was always
a midpoint of two consecutive grid points; this is one the examples in [Abg09] that the
author uses to check the rate of convergence of the proposed method. The difference is that
in [Abg09] the error in the l∞ norm is computed at the grid points in the interval[
13√2
, 12
]
,
54 4.3. COMPUTATIONAL RESULTS
−0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.42.5
2.55
2.6
2.65
2.7
2.75
2.8
2.85
2.9
2.95
3
Monotone
2ndupwind
Exact
Figure 4.2: Exact solution and solutions obtained with the monotone scheme and the 2nd order upwindfiltered scheme with 50 grid points for the first example of the eikonal equation.
instead of all the grid points as we do here. The author chooses that interval since it is an
interval where the solution is smooth but as we explained above we can look at the error
on all grid points and still obtain the high order convergence.
We consider as well two HJ equations. The first one given by H(p) = p2, a convex
Hamiltonian, with f(x) = ex and
u(x) =
−2ex2 + 20 x ∈ [−2, 0],
2ex2 + 16 x ∈ [0, 2].
The second one given by H(p) = cos(p)2 + |p|, a nonconvex Hamiltonian considered in
[Abg09], with u(x) = e−|x| and f(x) = cos(e−|x|)2 + e−|x|. The profile of both solutions is
depicted in Figure 4.1 and the Dirichlet boundary conditions are prescribed at x = ±2,
with the computational domain being [−2, 2]. For the nonconvex example, the factor√
h
in the filtered scheme (4.11) was replaced by h1
10 (see Remark 4.1).
The computational domain is discretized on a grid with N points and the singularity is
never a grid point.
Chapter 4. Filtered Schemes for Hamilton-Jacobi equations 55
4.3.2 Computational results in one dimension
In this subsection we discuss the computational results obtained in one dimension. The
main purpose is to demonstrate that the filtered scheme achieves the higher order accuracy
and that, in particular for the eikonal equation, the upwind filtered schemes achieve higher
order convergence rate as proved in subsection 4.2.7 for the (unfiltered) upwind schemes.
We organize the discussion in three parts: accuracy and behaviour, order of convergence
and upwind vs ENO. For the eikonal equation, we obtained results with the monotone
scheme (4.3) and the respective filtered schemes using as the accurate scheme the second
centered scheme and the second, third and forth order upwind and ENO schemes. For HJ
equations, we obtain results using the monotone scheme (4.6) and the respective filtered
schemes using as the accurate scheme the second order centered, upwind and ENO
schemes. Third order upwind and ENO filtered schemes were also used, but they did not
show any advantage over the second order schemes.
Errors and order, 1st ExampleN Monotone 2nd Upwind 3rd Upwind 4th Upwind64 4.465 × 10−2 - 1.141 × 10−3 - 8.532 × 10−5 - 2.646 × 10−6 -
128 2.223 × 10−2 0.99 2.908 × 10−4 1.95 1.076 × 10−5 2.95 1.700 × 10−7 3.92256 1.109 × 10−2 1.00 7.337 × 10−5 1.98 1.348 × 10−6 2.98 1.074 × 10−8 3.96512 5.538 × 10−3 1.00 1.842 × 10−5 1.99 1.687 × 10−7 2.99 6.745 × 10−10 3.981024 2.767 × 10−3 1.00 4.615 × 10−6 1.99 2.109 × 10−8 3.00 4.224 × 10−11 3.99
N 2nd centered 2nd ENO 3rd ENO 4th ENO64 6.553 × 10−4 - 7.660 × 10−4 - 2.780 × 10−5 - 5.561 × 10−7 -
128 1.559 × 10−4 2.05 1.918 × 10−4 1.97 3.546 × 10−6 2.94 3.544 × 10−8 3.93256 3.789 × 10−5 2.03 4.803 × 10−5 1.99 4.470 × 10−7 2.97 2.236 × 10−9 3.96512 9.451 × 10−6 2.00 1.201 × 10−5 1.99 5.608 × 10−8 2.99 1.404 × 10−10 3.981024 2.317 × 10−6 2.03 3.004 × 10−6 2.00 7.022 × 10−9 2.99 8.776 × 10−12 3.99
Table 4.1: Accuracy in the l∞ norm and order of convergence of the schemes for the first example of theeikonal equation.
Accuracy and behaviour of the filtered schemes.
We begin by comparing the accuracy of the monotone scheme with the filtered schemes
by looking at the error in the l∞ norm in Figure 4.5 and Tables 5.3, 5.4, 5.5, 5.6, 5.7. As
expected the filtered schemes have improved accuracy.
Once close to the solution, the filtered schemes behave as designed choosing to use
the accurate scheme whenever possible, i.e., whenever they interpolate the solution in a
56 4.3. COMPUTATIONAL RESULTS
Errors and order, 2nd ExampleN Monotone 2nd Upwind 3rd Upwind 4th Upwind64 1.997 × 10−1 - 8.011 × 10−3 - 3.642 × 10−4 - 1.766 × 10−5 -
128 9.984 × 10−2 0.99 2.042 × 10−3 1.95 4.716 × 10−5 2.92 1.162 × 10−6 3.88256 4.992 × 10−2 0.99 5.153 × 10−4 1.98 5.995 × 10−6 2.96 7.441 × 10−8 3.94512 2.496 × 10−2 1.00 1.294 × 10−4 1.99 7.555 × 10−7 2.98 4.706 × 10−9 3.971024 1.248 × 10−2 1.00 3.242 × 10−5 1.99 9.482 × 10−8 2.99 2.959 × 10−10 3.99
N 2nd centered 2nd ENO 3rd ENO 4th ENO64 6.358 × 10−3 - 3.983 × 10−3 - 1.705 × 10−3 - 1.492 × 10−3 -
128 1.570 × 10−3 2.00 1.018 × 10−3 1.95 2.764 × 10−4 2.60 1.823 × 10−3 -0.29256 3.859 × 10−4 2.01 2.573 × 10−4 1.97 4.899 × 10−5 2.48 2.499 × 10−4 2.85512 9.700 × 10−5 1.99 6.466 × 10−5 1.99 8.981 × 10−6 2.44 5.037 × 10−5 2.301024 2.410 × 10−5 2.01 1.621 × 10−5 1.99 1.422 × 10−6 2.66 4.332 × 10−5 0.22
Table 4.2: Accuracy in the l∞ norm and order of convergence of the schemes for the second example of theeikonal equation.
Errors and order, 3rd ExampleN Monotone 2nd Upwind 3rd Upwind 4th Upwind64 1.368 × 10−2 - 3.079 × 10−4 - 1.332 × 10−15 - 2.887 × 10−15 -
128 6.756 × 10−3 1.01 7.860 × 10−5 1.95 1.110 × 10−15 0.26 3.109 × 10−15 -0.11256 3.417 × 10−3 0.98 1.960 × 10−5 1.99 2.220 × 10−15 -0.99 4.219 × 10−15 -0.44512 1.703 × 10−3 1.00 4.924 × 10−6 1.99 2.887 × 10−15 -0.38 2.665 × 10−15 0.661024 8.521 × 10−4 1.00 1.232 × 10−6 2.00 5.107 × 10−15 -0.82 4.663 × 10−15 -0.81
N 2nd centered 2nd ENO 3rd ENO 4th ENO64 6.357 × 10−4 - 3.079 × 10−4 - 2.442 × 10−15 - 1.332 × 10−15 -
128 1.596 × 10−4 1.97 7.860 × 10−5 1.95 7.550 × 10−15 -1.61 4.441 × 10−15 -1.72256 3.950 × 10−5 2.00 1.960 × 10−5 1.99 2.109 × 10−14 -1.47 3.819 × 10−14 -3.09512 9.886 × 10−6 1.99 4.924 × 10−6 1.99 3.220 × 10−14 -0.61 1.134 × 10−13 -1.57
1024 2.192 × 10−6 2.17 1.232 × 10−6 2.00 5.818 × 10−14 -0.85 8.527 × 10−14 0.41
Table 4.3: Accuracy in the l∞ norm and order of convergence of the schemes for the third example of theeikonal equation.
Errors and order, 4th ExampleN Monotone 2nd centered 2nd Upwind 2nd ENO64 1.234 × 10−1 - 8.532 × 10−2 - 9.307 × 10−2 - 8.308 × 10−2 -128 6.106 × 10−2 1.00 4.226 × 10−2 1.00 4.179 × 10−2 1.14 4.132 × 10−2 1.00256 3.037 × 10−2 1.00 2.108 × 10−2 1.00 2.095 × 10−2 0.99 2.067 × 10−2 0.99512 1.515 × 10−2 1.00 1.054 × 10−2 1.00 1.044 × 10−2 1.00 1.057 × 10−2 0.97
1024 7.563 × 10−3 1.00 5.310 × 10−3 0.99 5.304 × 10−3 0.98 5.272 × 10−3 1.00
Table 4.4: Accuracy in the l∞ norm and order of convergence of the schemes for the fourth example(H(p) = p2).
Chapter 4. Filtered Schemes for Hamilton-Jacobi equations 57
Errors and order, 5th ExampleN Monotone 2nd centered 2nd Upwind 2nd ENO64 1.328 × 10−1 - 2.105 × 10−2 – 1.129 × 10−1 - 8.577 × 10−2 -
128 1.095 × 10−1 0.27 1.111 × 10−2 0.91 3.446 × 10−2 1.69 6.995 × 10−2 0.29256 8.855 × 10−2 0.31 4.365 × 10−3 1.34 1.379 × 10−2 1.31 5.072 × 10−2 0.46512 7.043 × 10−2 0.33 2.360 × 10−3 0.88 3.772 × 10−3 1.86 1.288 × 10−2 1.97
1024 5.401 × 10−2 0.38 2.693 × 10−3 -0.19 1.870 × 10−3 1.01 8.170 × 10−3 0.66
Table 4.5: Accuracy in the l∞ norm and order of convergence of the schemes for the fifth example (H(p) =cos(p)2 + |p|).
smooth region. Therefore, in the eikonal equation case, the monotone scheme ends up not
being used in the upwind and ENO filtered schemes since these schemes have a choice on
where to interpolate, choosing to always do so on the region where the solution is smooth.
This is not however the case when the 2nd order centered scheme is used as the accurate
scheme. In this case, the filtered scheme falls back to the monotone scheme on the two grid
points adjacent to the singularity. As for the HJ equations case, the forward and backward
approximation are both always used and thus near the singularity the filtered schemes fall
back to the monotone scheme.
Order of convergence.
We first discuss the eikonal equation case. Examining Figure 4.5 and Tables 5.3, 5.4, 5.5, we
conclude that all the upwind filtered schemes have convergence rate corresponding to the
order of accuracy of the accurate scheme, except in the last example, where for the 3rd and
4th order schemes we obtain machine accuracy. This exception is explained by the fact that
in this example the solution is piecewise cubic and therefore these schemes end up being
exact (interpolating a cubic polynomial with 4 or more points yields the exact same cubic
polynomial). Obtaining the higher order convergence is in accordance with Theorem 4.2.4
since for the upwind filtered schemes the accurate scheme is always active as mentioned
above. We should point out that this higher order of convergence was already possible
to obtain using ENO schemes as is depicted in Figure 4.5 (with the sole exception of the
4th order ENO scheme in the second example, which we discuss below). Moreover, the
filtered scheme using the second centered scheme also provided second order convergence
even though as pointed above it falls into the monotone scheme near the singularity, more
precisely on the two grid points that enclose it.
In the general case of the HJ equations, the results are not as clean. In the first example,
the order of convergence remains the same with the monotone scheme still being first
order convergent. As for the second example, where the Hamiltonian is not convex, the
58 4.3. COMPUTATIONAL RESULTS
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−2
−1.8
−1.6
−1.4
−1.2
−1
−0.8
−0.6
−0.4
−0.2
0
2nd upwind
2nd ENO
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−3
−2.5
−2
−1.5
−1
−0.5
0
3rd upwind
3rd ENO
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−4
−3.5
−3
−2.5
−2
−1.5
−1
−0.5
0
4th upwind
4th ENO
Figure 4.3: Active stencils in the accurate schemein the last iteration for the solutions of the secondexample considered: −i means that i points to theleft were used in the interpolation.
101
102
103
10−14
10−12
10−10
10−8
10−6
10−4
10−2
100
Monotone
Filtered (2nd)
Filtered (2ndUpwind)
Filtered (3rdUpwind)
Filtered (4thUpwind)
Filtered (2ndENO)
Filtered (3rdENO)
Filtered (4thENO)
101
102
103
10−14
10−12
10−10
10−8
10−6
10−4
10−2
100
Monotone
Filtered (2nd)
Filtered (2ndUpwind)
Filtered (3rdUpwind)
Filtered (4thUpwind)
Filtered (2ndENO)
Filtered (3rdENO)
Filtered (4thENO)
101
102
103
10−16
10−14
10−12
10−10
10−8
10−6
10−4
10−2
100
Monotone
Filtered (2nd)
Filtered (2ndUpwind)
Filtered (3rdUpwind)
Filtered (4thUpwind)
Filtered (2ndENO)
Filtered (3rdENO)
Filtered (4thENO)
Figure 4.4: Log-log plot of the errors for the one-dimensional examples of the eikonal equation.
Chapter 4. Filtered Schemes for Hamilton-Jacobi equations 59
101
102
103
10−3
10−2
10−1
100
Monotone
Filtered (2nd)
Filtered (2ndUpwind)
Filtered (2ndENO)
101
102
103
10−4
10−3
10−2
10−1
100
Monotone
Filtered (2nd)
Filtered (2ndUpwind)
Filtered (2ndENO)
Figure 4.5: Log-log plot of the errors for the one-dimensional examples of HJ equations.
monotone scheme is not even first order convergent as in all the other examples and we
see an increase in the order of convergence for both the second order upwind and ENO
filtered schemes. In general we do not expect this increase in the order of convergence of
the global accuracy since near the singularity we fall back into the monotone scheme.
Upwind vs ENO.
The ENO filtered schemes only outperformed the upwind filtered schemes in the first
example for the eikonal equation. In this example, both schemes have the same order of
convergence but with ENO schemes having a smaller asymptotic error constant, which
can be explained by the fact that the ENO schemes in this example tend to use centered
discretizations which have a smaller truncation error than the upwind discretizations. On
the other examples, the upwind filtered schemes always performed at least as good as its
ENO counterparts.
To finish the discussion, we now take a closer look at the second example for the eikonal
equation. In this, the fourth order ENO scheme does not have fourth order accuracy and
is in fact less accurate than the third order ENO scheme, which also does not have third
order accuracy. In this case, although never interpolating where the solution is singular,
the ENO scheme uses three different stencils (see Figure 4.3) which somehow seems to
prevent us to obtain the fourth order accuracy. Moreover, the second order ENO scheme
performs an interpolation where the solution is singular, although this does not affect the
order of convergence of the method (see Figure 4.3). This example illustrates the advantage
of using the upwind filtered scheme, which has a fixed stencil, over the ENO scheme,
which, while designed heuristically to choose the best stencil, may not always do so. It is
60 4.3. COMPUTATIONAL RESULTS
worth mentioning that the WENO schemes were introduced to improve the ENO schemes,
but these add another layer of complexity without any clear advantage over the filtered
upwind schemes.
4.3.3 Exact solutions in two dimensions
In this subsection we discuss the two dimensional examples. We consider three solutions
to the eikonal equation (4.1) with f ≡ 1, g ≡ 0 and Γ given by a circle, two points, and a
semicircle. Specifically, we have
1. Γ = (x, y) ∈ R2 | x2 + y2 = 1 ,
2. Γ =(
12, 1
2
)
,(
−12, −1
2
)
,
3. Γ = (x, y) ∈ R2 | (x2 + y2 = 1, x ≥ 0) ∨ (|y| ≤ 1, x = 0) .
We chose these examples because the corresponding solutions have varying degrees
of regularity. In the first, the solution is smooth (outside Γ). In the second we have a
singularity along the line (x, y) ∈ R2 : x = −y and therefore the solution is only piecewise
smooth outside (Γ). In the third, (outside Γ) the solution is smooth for x > 0 but only
Lipschitz continuous for x < 0. The exact solution is the distance function to the set Γ.
All computations are performed on the domain [−2, 2] × [−2, 2], which is discretized on
an N × N grid. We assume the exact solution to be known at the neighboring grid points
of Γ as discussed in subsection 4.2.3, except in the second example where we initialize
the solution where u < 0.1 in order to avoid dealing with the singularities at Γ (this is a
standard thing to do when studying the higher global accuracy of the methods [ABM+11]).
All solutions are displayed in Figure 4.6.
4.3.4 Computational results in two dimensions
In this subsection we discuss the computational results obtained in two dimensions. The
main purpose is to demonstrate that the filtered scheme achieves the higher order accuracy
in the regions where the solution is smooth. We organize the discussion in three parts:
accuracy and behaviour, order of convergence and upwind vs ENO. We obtained results
with the monotone scheme (4.5) and with the respective filtered schemes using as the
accurate scheme the second order centered, upwind and ENO schemes.
Chapter 4. Filtered Schemes for Hamilton-Jacobi equations 61
−2
−1
0
1
2
−2
−1
0
1
2
0
0.5
1
1.5
2
−2
−1
0
1
2
−2
−1
0
1
2
0
0.5
1
1.5
2
2.5
3
−2
−1
0
1
2
−2
−1
0
1
2
0
0.5
1
1.5
2
2.5
−2 −1 0 1 2−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
−2 −1 0 1 2−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
0.5
1
1.5
2
2.5
−2 −1 0 1 2−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
2.2
Figure 4.6: Profile and contour plots of the solutions of the three examples considered in two dimensions.
Errors and order, 1st ExampleN Monotone 2nd centered 2nd Upwind 2nd ENO64 2.167 × 10−2 - 9.034 × 10−4 - 1.107 × 10−3 - 5.284 × 10−4 -128 1.126 × 10−2 0.93 2.368 × 10−4 1.91 3.030 × 10−4 1.85 1.476 × 10−4 1.82256 5.661 × 10−3 0.99 5.964 × 10−5 1.98 7.627 × 10−5 1.98 3.766 × 10−5 1.96512 2.854 × 10−3 0.99 1.516 × 10−5 1.97 1.949 × 10−5 1.96 9.682 × 10−6 1.95
1024 1.432 × 10−3 0.99 3.893 × 10−6 1.96 4.903 × 10−6 1.99 2.444 × 10−6 1.98
Table 4.6: Accuracy and order of convergence of the schemes for the first example in two dimensions in thel∞ norm.
Errors and order, 2nd ExampleN Monotone 2nd centered 2nd Upwind 2nd ENO64 5.128 × 10−2 - 1.643 × 10−2 - 1.276 × 10−2 - 1.297 × 10−2 -128 2.663 × 10−2 0.93 1.016 × 10−2 0.69 9.837 × 10−3 0.37 9.514 × 10−3 0.44256 1.326 × 10−2 1.00 5.485 × 10−3 0.88 5.121 × 10−3 0.94 4.795 × 10−3 0.98512 6.640 × 10−3 1.00 3.019 × 10−3 0.86 2.600 × 10−3 0.98 2.402 × 10−3 0.99
1024 3.324 × 10−3 1.00 1.483 × 10−3 1.02 1.425 × 10−3 0.87 1.490 × 10−3 0.69
Table 4.7: Accuracy and order of convergence of the schemes for the second example in two dimensions inthe l∞ norm.
62 4.3. COMPUTATIONAL RESULTS
Errors and order, 2nd ExampleN Monotone 2nd centered 2nd Upwind 2nd ENO64 4.310 × 10−1 - 4.355 × 10−2 - 7.111 × 10−2 - 3.589 × 10−2 -128 2.202 × 10−1 0.96 1.331 × 10−2 1.69 1.967 × 10−2 1.83 1.002 × 10−2 1.82256 1.088 × 10−1 1.01 2.893 × 10−3 2.19 4.844 × 10−3 2.01 2.538 × 10−3 1.97512 5.420 × 10−2 1.00 9.942 × 10−4 1.54 1.233 × 10−3 1.97 6.506 × 10−4 1.96
1024 2.706 × 10−2 1.00 2.697 × 10−4 1.88 3.149 × 10−4 1.97 1.697 × 10−4 1.94
Table 4.8: Accuracy and order of convergence of the schemes for the second example in two dimensions inthe l1 norm.
Errors and order, 3rd ExampleN Monotone 2nd centered 2nd Upwind 2nd ENO64 5.771 × 10−2 - 9.083 × 10−3 - 9.342 × 10−3 - 8.811 × 10−3 -128 3.541 × 10−2 0.70 4.833 × 10−3 0.90 5.508 × 10−3 0.75 4.566 × 10−3 0.94256 2.117 × 10−2 0.74 2.399 × 10−3 1.00 3.344 × 10−3 0.72 2.605 × 10−3 0.81512 1.238 × 10−2 0.77 1.470 × 10−3 0.70 2.523 × 10−3 0.41 1.574 × 10−3 0.72
1024 7.112 × 10−3 0.80 1.024 × 10−3 0.52 1.517 × 10−3 0.73 1.055 × 10−3 0.58
Table 4.9: Accuracy and order of convergence of the schemes for the third example in two dimensions in thel∞ norm.
Accuracy and behaviour of the filtered schemes
We begin with the results presented in Figure 4.7 and Tables 4.6, 4.7, 4.9. It is clear the
solutions computed using the filtered schemes are more accurate.
The behaviour of the filtered schemes is very much like the one obtained in the one-
dimensional examples: in first example, the monotone scheme is never used since the
solution is smooth; in the second example, it is only used near the singularity at x = −y; in
the third example, it is only used near the corners of Γ.
Order of convergence
Unlike the one dimensional case for the eikonal equation, the order of convergence of
the error in the l∞ norm can be less than the formal order of accuracy of the accurate
schemes and will depend on the smoothness of the solutions. In the first example, the
solution is smooth and we obtain second order convergence in the l∞ norm (see Figure
4.7 and Table 4.6). This was expected since the “equivalent” fast marching method was
already proven second order convergent for smooth solutions in [ABM+11]. In the second
example, we have a shock of co-dimension 1 and therefore we get first order convergence
in the l∞ norm and second order in the l1 norm (see Figures 4.7, 4.8 and Tables 4.7,
4.8). We can still recover the second order of convergence in the l∞ norm by looking
Chapter 4. Filtered Schemes for Hamilton-Jacobi equations 63
away from the singularities (see Figure 4.9). As for the third example, we do not have
shocks, but the solution is nevertheless singular due to the corners in Γ which have a
rarefaction effect much like the ones in hyperbolic conversation laws (see [QS99]). For
instance, in the region (x, y) ∈ R2 : x < 0, y > 1 all characteristics emanate from the point
(0, 1) and so the errors incurred there will propagate out and pollute the solution. Thus
the error is globally first order in both the l∞ and l1 norm. However if we restrict the
errors to the region (x, y) ∈ R2 : x2 + y2 ≥ 1, x ≥ 0.1 where the solution is smooth we
do obtain second order convergence in the l∞ norm (see Figure 4.10). Finally, in region
(x, y) ∈ R2 : |y| ≤ 0.8, x ≤ 0, all the schemes were exact up to machine precision since
they are exact on flat regions.
Upwind vs ENO
Comparing the upwind schemes to the ENO schemes, we see that we obtained similar
results with the difference being a smaller asymptotical constant. This is explained by the
fact that ENO schemes tend to use centered discretizations which have a smaller truncation
error than the upwind discretizations.
Third order upwind and ENO filtered schemes were also used, but they did not show
any advantage over the second order schemes. We did not even obtain the third order
convergence for the first example even though the solution is smooth. This is most likely
related to a result proven in [ABM+11]. There the authors show that the “equivalent” third
order fast marching method is unstable. They also provide an alternative scheme which
uses full two-dimensional stencils and that it is provably third order globally convergent
in the l∞ norm for smooth solutions. We expect that if we use that scheme as our accurate
scheme we would obtain a filtered scheme with the same order of convergence.
4.4 CONC LUS IONS
We introduce filtered schemes for HJ equations, which allow us to construct convergent,
high order accurate finite difference schemes. These schemes are extremely flexible in the
choice of accurate scheme, and so they allow for a wide range of existing discretizations
(even unstable ones) to be used, while retaining the stability and convergence proof of the
monotone schemes.
Focusing on the special, but important case of the eikonal equation, we tested the
accuracy of several discretizations on solutions of varying regularity in one and two
dimensions. In one dimension, we used filtered central differences, filtered higher order
upwinding, and filtered ENO schemes. In each case we obtained higher accuracy, even in
64 4.4. CONCLUSIONS
102
103
10−6
10−4
10−2
Monotone
Filtered (2nd)
Filtered (2ndUpwind)
Filtered (2ndENO)
102
103
10−4
10−2
Monotone
Filtered (2nd)
Filtered (2ndUpwind)
Filtered (2ndENO)
102
103
10−4
10−2
Monotone
Filtered (2nd)
Filtered (2ndUpwind)
Filtered (2ndENO)
Figure 4.7: Log-log plot of the errors for the two-dimensional examples in the l∞ norm.
102
103
10−6
10−4
10−2
100
Monotone
Filtered (2nd)
Filtered (2ndUpwind)
Filtered (2ndENO)
102
103
10−4
10−2
100
Monotone
Filtered (2nd)
Filtered (2ndUpwind)
Filtered (2ndENO)
102
103
10−4
10−2
100
Monotone
Filtered (2nd)
Filtered (2ndUpwind)
Filtered (2ndENO)
Figure 4.8: Log-log plot of the errors for the two-dimensional examples in the l1 norm.
Chapter 4. Filtered Schemes for Hamilton-Jacobi equations 65
102
103
10−4
10−2
Monotone
Filtered (2nd)
Filtered (2ndUpwind)
Filtered (2ndENO)
Figure 4.9: Log-log plot of the errors for thesecond example in the l∞ norm in regions
(x, y) ∈ R2 : |x + y| > 0.1
102
103
10−6
10−4
10−2
Monotone
Filtered (2nd)
Filtered (2ndUpwind)
Filtered (2ndENO)
Figure 4.10: Log-log plot of the errors forthe third example in the l∞ norm in regions
(x, y) ∈ R2 : x2 + y2 ≥ 1, x ≥ 0.1
.
regions where the solution was not smooth. For the eikonal equation case we were able
to prove the higher order convergence. This result, although very special to the eikonal
equation, illustrates the potential accuracy of the method.
Due to the explicit nature of the filtered upwind schemes we were able to use the simple
but effective fast sweeping method to compute solutions. In the case of filtered ENO, a
slower iterative method was used. We also gave a comparison using filtered ENO schemes,
and found an example where the error for ENO was greater than its formal accuracy.
The convergence results in two dimensions were more complicated, but more generic,
in that we expect similar results on more general HJ equations. In this case, for smooth
solutions, we obtained second order accuracy. The same order of accuracy has been
previously obtained by several authors using more complicated schemes as opposed to
the simplicity of the upwind filtered schemes. In particular, our filtered upwind schemes
in two dimensions are still explicit, thus allowing the use of the fast sweeping method to
obtain solutions.
The schemes developed here are simple to implement, and allow an unrestricted choice
of higher order discretizations to be used. While we mainly focused on a particular type
equation (HJ equations), it should be clear that the filtered schemes can be used in much
wider context, while still retaining the advantages of accuracy, stability and convergence
to the viscosity solution of the monotone schemes.
CHAPTER 5
2-HESSIAN EQUATION
5.1 INTR ODUC TI ON
In this chapter, we study numerical approximations of the fully nonlinear elliptic partial
differential equation (PDE), the 2-Hessian equation in three dimensions,
S2[u] = uxxuyy + uxxuzz + uyyuzz − u2xy − u2
xz − u2yz. (5.1)
We introduce a monotone discretization and a convergence proof to the viscosity solution
is provided. We also build a second order accurate finite difference solver which, while
unstable if a simple iteration is used, can be modified to converge in practice. Numerical
results are presented on solutions with varying regularity.
We focus on the Dirichlet problem
S2[u] = f, in Ω,
u = g, on ∂Ω,(2H)
where Ω is a rectangular (three dimensional box) domain, which is natural when treating
computationally prescribed curvature problems. (For other topologies, different boundary
conditions need to be used. For the torus, periodic boundary conditions can be used. For
the sphere, it is more complicated, but it should be possible to patch together several cubic
domains to obtain this topology.)
The operator is not elliptic, unless an additional constraint is imposed. This condition
is explained in Proposition 5.2.9 and if we assume that f > 0, it reduces to
d2u
dv2+
d2u
dw2≥ 0, for every orthogonal triplet of vectors (v, w, z).
In other words, the two dimensional Laplacian restricted to every plane is positive for
the function u. Hence the discretizations of the operator must also enforce the condition
above. This means that either we are working with a family of inequality constraints,
which makes the discretization very challenging, or that we need to find a way to encode
the constraints in either the PDE or the solver. Here, we pursue the second option.
67
68 5.2. BACKGROUND ON THE EQUATION
5.2 BACKG ROUND ON THE EQUATI ON
In this section, we present the background analysis for the k-Hessian equations, with
particular focus on the 2-Hessian equation in the three dimensional case. We follow the
review by Wang [Wan09].
The k-Hessian equation can be written as
Sk[u] = f (5.2)
where 1 ≤ k ≤ n, Sk[u] = σk(λ(D2u)), λ(D2u) = (λ1, . . . , λn) are the eigenvalues of the
Hessian matrix D2u and
σk(λ) =∑
i1<···<ik
λi1 . . . λik
is the k-th elementary symmetric polynomial. It includes the Poisson equation (k = 1)
∆u = f,
and the Monge-Ampère equation (k = n)
det D2u = f,
as particular cases.
We are interested in the Dirichlet problem
Sk[u] = f, in Ω,
u = g, on ∂Ω.(kH)
5.2.1 Admissible functions and ellipticity
When k is even, the k-Hessian equation lacks uniqueness: if u solves the k−Hessian equa-
tion, so does −u. Thus an additional condition is needed to ensure solution uniqueness.
Moreover, when studying the Poisson equation it is customary to focus on the case f ≥ 0,
which is equivalent to look for solutions that are subharmonic since as a result a maximum
principle holds. In the case of the Monge-Ampère equation, one imposes instead the
additional constraint that u is convex, which is required for the ellipticity of the equation.
In both cases, it is thus necessary to restrict the solutions to an appropriate class of func-
tions in order to ensure the equation has interesting properties. The general case of the
k-Hessian is not different.
Set
Γk = λ ∈ Rn | σj(λ) > 0, j = 1, . . . , k . (5.3)
Chapter 5. 2-Hessian equation 69
Γk is a symmetric cone, meaning that any permutation of λ is in Γk. When k = 1, Γ1
is the half space λ ∈ Rn | λ1 + . . . + λn > 0. When k = n, Γn is the positive cone Γn =
λ ∈ Rn | λj > 0, j = 1, . . . , n. The result is a restriction to subharmonic functions for k = 1
and convex functions for k = n, as mentioned above. We define as well Sn(Γk) as the set of
n × n real symmetric matrices with eigenvalue vector belonging to Γk.
Definition 5.2.1. A function u ∈ C2(Ω) is k−admissible if λ(D2u(x)) ∈ Γk for all x ∈ Ω.
Proposition 5.2.2. The function F : Ω×R×Rn×Sn → R given by F (x, r, p, M) = −σk(λ(M))+
f(x), which corresponds to the k−Hessian equation (kH), is degenerate elliptic when restricted to
Sn(Γk).
Proof. See [Wan09] for a proof.
5.2.2 Viscosity Solutions
In chapter 2, we discussed viscosity solutions. However, the k-Hessian equation is only
elliptic for k-admissible functions. Therefore, the definition of viscosity solution must be
adapted.
Definition 5.2.3. A function u ∈ USC(Ω) is a viscosity subsolution of (kH) if for every k-
admissible φ ∈ C2(Ω), whenever, u − φ has a local maximum at x ∈ Ω then
−σk
(
λ(D2φ(x)))
+ f(x) ≤ 0, if x ∈ Ω,
min(
−σk
(
λ(D2φ(x)))
+ f(x), u(x) − g(x))
≤ 0, if x ∈ ∂Ω.
Similarly, a function u ∈ LSC(Ω) is a viscosity supersolution of (kH) if for every k-admissible
φ ∈ C2(Ω), whenever, u − φ has a local minimum at x ∈ Ω then
−σk
(
λ(D2φ(x)))
+ f(x) ≥ 0, if x ∈ Ω,
max(
−σk
(
λ(D2φ(x)))
+ f(x), u(x) − g(x))
≥ 0, if x ∈ ∂Ω.
Finally, we call u a viscosity solution of (kH) if u is both a viscosity subsolution and viscosity
supersolution of (kH).
The next proposition illustrates the consistency of the definition of viscosity solution
with that of classical solution.
Proposition 5.2.4. If u is a k-admissible classical solution of (kH), then u is a viscosity solution.
Conversely, if u is a viscosity solution of (kH), f > 0 and u ∈ C2(Ω), then u is a k-admissible
classical solution.
70 5.2. BACKGROUND ON THE EQUATION
Proof. We follow [Urb90]. We focus on the first assertion. Suppose u is a k-admissible
classical solution of (kH). Let φ ∈ C2(Ω) be a k-admissible test function and assume that
u − φ has a local maximum at x ∈ Ω. If x ∈ ∂Ω,
min(
−σk
(
λ(D2φ(x)))
+ f(x), u(x) − g(x))
≤ u(x) − g(x) = 0.
If x ∈ Ω, then ∇u(x) = ∇φ(x) and D2u(x) ≤ D2φ(x) and therefore
−σk
(
λ(D2φ(x)))
+ f(x) ≤ −σk
(
λ(D2u(x)))
+ f(x) = 0,
since (kH) is degenerate elliptic by Proposition 5.2.2 and u is an admissible classical
solution by assumption. Thus u is a viscosity subsolution. The proof that u is a viscosity
supersolution is similar and so we can conclude that u is a viscosity solution.
As for the second assertion, let x0 ∈ Ω and assume that assume that D2u(x0) /∈ Sn(Γk).
Since u ∈ C2(Ω), we have
u(x) = u(x0) + ∇uᵀ(x − x0) +1
2(x − x0)
ᵀD2u(x0)(x − x0) + O(
|x − x0|3)
Sn(Γk) is an open convex cone with vertex at the origin and contains Sn(Γn). Therefore,
since D2u(x0) /∈ Sn(Γk), there exists a unique α > 0 such that D2u(x0) + αI ∈ ∂Sn(Γk).
Consider the function φ given by
φ(x) = u(x0) + ∇uᵀ(x − x0) +1
2(x − x0)
ᵀD2u(x0)(x − x0) +1
2α |x − x0|2 .
φ is k-admissible and u − φ has a local maximum at x0. Hence, since u is a viscosity
subsolution,
−σk
(
λ(D2φ(x0) + αI))
+ f(x0) ≤ 0.
However, σk = 0 on ∂Sn(Γk) and f(x0) > 0, and so we obtain a contradiction.
The well-posedness and regularity of the equation was studied in [CNS85]. Here we
recall a well-posedness result.
Definition 5.2.5. We say that Ω ⊆ Rn is (k − 1)-convex if it satisfies
σk−1(κ) ≥ c0 > 0 on ∂Ω
for some positive constant c0 where κ = (κ1, . . . , κn−1) denote the principal curvatures of ∂Ω with
respect to its inner normal.
Chapter 5. 2-Hessian equation 71
Theorem 5.2.6. Assume that Ω is a bounded (k − 1)-convex domain in Rn with C3,1 boundary
∂Ω, g ∈ C3,1 (∂Ω) and f ∈ C1,1(Ω) with f ≥ f0 > 0. Then there is a unique k-admissible solution
u ∈ C3,α(Ω) to the Dirichlet problem (kH) for some α ∈ (0, 1).
5.2.3 2-Hessian equation
In this chapter, we focus now on the the three-dimensional case of the 2-Hessian equation
S2[u] = f, (5.4)
where
S2[u] = σ2(λ) = λ1λ2 + λ1λ3 + λ2λ3. (5.5)
The 2-Hessian operator, S2[u], can be characterized in terms of the Hessian matrix as the
next Proposition illustrates. This alternative formula will be of particularly useful to define
a naive finite difference discretization for the 2-Hessian operator.
Proposition 5.2.7. For u ∈ C2(Ω), we can write
S2[u] = c(
D2u)
= uxxuyy + uxxuzz + uyyuzz − u2xy − u2
xz − u2yz
where c(M), the sum of the principal minors of M , is given by
c(M) =1
2
(
trace(M)2 − trace(M2))
. (5.6)
Proof. For a 3 × 3 matrix M , the characteristic polynomial is given by
det(M) − c(M)λ + trace(M)λ2 − λ3
If λ1, λ2 and λ3 are the eigenvalues of M then
c(M) = λ1λ2 + λ1λ3 + λ2λ3.
Therefore, using (5.5), we conclude that
S2[u] = c(
D2u)
= uxxuyy + uxxuzz + uyyuzz − u2xy − u2
xz − u2yz
as desired.
The linearization of c(M) defined in (5.6), is given by:
∇c(M) · N = trace(M) trace(N) − trace(MN).
72 5.2. BACKGROUND ON THE EQUATION
We can apply it to obtain the linearization of the 2-Hessian operator, S2[u], for u ∈ C2(Ω),
∇S2[u] · ν = trace(D2u) trace(D2ν) − trace(D2uD2ν). (5.7)
Recall that a linear operator L[u] ≡ trace(A(x)D2u) is elliptic if the coefficient matrix A(x)
is positive definite.
Lemma 5.2.8. Let u ∈ C2(Ω). The linearization of the 2−Hessian operator (5.7) is elliptic if u is
2-admissible.
Proof. Without loss of generality, we choose coordinates such that D2u(x) is diagonal. We
can then rewrite the linearization of the 2-Hessian operator as
∇S2[u] · ν = trace(AD2ν)
where A = diag(λ2 + λ3, λ1 + λ3, λ1 + λ2). Hence, the linearization is elliptic if A is positive
definite, which is true if u is 2-admissible.
Remark 5.1. When the function u fails to be “strictly” 2-admissible, the linearization can be
degenerate elliptic, which affects the conditioning of the linear system (5.7). When u is not
2-admissible, the linear system can be unstable.
Our approach to build discretizations for the 2-Hessian operator relies on encoding the
2-admissibility of the solution into the PDE. In order to do so, an alternative characteri-
zation of the set Γ2, derived below, is used. Recall that by definition of Γk (see (5.3) with
k = 2), we have
Γ2 =
λ ∈ R3 | λ1 + λ2 + λ3 > 0, σ2(λ) > 0
.
Proposition 5.2.9. Let
Γ =
λ ∈ R3 | λ1 + λ2 > 0, λ1 + λ3 > 0, λ2 + λ3 > 0
. (5.8)
Then
Γ2 = Γ ∩ λ ∈ R3 | σ2(λ) > 0.
Proof. Proving the ⊇ part is straightforward. We then prove the inclusion ⊆. Suppose that
(λ1, λ2, λ3) ∈ Γ2. Without loss of generality we can assume that λ1 ≤ λ2 ≤ λ3. Thus, it is
sufficient to show that λ1 + λ2 > 0. Suppose that λ1 + λ2 ≤ 0. We consider two cases, each
leading to a contradiction.
• λ1 + λ2 = 0
Chapter 5. 2-Hessian equation 73
We have λ1λ2 ≤ 0. Hence
σ2(λ) = λ1λ2 + λ1λ3 + λ2λ3 = λ1λ2 + (λ1 + λ2)λ3 = λ1λ2 ≤ 0,
contradicting the assumption σ2(λ) > 0.
• λ1 + λ2 < 0
Since λ1 ≤ λ2, we have λ1 < 0. Moreover
σ2(λ) > 0 ⇐⇒ λ3(λ1 + λ2) > −λ1λ2 ⇐⇒ λ3 < − λ1λ2
λ1 + λ2
and
λ1 + λ2 + λ3 > 0 ⇐⇒ λ3 > −λ1 − λ2.
From the above two inequalities we get
−λ1 − λ2 < − λ1λ2
λ1 + λ2
which we can rewrite as
λ1(λ1 + λ2) + λ22 < 0.
Now, since λ1 < 0 and λ1 + λ2 < 0, the left-end side of the inequality must be positive and
we have thus derived a contradiction.
Using differentiation it is straight forward to show that the function σ2 is nondecreasing
on the set Γ, which gives some insight to why the set of admissible functions is the set of
functions where S2 is elliptic.
The constraint σ2(λ) ≥ 0 will be enforced automatically in our schemes by taking a
nonnegative f in the PDE (2H). Therefore it is sufficient to look at the set Γ as defined in
(5.8). We will refer to this restriction as plane-subharmonic since it corresponds to u being
subharmonic on every plane.
5.3 D IS C RETIZATION AND S OLVER S
In this section, we explain why the naive finite difference method fails in general. We
introduce explicit, semi-implicit, and Newton solvers for the naive finite difference method,
which perform better by enforcing the plane-subharmonic constraint. This is similar
to the solvers used in [BFO10] for the Monge-Ampère equation. Then we introduce a
discretization which is monotone and thus provably convergent.
74 5.3. DISCRETIZATION AND SOLVERS
While the monotone discretization is less accurate, it has the advantage that it gives a
globally consistent monotone discretization of the operator, meaning that we can apply the
operator to non-admissible functions. This is useful because it circumvents the need for
special initial data, and allows for the parabolic (time-dependent) operator to be defined
on an unconstrained class of functions.
In addition, we could combine the monotone discretization with the naive finite dif-
ference discretization to obtain provably convergent, accurate filtered finite difference
schemes (see section 3.4 for more details). This approach combines the advantages of both
schemes, with little additional effort. In this work, we were mainly interested in comparing
the performance of the two schemes, so we did not implement the filtered scheme.
5.3.1 Naive finite difference scheme
We begin by introducing a naive finite difference discretization of the 2-Hessian operator.
This is done by simply using standard finite differences to discretize the operator. Denote
by D2,hu the discretized Hessian using standard finite differences on a uniform grid with
grid spacing h, i.e.,
D2,huijk =
Dxxuijk Dxyuijk Dxzuijk
Dxyuijk Dyyuijk Dyzuijk
Dxzuijk Dyzuijk Dzzuijk
,
where, e.g.,
Dxxuijk =ui+1,j,k − 2ui,j,k + ui−1,j,k
h2,
Dxyuijk =ui+1,j+1,k + ui−1,j−1,k − ui−1,j+1,k − ui+1,j−1,k
4h2.
We then get the discrete version of the 2-Hessian operator S2[u] as
SA2 [u] = c
(
D2,hu)
. (2H)A
Since we are using centered finite differences, this discretization is consistent, and it is
second order accurate if the solution is smooth (hence the superscript A) as the Lemma
below shows. However, this scheme is not monotone due to the off-diagonal terms in
the cross derivatives uxy, uxz and uyz. Therefore the Barles and Souganidis theory [BS91],
discussed in chapter 3, does not apply and no convergence proof is available.
Lemma 5.3.1. Let x ∈ Ω be a reference point on the grid and φ be a C4 function that is defined
in a neighbourhood of the grid. Then the scheme SA2 [φ] defined in (2H)A approximates (2H) with
Chapter 5. 2-Hessian equation 75
accuracy
SA2 [φ](x) = S2[φ](x) + O(h2).
Proof. It follows from a standard Taylor series argument and the fact that all the standard
finite differences used are second order accurate.
5.3.2 Failure of the parabolic solver for the naive finite differences
In this section, we give a simple example to illustrate that the use of the naive finite
difference scheme (2H)A together with a parabolic solver fails to converge.
The parabolic solver is given by
un+1 = un − dt(−SA2 [u] + f). (5.9)
Consider the solution of (2H) in [0, 1]3, given by
u(x) =x
2
2, f(x) = 3.
The iteration is initialized with the exact solution with noise from a uniform distribution
U(−0.01, 0.01). The result after performing 25 iterations with the parabolic solver (5.9)
with time step dt = dx4 and the initial guess are illustrated in Figure 5.1. Regardless of the
time step choosen (dt = dx4/10 and dt = dx4/100 were also used), after a sufficient number
of iterations the solution behaves like in the example of Figure 5.1, until it eventually
blows up. This tells us that the instability of the parabolic solver is inherent from the
discretization rather than being the result of a poorly chosen time step. This instability is
due to the fact that there is no mechanism to pick the right solution. The discretization,
being a quadratic equation as we will see below, has two solutions: the 2-admissible
solution we are looking for and the negative of this.
5.3.3 Solvers for the naive finite difference scheme
In this section we present three different solvers for the naive finite difference scheme:
a Jacobi type solver obtained by solving the discretization for the reference variable; a
semi-implicit solver based on an identity that relates the Laplacian and the 2-Hessian
operator; a Newton solver.
76 5.3. DISCRETIZATION AND SOLVERS
00.2
0.40.6
0.81
0
0.5
1
0
0.5
1
1.5
00.2
0.40.6
0.81
0
0.5
1
0
0.5
1
1.5
Figure 5.1: Failure of the parabolic solver using the naive finite difference scheme: section z = 0.9 of theinitial guess (left) and the solution after 25 iterations (right).
5.3.3.1 Jacobi solver
The accurate discretization of (2H) leads to a quadratic equation for the reference variable
at each grid point. To see this we introduce the notation
a1 =ui+1,j,k + ui−1,j,k
2a2 =
ui,j+1,k + ui,j−1,k
2a3 =
ui,j,k+1 + ui,j,k−1
2
a4 =ui+1,j+1,k + ui−1,j−1,k
2a5 =
ui−1,j+1,k + ui+1,j−1,k
2a6 =
ui+1,j,k+1 + ui−1,j,k−1
2
a7 =ui−1,j,k+1 + ui+1,j,k−1
2a8 =
ui,j+1,k+1 + ui,j−1,k−1
2a9 =
ui,j+1,k−1 + ui,j−1,k+1
2
(5.10)
Using (2H)A, SA2 [u] = f can be rewritten as
4
h4
∑
i1<i2≤3
(ai1 − uijk)(ai2 − uijk)
= fijk +1
4h4
4∑
p=2
(a2p − a2p+1)2.
Solving for uijk and selecting the smaller root (in order to select the locally more plane-
subharmonic solution), we obtain
uijk =a1 + a2 + a3
3− 1
12
√
√
√
√8∑
i1<i2≤3
(ai1 − ai2)2 + 34∑
p=2
(a2p − a2p+1)2 + 12fijkh4. (5.11)
We can now use a Jacobi iteration to find the fixed point of (5.11). Notice that the
plane-subharmonic constraint is not enforced beyond the selection of the smaller root
in (5.11).
Chapter 5. 2-Hessian equation 77
Remark 5.2. Formula (5.11) can be rewritten as
uijk =a1 + a2 + a3
3− h2
6
√
trace(D2,huijk)2 + 3(
fijk − S2,Ah [u]
)
.
Remark 5.3. Formula (5.11) can also be used in a Gauss-Seidel iteration, which should
converge faster than the Jacobi iteration. We chose not to implement it here since all
computational results were obtained in MATLAB, which is known to be slow with loops.
In order to prove the convergence of the above solver, it would be sufficient to show
that it is monotone, which in this case is the same as showing that the value uijk is a
nondecreasing function of the neighboring values [Obe06a]. However, this is not the case
for (5.11).
5.3.3.2 Semi-implicit solver
The next solver we discuss is a semi-implicit one, which involves solving a Laplace
equation at each iteration.
We begin with the following identity for the Laplacian in three dimensions:
|∆u| =√
(∆u)2 =√
u2xx + u2
yy + u2zz + 2uxxuyy + 2uxxuzz + 2uyyuzz.
If u solves the 2-Hessian equation, then
|∆u| =√
(∆u)2 =√
u2xx + u2
yy + u2zz + 2u2
xy + 2u2xz + 2u2
yz + 2f =√
|D2u|2 + 2f.
This leads to a semi-implicit scheme for solving the 2-Hessian equation given by
∆un+1 =√
|D2un|2 + 2f. (5.12)
Note that if u is a 2-admissible function, then ∆u ≥ 0, a condition the scheme enforces.
A good initial value for the iteration is given by the solution of
∆u0 =√
2f.
5.3.3.3 Newton solver
To solve the discretized equation SA2 [u] = f we can also use a damped Newton iteration
un+1 = un − αvn,
where 0 < α ≤ 1. The damping parameter α is chosen at each step to ensure that the
residual∥
∥
∥SA2 [un] − f
∥
∥
∥ is decreasing. (In practice we can often take α = 1, but damping is
78 5.3. DISCRETIZATION AND SOLVERS
sometimes needed.) The corrector vn solves the linear system
(
∇uSA2 [un]
)
vn = SA2 [un] − f.
To setup the above equation we need the Jacobian of the scheme, which is given by
∇uSA2 [u] =
∑
ν1,ν2∈x,y,z,ν1 6=ν2
(Dν1ν1u)Dν2ν2 − (Dν1ν2u)Dν1ν2 .
Notice that this corresponds to the discrete version of the linearization of the 2-Hessian
equation (5.7).
5.3.4 Monotone finite difference scheme
In this section we construct a monotone finite difference scheme. As we saw before, the
naive approach of simply using standard finite differences for the terms in the Hessian
matrix will not work because the cross derivative terms uxy, uxz and uyz are not monotone.
Instead the idea is to use wide stencils and a rotated coordinate system in which the
Hessian matrix is diagonal. However, this coordinate system must be found in a monotone
way. This is achieved in different steps. First, we extend the function σ2 (5.5) to be
nondecreasing in R3. This then allows us to find an equivalent expression for the 2-Hessian
operator S2[u] which can not only be discretized in a monotone manner but also encodes
the 2-admissibility constraint into the operator. Finally, we present the monotone finite
difference scheme.
We start by finding a nondecreasing extension of σ2 from Γ to R3. The reason for this is
that since we know that the eigenvalues of admissible solutions u belong to the set Γ, we
are free to redefine σ2 outside of Γ, which will prove useful later to ensure convergence.
Lemma 5.3.2. The function σ2 = f sort where sort denotes the sorting function and f is given
by
f(x, y, z) = x max(y, |x|) + x max(z, |x|) + max(y, |x|) max(z, |x|)
extends σ2 on Γ and is nondecreasing in R3.
Proof. Without loss of generality, we assume that x ≤ y ≤ z since sorting the values is
monotone. Moreover, we can rewrite f as
f(x, y, z) = max (y + x, |x| + x) max (z + x, |x| + x) − x2.
Suppose (x, y, z) ∈ Γ, then we recover σ2(x, y, z).
Chapter 5. 2-Hessian equation 79
Next we show that σ2 is nondecreasing as a function of (x, y, z). We have two cases to
consider:
• x + y ≥ 0
Since x ≤ y ≤ z, (x, y, z) ∈ Γ and so we recover σ2 which we know to be a nondecreasing
function in Γ.
• x + y < 0
Since x ≤ y ≤ z, x < 0. We then get σ2(x, y, z) = −x2, which is increasing since x < 0.
Hence σ2 is nondecreasing.
Now, we derive an alternative formula for the 2-Hessian operator. This, together with
the extension of σ2 derived in Lemma 5.3.2 will allow us to derive an expression for the
2-Hessian operator that can be discretized in a monotone way and that encodes the 2-
admissibility constraint. The idea is to use a matrix identity, following the ideas in [FO11b]
for the Monge-Ampère equation.
Lemma 5.3.3. Let M be a 3 × 3 symmetric matrix and V be the set of all orthonormal bases of R3:
V =
(ν1, ν2, ν3) | νi ∈ R3, νi ⊥ νj if i 6= j, ‖νi‖2 = 1
.
Then
c(M) = min(ν1,ν2,ν3)∈V
σ2
(
νT1 Mν1, νT
2 Mν2, νT3 Mν3
)
. (5.13)
Proof. First note that trace(M) is invariant over conjugation OT MO by orthogonal matrices
O. Second note that trace(M2) =∑
ij m2ij ≥ ∑
i m2ii with equality when M is diagonal.
Hence, for any orthogonal matrix O, we have
trace(M)2 − trace(M2) = trace(OT MO)2 − trace(OT M2O)
= trace(OT MO)2 − trace(
(OT MO)2)
≤ trace(OT MO)2 −∑
i
(OT MO)2ii.
Therefore
2c(M) = minOT O=I,
R=OT MO
(
∑
i
rii
)2
−∑
i
r2ii
,
which can be rewritten as
c(M) = minOT O=I,
R=OT MO
σ2(diag(R)), (5.14)
80 5.3. DISCRETIZATION AND SOLVERS
where diag(R) = (r11, r22, r33) is the vector formed by the elements in the diagonal of the
matrix R and σ2 is defined by (5.5).
We can now use Lemma 5.3.3 to characterize the 2-Hessian operator of a C2 function
by expressing it in terms of second directional derivatives of u as follows:
S2[u] = min(ν1,ν2,ν3)∈V
σ2
(
∂2u
∂ν21
,∂2u
∂ν22
,∂2u
∂ν23
)
. (5.15)
The equation S2[u] = f is elliptic only on the space of 2-admissible functions (see
Proposition 5.2.2) and so the 2-admissibility of the solution needs to be treated as an
additional constraint. As mentioned before, there are two possible approaches: develop
numerical methods that enforce this constraint or absorb the constraint into the PDE
operator to produce an equivalent equation that is globally elliptic and will automatically
select the 2-admissible function. We follow the second approach by considering the
equation S2[u] = f where
S2[u] = min(ν1,ν2,ν3)∈V
σ2
(
∂2u
∂ν21
,∂2u
∂ν22
,∂2u
∂ν23
)
. (5.16)
Thus we are also interested in the Dirichlet problem
S2[u] = f, in Ω,
u = g, on ∂Ω.(2H)
We show that S2 is globally elliptic.
Proposition 5.3.4. The function F : Ω × R × Rn × Sn → R given by
F (x, r, p, M) = − min(ν1,ν2,ν3)∈V
σ2 (νᵀ
1 Mν1, νᵀ
2 Mν2, νᵀ
3 Mν3) + f(x),
which corresponds to (2H), is degenerate elliptic.
Proof. We first notice that we can rewrite F as
F (x, r, p, M) = max(ν1,ν2,ν3)∈V
−σ2 (νᵀ
1 Mν1, νᵀ
2 Mν2, νᵀ
3 Mν3) + f(x)
Let M, N ∈ Sn be such that M N and (ν1, ν2, ν3) ∈ V . Then, νᵀ
i Nνi ≤ νᵀ
i Mνi for i = 1, 2, 3
by definition of M N . Hence, since σ2 is nondecreasing in R3 by Lemma 5.3.2,
σ2 (νᵀ
1 Nν1, νᵀ
2 Nν2, νᵀ
3 Nν3) ≤ σ2 (νᵀ
1 Mν1, νᵀ
2 Mν2, νᵀ
3 Mν3)
Chapter 5. 2-Hessian equation 81
and therefore F (x, r, p, M) ≤ F (x, r, p, N) as desired.
Unlike (2H), (2H) is globally degenerate elliptic and so no additional care is needed in
the definition of viscosity solution. We simply use the definition given in chapter 2.
Proposition 5.3.5. Suppose f > 0. Then u is a viscosity solution of (2H) if and only if u is a
viscosity solution of (2H).
Proof. The proof follows from Lemmas 5.3.6 and 5.3.7 below.
Lemma 5.3.6. Suppose f > 0. Then u is a viscosity subsolution of (2H) if and only if u is a
viscosity subsolution of (2H).
Proof. Suppose u is a viscosity subsolution of (2H). Let φ ∈ C2(Ω) be a test function and
x0 ∈ Ω be a local maximum of u − φ.
We consider first the case where x0 ∈ Ω. Then, if φ is 2-admissible, i.e., D2φ(x0) ∈ S3(Γ2),
then
S2[φ](x0) = S2[φ](x0) ≥ f(x0)
since u is a viscosity subsolution of (2H).
If φ is not a 2-admissible function, i.e., D2φ(x0) /∈ S3(Γ2), there exists a unique α > 0
such that D2φ(x0) + 2αI ∈ ∂S3(Γ2). Let φ(x) = φ(x) + α |x − x0|2. Then u − φ has a local
maximum at x0 and so
S2[φ](x0) ≥ f(x0) > 0.
Since S2[φ](x0) = 0 due to the choice of α, the above inequality gives a contradiction.
Suppose now that x0 ∈ ∂Ω. If D2φ(x0) ∈ S3(Γ2), then S2[φ](x0) = S2[φ](x0) and so
min(
−S2[φ](x0) + f(x0), u(x0) − g(x0))
= min (−S2[φ](x0) + f(x0), u(x0) − g(x0)) ≤ 0.
since u is a viscosity subsolution of (2H). Otherwise, if D2φ(x0) /∈ S3(Γ2), then, as before,
there exists a unique α > 0 such that D2φ(x0)+2αI ∈ ∂S3(Γ2). Let φ(x) = φ(x)+α |x − x0|2.
Then u − φ has a local maximum at x0 and so
min(
−S2[φ](x0) + f(x0), u(x0) − g(x0))
≤ 0
If min(
−S2[φ](x0) + f(x0), u(x0) − g(x0))
= u(x0) − g(x0), then u(x0) − g(x0) ≤ 0 and so
min(
−S2[φ](x0) + f(x0), u(x0) − g(x0))
≤ u(x0) − g(x0) ≤ 0
Otherwise, min(
−S2[φ](x0) + f(x0), u(x0) − g(x0))
= −S2[φ](x0) + f(x0), in which case
S2[φ](x0) ≥ f(x0) > 0. Then, as before, we have derived a contradiction since S2[φ](x0) = 0.
82 5.3. DISCRETIZATION AND SOLVERS
Suppose now that u is a viscosity subsolution of (2H). Let φ ∈ C2(Ω) be a 2-admissible
test function and x0 ∈ Ω be a local maximum of u − φ. Then S2[φ] = S2[φ] and so
−S2[φ](x0) + f(x0) = −S2[φ](x0) + f(x0) ≤ 0
if x ∈ Ω and
min (−S2[φ](x0) + f(x0), u(x0) − g(x0)) = min(
−S2[φ](x0) + f(x0), u(x0) − g(x0))
≤ 0.
if x ∈ ∂Ω, since φ is 2-admissible and u is a viscosity subsolution of (2H).
Lemma 5.3.7. Suppose f > 0. Then u is a viscosity supersolution of (2H) if and only if u is a
viscosity supersolution of (2H).
Proof. Suppose u is a viscosity supersolution of (2H). Let φ ∈ C2(Ω) be a test function and
x0 ∈ Ω be a local minimum of u − φ. Then, if D2φ(x0) ∈ S3(Γ2), then S2[φ](x0) = S2[φ](x0)
and so, since u is a viscosity supersolution of (2H),
−S2[φ](x0) + f(x0) = −S2[φ](x0) + f(x0) ≥ 0
if x ∈ Ω, and
max(
−S2[φ](x0) + f(x0), u(x0) − g(x0))
= max (−S2[φ](x0) + f(x0), u(x0) − g(x0)) ≥ 0.
if x ∈ ∂Ω.
Otherwise, D2φ(x0) /∈ S3(Γ2) then
S2[φ](x0) ≤ 0 < f(x0)
by definition of σ2. Thus, we can conclude that u is a viscosity supersolution of (2H).
Suppose now that u is a viscosity supersolution of (2H). Let φ ∈ C2(Ω) be a 2-admissible
test function and x0 ∈ Ω be a local minimum of u − φ. Then S2[φ](x0) = S2[φ](x0) and so,
since φ is 2-admissible and u is a viscosity supersolution of (2H),
−S2[φ](x0) + f(x0) = −S2[φ](x0) + f(x0) ≥ 0.
if x ∈ Ω and
max (−S2[φ](x0) + f(x0), u(x0) − g(x0)) = max(
−S2[φ](x0) + f(x0), u(x0) − g(x0))
≥ 0.
if x ∈ ∂Ω as desired.
Chapter 5. 2-Hessian equation 83
We can finally present the monotone discretization of the 2-Hessian operator. Here we
are interested in solving problems on the three-dimensional box Ω = (a, b)3 and so the
computation domain is D = [a, b]3 and we take Gh to be a N × N × N uniform grid with
h = b−aN
, i.e.,
Gh =
(a + ih, a + jh, a + kh) | i, j, k ∈ 0, 1, . . . , N, h =b − a
N
.
In addition, we set GhV = Ω ∩ Gh and ∂Gh = Gh \ Gh
V . Define as well the distance of a grid
point x ∈ GhV to the boundary of the grid as
d(x, ∂Gh) = miny∈∂Gh
‖x − y‖∞ .
Notice that since we have a uniform grid d(x, ∂Gh) will be a multiple of h.
As for the second order derivatives, we approximate them using centered finite differ-
ences which leads to a spatial discretization with parameter h:
∂2u
∂ν2(xi) = Dννu(xi) + O(|ν|2 h2),
where Dνν is the finite difference operator for the second directional derivative in the
direction ν which lies on the finite difference grid and are given by
Dννu(xi) =1
|ν|2h2(u(xi + hν) + u(xi − hν) − 2u(xi)).
Since we lie on a grid, we consider only a finite number possible directions ν, which
introduces the directional discretization with parameter dθ. We denote by Gnθthe set of
orthogonal bases available on the grid for a stencil with width nθ. Setting
V1 =
ν ∈ Z3 | |νi| ≤ 1, ‖ν‖ 6= 0
and, for nθ ≥ 2,
Vnθ=
ν ∈ Z3 | |νi| ≤ nθ, ∀|t|<1 tν /∈ Vnθ−1
,
we let
Gnθ=
(ν1, ν2, ν3) ∈ V3nθ
| νi ⊥ νj if i 6= j
.
Finally, we can define the monotone scheme, with a stencil of width nθ, as
SM2 [u](x) = min
(ν1,ν2,ν3)∈GW (x)
σ2 (Dν1ν1u(x), Dν2ν2u(x), Dν3ν3u(x)) , (2H)M
84 5.3. DISCRETIZATION AND SOLVERS
where W (x) denotes the width of the stencil at x ∈ GhV and is given by
W (x) = min
nθ,d(x, ∂Gh)
h
.
We define dθ, the angular resolution, as
dθ(x) = max(w1,w2,w3)∈V
min(ν1,ν2,ν3)∈GW (x)
max
arccos
(
wT1 ν1
‖ν1‖
)
, arccos
(
wT2 ν2
‖ν2‖
)
, arccos
(
wT3 ν3
‖ν3‖
)
.
This means that near the boundary we will have a lower angular resolution since we use
narrower stencils. This can be avoided with the use of additional boundary values at the
expense of a lower order accuracy in space (because the distances to the reference point
are not the same).
We will refer to the monotone schemes with respect to the number of points in the
stencil. For instance, the monotone scheme with the stencil of length 1 (i.e., nθ = 1) has
nS + 1 = 27 points.
Remark 5.4. Given that σ2 is a symmetric function when implementing the monotone
scheme away from the boundary we do not need to look into all the triplets in Gnθ. For
instance, for nθ = 1 we only need to look for the triplets in Table 5.1.
v1 v2 v3
(1, 1, 0) (1, −1, 0) (0, 0, 1)
(1, 0, 1) (1, 0, −1) (0, 1, 0)
(1, 0, 0) (0, 1, 1) (0, 1, −1)
(1, 0, 0) (0, 1, 0) (0, 0, 1)
Table 5.1: Elements of G1 up to permutations.
nθ 1 2 3 4 5 6
nS 26 98 290 579 1155 1731
Table 5.2: nS is the number of ν directions available in the stencil, i.e., nS = #Vnθ
We now give the proof of the convergence of the monotone scheme. In order to do that,
we first need to define our scheme at the boundary, which in this case is immediate since
the grid points are aligned with the boundary. Set
F M [u](x) =
−SM2 [u](x) + f(x), if x ∈ Gh
V ,
u(x) − g(x), if x ∈ ∂Gh.(M)
Chapter 5. 2-Hessian equation 85
Figure 5.2: Elements of V1 (solid) and elements of V2 \ V1 (dashed).
Lemma 5.3.8. The finite difference scheme F M [u], given by (M), is elliptic (and therefore mono-
tone).
Proof. From the definition, the discrete second directional derivatives Dνν are nondecreas-
ing functions of the differences between neighbouring values and reference values, uj − ui,
where uj is one of the neighbouring values of ui in the direction ν. The scheme SM2 [u]
(2H)M is a nondecreasing combination of the operators min and σ2 (the latter proved
in Lemma 5.3.2 to be nondecreasing) applied to Dνν , and so it is also a nondecreasing
function of the differences between neighbouring values and reference values. Thus
−SM2 [u](x) + f(x) is elliptic according to Definition 3.2.1. It is also clear that u − g is elliptic.
Hence, F M [u] is elliptic and its monotonicity follows from from Proposition 3.2.2.
Lemma 5.3.9. Let x0 ∈ Ω be a reference point on the grid and φ be a smooth function that is
defined in a neighborhood of the grid. Then the scheme SM2 [φ] defined in (2H)M approximates (2H)
with accuracy
SM2 [φ](x0) = S2[φ](x0) + O((nθh)2 + dθ(x0)).
Proof. From a simple Taylor series computation we have
Dννφ(x0) =∂2φ
∂ν2(x0) + O((nθh)2).
86 5.3. DISCRETIZATION AND SOLVERS
We will omit the dependence on x0 to simplify the notation.
By definition
S2[φ] = min(ν1,ν2,ν3)∈V
σ2
(
∂2φ
∂ν21
,∂2φ
∂ν22
,∂2φ
∂ν23
)
= σ2
(
∂2φ
∂v21
,∂2φ
∂v22
,∂2φ
∂v23
)
,
where the vj are orthogonal unit vectors, which may not be in the set of grid vectors G. We
know by definition of dθ that
min(ν1,ν2,ν3)∈G
max
arccos
(
vT1 ν1
‖ν1‖
)
, arccos
(
vT2 ν2
‖ν2‖
)
, arccos
(
vT3 ν3
‖ν3‖
)
≤ dθ.
Let then w ∈ G where the above min is attained. Then the angle between between each vj
and wj is less or equal than dθ and so there is dvj such that
vj + dvj =wj
‖wj‖
with ‖dvj‖ = O(dθ).
Now we consider the discretized problem
SM2 [φ] = min
(ν1,ν2,ν3)∈Gσ2 (Dν1ν1φ, Dν2ν2φ, Dν3ν3φ)
≤ σ2 (Dw1w1φ, Dw2w2φ, Dw3w3φ)
= σ2
(
∂2φ
∂w21
,∂2φ
∂w22
,∂2φ
∂w23
)
+ O((nθh)2)
= σ2
(
∂2φ
∂v21
,∂2φ
∂v22
,∂2φ
∂v23
)
+ O((nθh)2 + dθ)
= min(ν1,ν2,ν3)∈V
σ2
(
∂2φ
∂ν21
,∂2φ
∂ν22
,∂2φ
∂ν23
)
+ O((nθh)2 + dθ)
= S2[φ] + O((nθh)2 + dθ),
where we used the fact that∂2φ
∂w2j
=∂2φ
∂v2j
+ O(dθ).
In addition, since the set of grid vectors G is a subset of the set of all orthogonal vectors
V up to scaling, we find that
SM2 [φ] = min
(ν1,ν2,ν3)∈Gσ2 (Dν1ν1φ, Dν2ν2φ, Dν3ν3φ)
≥ min(ν1,ν2,ν3)∈V
σ2 (Dν1ν1φ, Dν2ν2φ, Dν3ν3φ)
Chapter 5. 2-Hessian equation 87
= min(ν1,ν2,ν3)∈V
σ2
(
∂2φ
∂ν21
,∂2φ
∂ν22
,∂2φ
∂ν23
)
+ O((nθh)2)
= S2[φ] + O((nθh)2),
Combining the two inequalities deduced above, we conclude the proof.
Lemma 5.3.10. Assume g ∈ C(∂Ω) and f ∈ C(Ω). Then F M [u], given by (M), is consistent
with (2H) as h → 0, nθ → ∞ and nθh → 0.
Proof. As nθ → ∞, dθ → 0. Hence the consistency of F M [u] in Ω follows from the previous
Lemma.
Consider x ∈ ∂Ω, smooth φ and sequences hn → 0, yn ∈ Gh ∩ ∂Ω, zn ∈ Gh ∩ Ω such that
yn, zn → x. For sequences that approach x along the boundary we have
limn→∞,ξ→0
F M(yn, φ(yn) + ξ, φ(yn) − φ(·)) = limn→∞,ξ→0
(φ(yn) + ξ − g(yn)) = φ(x) − g(x).
For sequences that approach x from the interior, we can use the consistency and continuity
of the interior approximation to calculate
limn→∞,ξ→0
F M(zn, φ(zn) + ξ, φ(zn) − φ(·)) = −S2[φ](x) + f(x).
Combining these results yields
lim suph→0,y∈Gh→x,ξ→0
F M(y, φ(y) + ξ, φ(y) − φ(·)) = maxφ(x) − g(x), −S2[φ](x) + f(x)
= G∗(x, φ(x), ∇φ(x), D2φ(x))
and similarly for the limit inferior condition.
Lemma 5.3.11. Assume that Ω is a bounded domain. Suppose that g ∈ C(∂Ω) and f ∈ C(Ω)
with minx∈Ω f(x) = σ > 0. Then F M [u], given by (M), is stable.
Proof. F M is consistent by Lemma 5.3.10. F M is elliptic by Lemma 5.3.8. F M is Lipschitz
by construction as it involves only addition, multiplication, and computing the minimum
of operators. Then by Lemma 3.2.8, we only need to show that there exists a strict classical
subsolution and supersolution of (2H), to conclude that F M is stable.
We propose the function w(x) = A ‖x‖22 + B with
0 < A ≤ σ
24and B ≥ max
x∈∂Ω
−A ‖x‖22 + g(x)
+σ
2
as strict classical supersolution.
88 5.3. DISCRETIZATION AND SOLVERS
Clearly w is 2-admissible since A > 0. At interior points, we substitute w into the PDE
to obtain
−S2[w](x) + f(x) = −12A + f(x) ≥ −σ
2+ σ =
σ
2
due to the choice of A. At boundary points we have
max
F (x, w(x), ∇w(x), D2w(x)), w(x) − g(x)
≥ w(x) − g(x)
= A ‖x‖22 + B − g(x)
≥ maxy∈∂Ω
−A ‖y‖22 + g(y)
+σ
2−(
−A ‖x‖22 + g(x)
)
≥ σ
2
which establishes w as a strict supersolution.
Now, we propose the function w(x) = A ‖x‖22 + B as a strict classical subsolution with
A ≥ Mf
6and B ≤ min
x∈∂Ω
−A ‖x‖22 + g(x)
− Mf ,
where Mf = maxx∈Ω f(x).
Clearly w is 2-admissible since A > 0. At interior points, we substitute w into the PDE
to obtain
−S2[w](x) + f(x) = −12A + f(x) ≤ −2Mf + Mf = −Mf
due to the choice of A. At boundary points we have
min
F (x, w(x), ∇w(x), D2w(x)), w(x) − g(x)
≤ w(x) − g(x)
= A ‖x‖22 + B − g(x)
≤ miny∈∂Ω
−A ‖y‖22 + g(y)
− Mf −(
−A ‖x‖22 + g(x)
)
≤ − Mf
which establishes w as a strict subsolution.
Theorem 5.3.12. Consider the Dirichlet problem (2H) on a bounded domain Ω where g ∈ C(∂Ω)
and f ∈ C(Ω) with minx∈Ω f(x) = σ > 0. Assume that it has a strong comparison principle (see
Definition 2.3.7). Let F M be the monotone scheme (M) and Uh,nθ ∈ C(Gh) be any of its solutions.
Then
limh→0,nθ→∞,hnθ→0
uh,nθ(x) = u(x), for all x ∈ Ω
Chapter 5. 2-Hessian equation 89
where uh,nθ is the piecewise constant extension of Uh,nθ (3.4) and u is the viscosity solution of
(2H).
Proof. F M is consistent by Lemma 5.3.10. F M is monotone by Lemma 5.3.8. F M is stable by
Lemma 5.3.11. The result then follows from Theorem 3.1.5 and the equivalence of viscosity
solutions of (2H) and (2H) proved in Proposition 5.3.5.
Remark 5.5. The assumption that a strong comparison principle holds for (2H) was dis-
cussed in section 2.3.
5.3.5 Solvers for the monotone finite difference scheme
In this section we present two solvers for the monotone finite difference scheme.
5.3.5.1 Parabolic solver
Using the monotone discretization F M [u], the simplest solver for the 2-Hessian equation is
to use the fixed point method
un+1 = un − dtF M [un] (5.17)
which corresponds to the discrete version of the parabolic equation ut = S2[u] − f using a
forward Euler step. The fixed point iteration will be a contraction in the maximum norm
provided that we choose dt small enough, as dictated by the nonlinear CFL condition
[Obe06a], which in this case means dt = O(h4) (see section 3.2 for more details). This will
make the solver very slow. However, since F M is globally degenerate elliptic, this is a
global solver, meaning that it will converge regardless of the initial guess we choose.
5.3.5.2 Newton solver
As with the standard finite difference scheme, one can also use a (damped) Newton solver.
In this case the Jacobian for the monotone discretization is obtained using Danskin’s
Theorem [Ber03] and the product rule:
∇uSM2 [u] =
−2(Dν∗
1 ν∗
1u)Dν∗
1 ν∗
1, if Dν∗
1 ν∗
1u + Dν∗
2 ν∗
2u < 0,
∑
ν1,ν2∈ν∗
1 ,ν∗
2 ,ν∗
3 ,ν1 6=ν2
(Dν1ν1u)Dν2ν2 , otherwise,
where ν∗j are the directions active in the minimum in (2H)M , with Dν∗
1 ν∗
1u ≤ Dν∗
2 ν∗
2u ≤
Dν∗
3 ν∗
3u. Unlike the previous solver, this is a local solver, meaning that we need a good
initial guess in order to have convergence.
90 5.4. COMPUTATIONAL RESULTS
5.4 COMP UTATIONAL R ES ULTS
In this section we summarize the results of a number of different examples using the
solvers described in the previous section. The computations are performed on a N ×N ×N
grid on the cube [0, 1]3. Unless otherwise mentioned, all solvers were initialized with an
initial guess provided by the explicit method (5.11), which we iterate until∣
∣
∣SA2 [un] − f
∣
∣
∣ <
10−1. The initial guess for the explicit method (5.11) was the exact solution with some
noise from a uniform distribution. As stopping criteria for the Newton solver we used∣
∣
∣SH2 [un] − f
∣
∣
∣ < 10−10 where H ∈ A, M. Solutions were also computed using (5.11) and
(5.12) with very similar results to the ones provided by the Newton solver being obtained.
For that reason, we chose not to display them here.
Remark 5.6. Notice that at points near the boundary of the domain, the scheme uses a
narrower stencil. Since we are interested in comparing the different wide stencils and to
avoid the lower angular resolution at the boundary in the computational results, we set
the exact solution at those points. However it is important to point out the lower angular
resolution near the boundary can be avoided by considering extra boundary points. By
doing so we can maintain the angular resolution of the wide stencil at the expense of a
lower spatial resolution since the stencil loses its symmetry.
Example 5.1 (Quadratic function). We consider the case where u is a non-convex (but
2-admissible function) given by
u(x) = x21 − 1
2x2
2 + 2x23, f(x) = 2. (5.18)
with x = (x1, x2, x3). In Table 5.3, we compare the results obtained using standard finite
differences and the monotone schemes with different stencil sizes. For this example, we
used the Newton solver for all schemes.
All methods provide machine accuracy which is expected since the standard finite
differences are exact for quadratic functions and the monotone schemes computed the
desired directional derivative.
Example 5.2 (smooth convex radial function). We consider now the case where u is given
by
u(x) = exp
(
‖x − x0‖2
2
)
, f(x) = (3 + 2‖x − x0‖2) exp(‖x − x0‖2). (5.19)
The maximum errors are given in Table 5.4. As in the previous example we used the
Newton solver for all schemes.
Chapter 5. 2-Hessian equation 91
Errors and order, 1st ExampleN Standard Monotone (27-point) Monotone (99-point) Monotone (291-point)15 4.441 × 10−16 4.441 × 10−16 4.441 × 10−16 4.441 × 10−16
20 4.441 × 10−16 8.882 × 10−16 8.882 × 10−16 6.661 × 10−16
25 4.441 × 10−16 8.882 × 10−16 8.882 × 10−16 8.882 × 10−16
30 4.441 × 10−16 1.332 × 10−15 8.882 × 10−16 8.882 × 10−16
35 4.441 × 10−16 1.332 × 10−15 8.882 × 10−16 1.110 × 10−15
Table 5.3: Accuracy in the l∞ norm of the schemes for the first example using the Newton solver.
The standard finite differences provided second order convergence, which was expected
since the solution is smooth. The monotone schemes provided only first order convergence
(or close to it).
Errors and order, 2nd ExampleN Standard Monotone (27-point) Monotone (99-point) Monotone (291-point)15 2.393 × 10−4 - 3.472 × 10−4 - 2.167 × 10−4 - 1.302 × 10−4 -20 1.298 × 10−4 2.00 2.225 × 10−4 1.46 1.518 × 10−4 1.17 1.034 × 10−4 0.7525 8.197 × 10−5 1.97 1.650 × 10−4 1.28 1.165 × 10−4 1.13 8.552 × 10−5 0.8130 5.607 × 10−5 2.01 1.346 × 10−4 1.08 9.357 × 10−5 1.16 7.216 × 10−5 0.9035 4.091 × 10−5 1.98 1.259 × 10−4 0.42 7.809 × 10−5 1.14 6.247 × 10−5 0.91
Table 5.4: Accuracy in the l∞ norm and order of convergence of the schemes for the second example usingthe Newton solver.
Example 5.3 (smooth non-convex radial function). We consider now the case where u is
given by
u(x) = exp(
2x21 − x2
2 + 4x23
)
, f(x) = 8(
1 + 12x21 + 6x2
2 + 16x23
)
exp(
4x21 − 2x2
2 + 8x23
)
.
(5.20)
The maximum errors are given in Table 5.5. Once again the solutions were computed
with a Newton solver for all schemes.
The standard finite differences demonstrates again second order convergence. For
the monotone schemes, the error tappers off with the grid size and we only see an error
reduction by considering wider stencils. This tells us that the directional resolution
error dominates the spatial resolution error. It is important to point out that this does
not contradict our theoretical results since the only thing we proved was that we have
convergence as both h and dθ go to 0, which we observe here.
92 5.4. COMPUTATIONAL RESULTS
Errors and order, 3rd ExampleN Standard Monotone (27-point) Monotone (99-point) Monotone (291-point)15 3.028 × 10−4 - 3.287 × 10−2 - 1.110 × 10−2 - 5.044 × 10−3 -20 1.669 × 10−4 1.95 3.312 × 10−2 -0.02 1.211 × 10−2 -0.29 5.617 × 10−3 -0.3525 1.052 × 10−4 1.98 3.305 × 10−2 0.01 1.260 × 10−2 -0.17 5.920 × 10−3 -0.2230 7.218 × 10−5 1.99 3.311 × 10−2 -0.01 1.306 × 10−2 -0.19 6.396 × 10−3 -0.4135 5.262 × 10−5 1.99 3.302 × 10−2 0.02 1.339 × 10−2 -0.16 6.703 × 10−3 -0.29
Table 5.5: Accuracy in the l∞ norm and order of convergence of the schemes for the third example using theNewton solver.
Example 5.4 (smooth non-convex radial function). We consider another example of smooth
radial function which is non convex but 2-admissible:
u(x) = log(2 + ‖x‖2), f(x) = −4(−6 + ‖x‖2)
(2 + ‖x‖2)3. (5.21)
The maximum errors are given in Table 5.6. Once again the solutions were computed
with a Newton solver, regardless of the scheme.
As in the previous example, standard finite differences provide second order conver-
gence and only with wider stencils we see a decrease in error with the grid size. Moreover,
the monotone schemes with wider stencils also exhibit second order convergence (before
it tappers off in the case of the 99-point stencil).
Errors and order, 4th ExampleN Standard Monotone (27-point) Monotone (99-point) Monotone (291-point)15 4.723 × 10−5 - 1.664 × 10−3 - 3.882 × 10−4 - 4.909 × 10−4 -20 2.564 × 10−5 2.00 1.668 × 10−3 -0.01 1.787 × 10−4 2.54 2.500 × 10−4 2.2125 1.615 × 10−5 1.98 1.674 × 10−3 -0.01 1.007 × 10−4 2.46 1.462 × 10−4 2.3030 1.111 × 10−5 1.98 1.672 × 10−3 0.01 8.617 × 10−5 0.82 9.063 × 10−5 2.5335 8.052 × 10−6 2.02 1.670 × 10−3 0.01 9.620 × 10−5 -0.69 6.506 × 10−5 2.08
Table 5.6: Accuracy in the l∞ norm and order of convergence of the schemes for the fourth example usingthe Newton solver.
Example 5.5 (non smooth convex function). We consider now the case where u is given by
u(x) =1
2
(
(‖x − x0‖ − 0.2)+)2
, f(x) =
(
3 +1
25‖x − x0‖2− 4
5‖x − x0‖
)
1‖x−x0‖>0.2(x).
(5.22)
The maximum errors are given in Table 5.7. Due to its degenerate ellipticity, the
monotone schemes required the use of the damped Newton solver.
Chapter 5. 2-Hessian equation 93
Despite the lack of smoothness of the solution, the Newton solver with standard finite
differences still converged. As for the monotone scheme, there was a significant increase
in the number of iterations required: the wider the stencil, the more iterations required
(around 10 times more iterations when compared to the Newton solver for the naive finite
differences in the worst cases).
For the 291-stencil, as in Example 5.3, the error tapers off, indicating that the directional
resolution error dominates the spatial error and, again, we still see the convergence as both
h and dθ go to 0.
Errors and order, 5th ExampleN Standard Monotone (27-point) Monotone (99-point) Monotone (291-point)15 7.580 × 10−4 - 2.261 × 10−3 - 7.707 × 10−4 - 5.086 × 10−4 -20 6.506 × 10−4 0.50 2.329 × 10−3 -0.10 7.235 × 10−4 0.21 1.924 × 10−4 3.1825 3.353 × 10−4 2.84 2.057 × 10−3 0.53 5.871 × 10−4 0.89 1.758 × 10−4 0.3930 3.032 × 10−4 0.53 2.156 × 10−3 -0.25 5.431 × 10−4 0.41 2.197 × 10−4 -1.1835 2.129 × 10−4 2.22 2.018 × 10−3 0.42 5.159 × 10−4 0.32 2.351 × 10−4 -0.43
Table 5.7: Accuracy in the l∞ norm and order of convergence of the schemes for the fifth example using theNewton solver.
Example 5.6 (example with blow-up). We considered as well the case
u(x) = −√
3 − ‖x‖2, f(x) = − −9 + ‖x‖2
(−3 + ‖x‖2)2. (5.23)
The maximum errors are given in Table 5.8. All solutions were computed with a Newton
solver.
Notice that f is unbounded at the boundary point (1, 1, 1) and u will be singular
at that point. Despite that the Newton solver still converged, but with a smaller rate
of convergence (approximately 0.3). It is important to observe that in the case of the
Monge-Ampère, the Newton solver failed to converge in the analogue example. This
may be because the Monge-Ampère equation is more strongly nonlinear than the 2-
Hessian equation. The better accuracy of the wider monotone schemes is explained by
the fact that the exact solution is prescribed at more grid points near the boundary of the
(computational) domain, in particular, where u is singular and f is unbounded.
Example 5.7. We consider as well the example with f ≡ 1 and zero Dirichlet boundary
conditions (g ≡ 0 ). No exact solution is known. In Figure 5.3, we illustrate some of the
surface plots of the level sets u = c of the solution with the standard finite differences
94 5.4. COMPUTATIONAL RESULTS
Errors and order, 6th ExampleN Standard Monotone (27-point) Monotone (99-point) Monotone (291-point)15 1.104 × 10−3 - 5.627 × 10−3 - 5.600 × 10−4 - 3.026 × 10−4 -20 1.096 × 10−3 0.02 5.224 × 10−3 0.24 4.229 × 10−4 0.92 2.628 × 10−4 0.4625 1.054 × 10−3 0.17 4.891 × 10−3 0.28 3.454 × 10−4 0.87 2.344 × 10−4 0.4930 1.007 × 10−3 0.24 4.698 × 10−3 0.21 2.921 × 10−4 0.88 2.102 × 10−4 0.5835 9.621 × 10−4 0.29 4.612 × 10−3 0.12 2.538 × 10−4 0.89 1.906 × 10−4 0.62
Table 5.8: Accuracy in the l∞ norm and order of convergence of the schemes for the sixth example using theNewton solver.
and monotone scheme where c ∈ −0.01, −0.03, −0.07. Note that the zero level set
(c = 0) is the boundary of the cube [0, 1]3 where the zero Dirichlet boundary conditions
are prescribed. The surface plots become spheres as c decreases, with c = −.01 being the
only where there is a tangible difference between the two schemes, most likely due to the
expected higher accuracy from the standard finite differences. In Figure 5.4, we plot the
curve u(t, t, t) with t ∈ [0, 1] and see that there’s a small difference between the solutions
from the standard finite differences and the monotone scheme.
Figure 5.3: Surface plots of the level sets of the solution to Example 5.7 on a 30 × 30 × 30 grid with thenaive finite differences (left) and the 27-point monotone scheme (right).
Example 5.8. We consider as well the example with f ≡ 1 and zero Dirichlet boundary
Chapter 5. 2-Hessian equation 95
0 0.2 0.4 0.6 0.8 1−0.12
−0.1
−0.08
−0.06
−0.04
−0.02
0
Newton (J)
Newton (M 27−point)
Figure 5.4: Plot of the curves t 7→ u(t, t, t) of the solution of Example 5.7 on a 30 × 30 × 30 grid.
conditions (g ≡ 0) but with a different domain Ω = Ω1 ∪ Ω2 where
Ω1 = (x, y, z) ∈ R3 : (x − 0.35)2 + (y − 0.35)2 + (z − 0.5)2 < 0.32,
Ω2 = (x, y, z) ∈ R3 : (x − 0.65)2 + (y − 0.65)2 + (z − 0.5).2 < 0.32.
No exact solution is known. To implement the boundary conditions in this case, since
the domain is not a cube, we set zero Dirichlet boundary conditions on the cube [0, 1]3
and defined f as the indicator function of Ω. In Figure 5.5, we illustrate some of the
surface plots of the level sets u = c of the solution with the standard finite differences and
monotone scheme with c ∈ 0, −0.01, −0.02, −0.03, −0.035, −0.039. In this case the zero
level set is not convex and the level sets u = c become more convex as c decreases. In this
case the difference between the standard finite differences and monotone scheme is even
smaller than in Example 5.7, as we can see in Figure 5.6, where we plot the curve u(t, t, t)
with t ∈ [0, 1].
5.5 CONC LUS IONS
The 2-Hessian equation is a fully nonlinear PDE which is elliptic provided the solutions
are restricted to a convex cone (we refer to them as plane-subharmonic functions). It is
natural to compare this equation with the Monge-Ampère PDE, which is elliptic on the
96 5.5. CONCLUSIONS
Figure 5.5: Surface plots of the level sets of the solution to Example 5.8 on a 30 × 30 × 30 grid with thenaive finite differences (left) and the 27-point monotone scheme (right).
0 0.2 0.4 0.6 0.8 1−0.04
−0.035
−0.03
−0.025
−0.02
−0.015
−0.01
−0.005
0
Newton (J)
Newton (M 27−point)
Figure 5.6: Plot of the curves t 7→ u(t, t, t) of the solution of Example 5.8 on a 30 × 30 × 30 grid.
Chapter 5. 2-Hessian equation 97
cone of convex functions and has been extensively studied numerically. In comparison,
the 2-Hessian equation is more challenging because the constraints for ellipticity are less
restrictive.
We gave two different discretizations for the 2-Hessian equation in the three-dimensional
case: a naive one obtained by simply using standard finite differences to discretize the
Hessian and a monotone discretization that takes advantage of a characterization of the
operator using a matrix inequality. Computational results were provided using exact
solutions of varying regularity and shape, from smooth to non differentiable, and from
convex to non-convex.
The naive discretization failed using a standard parabolic solver since there was no
mechanism for selecting the correct plane-subharmonic solution. Two alternative solvers
were presented, which enforced the plane-subharmonic restriction and for which experi-
mental results on a variety of solutions demonstrated that the method appears to converge.
Additionally, a Newton solver was also implemented, converging for all examples con-
sidered, even for degenerate ones or with singular right-hand sides, whenever initialized
with a good initial guess. For smooth examples, we obtained second order convergence.
The monotone discretization is stable and provably convergent but less accurate since it
requires the use of a wide stencil that introduces a directional resolution error. Numerical
examples show that the directional resolution easily dominates the spacial resolution, a
natural consequence of the three dimensional setting.
Moreover, one could have implemented filtered schemes which would provide schemes
that are provably convergent but with greater accuracy than the monotone schemes.
However, we did not implement them here, since our main goal was to compare the two
different discretizations presented and, moreover, the accurate scheme by itself proved to
be convergent for all the examples considered, even degenerate ones.
In this work, we chose the box domain since it is easier to deal with computationally as
the boundary conditions are easily implemented. Dealing with more complex boundaries
requires additional work. It is challenging to obtain higher order at the boundary while
maintaining second order directional derivatives. A natural approach would be a com-
bination of filtered schemes at the boundary and multi-scale grids [OZ15]. Unstructured
grids are another possibility, having been used successfully to solve several fully nonlinear
elliptic equations in two dimensions [Fro15].
CHAPTER 6
AFFINE CURVATURE
6.1 INTR ODUC TI ON
The planar motion of level sets by affine curvature is governed by the nonlinear partial
differential equation (PDE)
ut = Aff [u] := |∇u| (k[u])1/3 . (AC)
Here u : R2 → R, ∇u = (ux, uy) denotes the gradient of u, and k[u] denotes the curvature of
the level set of u
k[u] = div
(
∇u
|∇u|
)
=uxxu2
y − 2uxuyuxy + uyyu2x
(u2x + u2
y)3/2. (6.1)
The affine curvature PDE is closely related to the well known PDE for motion of level sets
by mean curvature
ut = ∆1u := |∇u| k[u] (MC)
studied in the seminal article [OS88]. However, the PDE (AC) exhibits instabilities not
found in the mean curvature PDE (MC), as demonstrated below. In order to resolve these
instabilities, we propose a Lipschitz regularization of the operator.
The regularized operator is also a geometric PDE, and viscosity solutions converge to
solutions of the affine curvature PDE in the limit as the regularization parameter goes to
zero [Gig06, Theorem 4.6.1]. The advantage of the regularized operator is that it allows us
to build stable, convergent explicit solvers which are not otherwise available. Moreover,
the resulting discretization can be combined into a filtered scheme that achieves the
higher accuracy of the otherwise unstable standard finite difference scheme. In addition,
the numerical solutions exhibit the affine invariance and morphology properties of the
evolution.
Our approach to the convergent discretization
In this paragraph we present an overview of our approach to build an elliptic discretization
for clarity. The details and supporting theory can be found in the sections that follow.
Since elliptic discretizations are available for ∆1u, rather than for k[u], we rewrite Aff [u] in
99
100 6.1. INTRODUCTION
terms of |∇u| and ∆1u and get
Aff [u] = A(|∇u| , ∆1u) = (|∇u|2 ∆1u)1/3, where A(p, q) =(
p2q)1/3
. (6.2)
The goal is to make use of available elliptic discretizations of ± |∇u| and ∆1u to build an
elliptic discretization of the full operator Aff [u]. However, simply inserting these operators
into the function A(p, q) is not sufficient: elliptic schemes are built by composing nonde-
creasing maps with elliptic operators, and A(p, q) fails to be nondecreasing. Furthermore,
explicit time discretizations require Lipschitz continuous operators, and A(p, q) also fails
to be Lipschitz continuous.
Since the properties of the nonlinear and singular function A(p, q) are so important, we
first study a model equation in one-dimension. Define
Aff 1D[u] := A(ux, uxx) =(
|ux|2 uxx
)1/3(6.3)
so that the Aff 1D[u] operator has the same homogeneity in first and second derivatives
as the Aff [u] operator. Like the higher dimensional PDE, the model equation exhibits the
instability of standard finite differences.
To build an elliptic scheme for the one dimensional equation, what is needed is a
nondecreasing representation of the function A(p, q), which is consistent with Aff 1D[u].
Furthermore, we need a Lipschitz continuous approximation of A(p, q), with Lipchitz
constant Kh, to build a monotone discretization of the time dependent PDE, using a time
step dt ≤ 1/Kh. Once this modified function is available, we proceed by inserting the
discretization of the two dimensional operators into the modified function, which results
in a convergent scheme.
Novelty of the work.
The scheme presented here uses the level set representation (AC). Thus it moves every
level set by the evolution, unlike thresholding methods, but requires a much smaller time
step. It is monotone, a desirable property for the convergence but also for its applications.
Moreover, it is shown numerically that it captures the affine invariance and morpholog-
ical properties. Moreover, a standard finite difference method is also considered, but
computational examples demonstrate its lack of stability.
Chapter 6. Affine curvature 101
6.2 THE AFFINE C URVATURE PDE
6.2.1 Definition of viscosity solutions
Although the theory of viscosity solutions discussed in chapter 2 is easily extended to
parabolic equations, due to their specific structure (linear dependence on ut), we present
here the specific viscosity solution definition for the affine curvature evolution. We follow
the book [Gig06]. See also the book [Cao03] for viscosity solutions for geometric evolution
equations, and numerical methods. The article [ES99] focuses on the mean curvature
equation.
First, we show the degenerate ellipticity of the affine curvature operator.
Proposition 6.2.1. The operator −Aff [u] is degenerate elliptic.
Proof. We start with the observation that using (6.1), we can write
∆1u = utt, t =(−uy, ux)
(u2x + u2
y)1/2,
where t is the (Euclidean) unit tangent. Then it is clear that −∆1 is degenerate elliptic.
Using the representation (6.2) the degenerate ellipticity of −Aff follows.
We now give the definition of viscosity solutions. Let Ω ⊆ R2 be a domain domain and
T > 0. We are interested in the Cauchy problem
ut = Aff [u] in (x, t) ∈ Ω × (0, T ),
B(x, p) := ν(x)ᵀp = 0 on (x, t) ∈ ∂Ω × (0, T ),
u(x, 0) = u0(x) in x ∈ Ω.
(6.4)
where ν is the outward unit normal of ∂Ω.
Remark 6.1. We will also be interested in the static equation Aff [u] = f . For the definition
of viscosity solution and comparison principle see the discussion in chapter 2.
Definition 6.2.2. A function u ∈ USC(Ω × [0, T ]) is called a viscosity subsolution of (6.4) if
u(x, 0) ≤ u0(x) for x ∈ Ω and for any ϕ ∈ C2(Ω × [0, T ]) such that u − ϕ has a local maximum
at (x, t) ∈ Ω × (0, T ) then
ϕt(x, t) − Aff [ϕ](x, t) ≤ 0 if x ∈ Ω,
minϕt(x, t) − Aff [ϕ](x, t), B(x, ∇φ(x, t)) ≤ 0 if ∈ ∂Ω.
102 6.2. THE AFFINE CURVATURE PDE
Similarly, u ∈ LSC(Ω × [0, T ]) is called a viscosity supersolution of (6.4) if u(x, 0) ≥ u0(x) for
x ∈ Ω and for any ϕ ∈ C2(Ω × [0, T ]) such that u − ϕ has a local minimum at (x, t) ∈ Ω × (0, T )
then
ϕt(x, t) − Aff [ϕ](x, t) ≥ 0 if x ∈ Ω,
maxϕt(x, t) − Aff [ϕ](x, t), B(x, ∇φ(x, t)) ≥ 0 if ∈ ∂Ω.
Finally, we call u a viscosity solution of (6.4) if u is both a viscosity subsolution and viscosity
supersolution.
We have the following uniqueness result.
Theorem 6.2.3 (Comparison Principle). Suppose that Ω is convex with C2 boundary ∂Ω. Let u
and v be a viscosity subsolution and supersolution of (6.4), respectively. Then u ≤ v in Ω × (0, T ).
Proof. See Theorem 3.7.1 in [Gig06].
6.2.2 Invariance properties of the PDE
We now study some properties of (AC), starting with the following Lemma.
Lemma 6.2.4. Let u ∈ C2(R2). Then the operator Aff has the following properties:
i) Rescaling: for h > 0, define v(x, y) := u(x/h, y/h)
Aff [v] = h−4/3Aff [u];
ii) Morphology: for g ∈ C1(R),
Aff [g u] = g′(u)Aff [u];
iii) Affine Invariance: for any affine map φ(x) = Ax + b,
Aff [u φ] = (det A)2/3Aff [u] φ.
Remark 6.2. For ∆1u, rescaling gives ∆1v = h−2∆1u and invariance only holds for orthogo-
nal transformations.
Proof. i) Writing, for |∇u| 6= 0, k[u] = 1|∇u|
(
tr(D2u) − ∇u ᵀD2u ∇u|∇u|2
)
, allows us to write
(Aff [u])3 = |∇u|2 tr(D2u)−∇uᵀ D2u∇u. Then for v(x, y) := u(x/h, y/h) we have ∇v = 1h∇u
and D2v = 1h2 D2u, and property i) follows.
Property ii) follows from the fact that the PDE has the structure of a level set PDE,
[Gig06], so the level sets are invariant under relabelling.
Chapter 6. Affine curvature 103
Finally, we prove iii). Setting v(x, y) = u(φ(x, y)), we have
∇v = A ∇u and D2v = AᵀD2u A. (6.5)
Moreover,
Aff [u] =(
|∇u|2 tr(D2u) − ∇uᵀ D2u ∇u)1/3
=(
uxxu2y − 2uxuyuxy + uyyu2
x
)1/3
Now, using this formula, we compute Aff [v], after which the derivatives of v are replaced
by derivatives of u using (6.5). The proof then follows from an elementary but lengthy
computation.
Theorem 6.2.5. Consider Φt : u0 7→ u(·, t) the solution map of (AC). Then Φt satisfies the
following properties:
i) Monotonicity: u ≤ v ⇒ Φt(u) ≤ Φt(v);
ii) Morphology/Relabelling: for any monotone scalar function g,
Φt(g u) = g Φt(u);
iii) Affine Invariance: for any affine map φ(x) = Ax + b,
Φt(u φ) =(
Φt(det A)2/3(u))
φ.
Remark 6.3. Note that our rescaling factor property iii) differs from the one in [Moi98],
however both formulas agree (and give unity) for special affine transformations which
have determinant 1.
Proof. We establish the three properties one by one.
i) Monotonicity: It follows easily from the fact that (AC) is an elliptic PDE and thus
satisfies a comparison principle.
ii) Morphology: Let g be a monotone scalar function and u be the solution of (AC) with
initial condition u(·, 0) = u0. We want to show that v := g u is the solution of (AC)
with initial condition v(·, 0) = g u0. Formally, it is enough to observe that
vt = g′(u)ut, Aff [v] = g′(u)Aff [u],
where the second equality follows by Lemma 6.2.4.
104 6.2. THE AFFINE CURVATURE PDE
iii) Affine Invariance: Let φ be an affine transformation with φ(x) = Ax + b. Let u be the
solution of (AC) with initial condition u(·, 0) = u0. We want to show that
v(x, y, t) := u(φ(x, y), t(det A)2/3)
is the solution of (AC) with initial condition v(·, 0) = u0 φ. We have
vt = (det A)2/3ut,
which together with Lemma 6.2.4 iii), is enough to conclude the proof.
6.2.3 An exact solution
We now give an example of an exact solution for (AC). See also [Gig06, Section 1.7.4].
Lemma 6.2.6. Given a, b > 0, let u : R2 × [0, ∞) be given by the shifted ellipse
u(x, y, t) = t +3
4(ab)2/3
(
(
x
a
)2
+(
y
b
)2)2/3
= t +3
4
(
b
ax2 +
a
by2
)2/3
.
Then u is a classical solution of (AC). Moreover, we conclude that ellipses remain ellipses of fixed
eccentricity under the motion by affine curvature.
Proof. Define
φ(s) =3(ab)2/3
4s2/3 and S(x, y) =
(
x
a
)2
+(
y
b
)2
.
We can then rewrite u as u(x, y, t) = t + (φ S)(x, y).
The proof then follows by showing that u is a solution of (AC) if and only if
1 = φ′(S) |∇S| div
(
∇S
|∇S|
)1/3
. (6.6)
Indeed, we have that
∇S(x, y) = 2(
x
a2,
y
b2
)
, |∇S| = 2ρ and ρ =
√
(
x
a2
)2
+(
y
b2
)2
.
Moreover,
div
(
∇S
|∇S|
)
=1
a2ρ− (x/a2)
2
a2ρ3+
1
b2ρ− (y/b2)
2
b2ρ3=
(y/b2)2
a2ρ3+
(x/a2)2
b2ρ3=
S(x, y)
a2b2ρ3.
Chapter 6. Affine curvature 105
Then (6.6) is equivalent to
1 = φ′(S)2ρ
(
S
a2b2ρ3
)1/3
.
To conclude the proof it is now enough to observe that φ is the solution of the equation
above together with φ(0) = 0.
6.3 NUMERIC AL METHODS FOR THE MODEL EQUATION
In this section we build convergent discretizations to the one dimensional model of our
equation defined by
Aff 1D[u] := A(ux, uxx) =(
u2x uxx
)1/3= f, x ∈ (−1, 1), (AC-1D)
as well as the parabolic PDE,
ut = Aff 1D[u] − f, for (x, t) ∈ (−1, 1) × (0, ∞),
along with initial and boundary conditions. In order to do so we need to build elliptic
discretizations for Aff 1D, which is achieved by studying the nonlinear function A(p, q). A
naive approach would suggest to simply substitute the elliptic discretizations for |ux| and
uxx into A(p, q). However, this is not possible since A(p, q) 6∈ ND(R2): dA/dp = 2/3(q/p)1/3
and so A fails to be nondecreasing when pq < 0.
Our approach is the following. First, in subsection 6.3.1, we write A(p, q) as a sum
of two nondecreasing functions in terms of |p|, − |p|, q. These terms are replaced by |uhx|,
−|uhx|, uh
xx which are elliptic and consistent and thus a consistent and elliptic discretization
for Aff 1D is built. Then, in subsection 6.3.2, we present a Lipschitz regularization of the
function A(p, q) and proceed similarly. The Lipschitz regularization is needed to build
a provably convergent explicit scheme for the parabolic PDE as discussed in section 3.2.
Finally, in subsection 6.3.3, we present the convergence results.
6.3.1 An elliptic discretization of the one dimensional operator
In the next lemma, we decompose A(p, q) into the sum of two nondecreasing functions.
Lemma 6.3.1. Define A+(p, q) = A(p+, q+) and A−(p, q) = A(p−, q−). Then A+ ∈ ND+(R2),
A− ∈ ND−(R2) and
−A(p, q) = A(p, −q) = A+(|p|, −q) + A−(−|p|, −q), for all p, q
106 6.3. NUMERICAL METHODS FOR THE MODEL EQUATION
Proof. The fact that A can be decomposed follows from checking two cases, depending on
the sign of q.
Next we establish that A+ ∈ ND+(R2) and A− ∈ ND−(R2). First it is clear that A+ ≥ 0
and that A− ≤ 0. Compute the partial derivatives,
A+p =
2
3
(
q+
p+
)1/3
≥ 0, A+q =
1
3
(
p+
q+
)2/3
≥ 0,
So A+ is nondecreasing in each variable, which is enough to show A+ ∈ ND. Similarly,
consider A−(p, q). Again taking partial derivatives, we see that A− is nondecreasing in
each variable, so A− ∈ ND.
Lemma 6.3.2. Define the finite difference operator
−Aff 1D,e[u] = A+(
∣
∣
∣uhx
∣
∣
∣
+, −uh
xx
)
+ A−(
−∣
∣
∣uhx
∣
∣
∣
−, −uh
xx
)
(AC)1D,e
where the finite difference operators involved are defined in section 3.3. Then −Aff 1D,e is elliptic
and consistent with the −Aff 1D.
Proof. The operators |uhx|+, −|uh
x|−, −uhxx are elliptic. Then, due to Lemma 6.3.1, the operator
−Aff 1D,e[u] consists of the composition of the nondecreasing functions A+ and A− with the
three preceding operators, with the first two taking the place of |p| and −|p| and the last one
taking the place of −q. Since the operators are elliptic and the functions are nondecreasing,
−Aff 1D,e[u] is elliptic. Consistency follows from the consistency of each of the schemes
used.
Remark 6.4 (Accuracy). One challenge in forming a discretization of (AC-1D) is that the
accuracy breaks down near uxx = 0. For example, if we use second order accurate
approximations for uxx, the accuracy of finite differences for the operator is only O(h2/3).
Indeed, consider the expression
A(p + hk, q + h2) = ((p + hk)2(q + h2))1/3
where k = 1 or 2. If p 6= 0, q = 0 we get
A(p + hk, q + h2) = h2/3(p + hk)2/3 = O(h2/3).
So regardless of the accuracy of the discretization of the ux term, the overall accuracy of
the scheme cannot be better than O(h2/3). In fact, a similar argument shows that the order
of accuracy is 2k/3 near p = 0, q 6= 0. Our elliptic discretization, which uses k = 1, will
have order of accuracy 2/3.
Chapter 6. Affine curvature 107
6.3.2 Lipschitz regularization of the one dimensional operator
The function A(p, q) fails to be Lipschitz continuous near the axis p = 0, q = 0. Hence
the elliptic scheme presented in the previous section is not Lipschitz, a property that is
required in order to build a monotone convergent scheme for the time dependent problem.
Thus, using the notation for the sign function sgn(q) = q/|q| for q 6= 0 and sgn(0) = 0
otherwise, we regularize A(p, q) as follows.
Definition 6.3.3. Define for K = K(δ) > 0, L = L(δ) > 0, the regularized function
Aδ(p, q) = sgn(q) min(|A(p, q)|, K|p|, L|q|). (6.7)
Naturally, we can then define the regularized PDE operator as
Aff 1D,δ[u] = Aδ(ux, uxx).
Defining an elliptic and consistent scheme for Aff 1D,δ[u] is accomplished by noticing
that the discretization (AC)1D,e generalizes when we replace A with the regularized version
Aδ in each term: just like for A(p, q) in Lemma 6.3.1, we can decompose Aδ(p, q) into the
sum of two nondecreasing functions. This is achieved in the following lemma.
Lemma 6.3.4. Define Aδ,+(p, q) = Aδ(p+, q+) and Aδ,−(p, q) = Aδ(p−, q−). Then
−Aδ(p, q) = Aδ,+(|p|, −q) + Aδ,−(−|p|, −q),
where Aδ,+ ∈ ND+(R2) and Aδ,− ∈ ND−(R2).
Proof. To prove the decomposition of Aδ we have to consider two cases: q ≥ 0 and q < 0.
Suppose first that q ≥ 0. Then Aδ,+(|p|, −q) = 0 since (−q)+ = 0. Therefore
Aδ,+(|p|, −q) + Aδ,−(−|p|, −q) = Aδ(−|p|, −q)
= − sgn(q) min (|A(p, q)|, K|p|, L|q|)= −Aδ(p, q).
Assume now that q < 0. Then Aδ,−(−|p|, −q) = 0 since (−q)− = 0. Hence
Aδ,+(|p|, −q) + Aδ,−(−|p|, −q) = Aδ(|p|, −q)
= sgn(−q) min (|A(p, q)|, K|p|, L|q|)= −Aδ(p, q).
108 6.3. NUMERICAL METHODS FOR THE MODEL EQUATION
To show that Aδ,+ is nondecreasing, we start by rewriting it as
Aδ,+(p, q) = min(
A+(p, q), Kp+, Lq+)
.
The result then follows by noticing that min, A+, Kp+, Lq+ are all nondecreasing. Similarly,
Aδ,− is also nondecreasing, which follows from rewriting it as
Aδ,−(p, q) = max(
A−(p, q), Kp−, Lq−)
.
and noticing that max, A−, Kp−, Lq− are all nondecreasing.
Ellipticity and consistency of the regularized scheme with respect to −Aff 1D,e,δ[u]
follows now easily, just like in Lemma 6.3.2. For this reason we omit the proof.
Lemma 6.3.5. For K = K(δ), L = L(δ) > 0, define the finite difference scheme
− Aff 1D,e,δ[u] = Aδ,+(
∣
∣
∣uhx
∣
∣
∣
+, −uh
xx
)
+ Aδ,−(
−∣
∣
∣uhx
∣
∣
∣
−, −uh
xx
)
. (AC)1D,e,δ
Then −Aff 1D,e,δ[u] is elliptic and consistent with −Aff 1D,δ[u].
Proving consistency of the regularized scheme with respect to −Aff 1D[u] requires extra
work as the parameters K, L need to be chosen carefully. The next theorem summarizes
the results proven in the lemmas that follow.
Theorem 6.3.6. Asssume K = h−1/3 and L = h−4/3. Let x ∈ Ω be a reference point on the grid
and φ be a smooth function that is defined in a neighborhood of the grid. Then the scheme Aff 1D,e,δ
defined by (AC)1D,e,δ is consistent with Aff 1D and has accuracy
Aff 1D,e,δ[φ](x) − Aff 1D[φ](x) = O(h2/3).
Moreover, Aff 1D,e,δ is Lipschitz continuous with constant Ch given by
Ch = h−4/3 + 2h−10/3. (6.8)
We start by showing that our regularization of A is indeed Lipschitz continuous.
Lemma 6.3.7. Suppose K√
L ≥ 1. Then
∣
∣
∣
∣
∣
d
dpAδ(p, q)
∣
∣
∣
∣
∣
≤ K,
∣
∣
∣
∣
∣
d
dqAδ(p, q)
∣
∣
∣
∣
∣
≤ L,
Chapter 6. Affine curvature 109
where the derivatives exist, and Aδ(p, q) is Lipschitz continuous. Moreover,
∣
∣
∣Aδ(p, q) − A(p, q)∣
∣
∣ ≤ max
(
4
27K2|q|, 2
3√
3L|p|)
, for all p, q.
Proof. First we determine the sets where each term in the minimum is active. We claim
that
Aδ(p, q) =
Lq, |q| ≤ L−3/2|p|,sgn(q)K|p|, |q| ≥ K3|p|,(p2q)1/3 otherwise.
Indeed, since K√
L ≥ 1, the claim follows from
|q| ≤ L−3/2|p| ⇒ K|p| ≥ L|q| and |q| ≥ K3|p| ⇒ L|q| ≥ K|p|.
and
L|q| ≤ |A(p, q)| ⇔ |q| ≤ L−3/2|p| and K|p| ≤ |A(p, q)| ⇔ K3|p| ≤ |q|.
We continue the proof by proving the derivate estimates. Computing the derivative
with respect to p gives
d
dpAδ(p, q) =
0, |q| ≤ L−3/2|p|sgn(q)K sgn(p), |q| ≥ K3|p|23(q/p)1/3 otherwise
In the third case, since |q/p| ≤ K3, the partial derivative is bounded by 23K, and so we can
conclude that |dAδ/dp| ≤ K.
Similarly, computing
d
dqAδ(p, q) =
L, q ≤ L−3/2p
0, q ≥ K3p
13(pq−1)2/3 otherwise
In the third case, since |p/q| ≤ L3/2 the value is bounded by 13L, and the global bound
holds.
Finally, we prove the estimate on the error introduced by the regularization. Since both
A and Aδ are even functions with respect to p and odd with respect to q, we can assume
without loss of generality that p and q are both positive. We have two cases to consider:
|q| ≥ K3|p| and |q| ≤ L−3/2|p| which is where A and Aδ differ.
110 6.3. NUMERICAL METHODS FOR THE MODEL EQUATION
If |q| ≥ K3|p| then∣
∣
∣Aδ(p, q) − A(p, q)∣
∣
∣ ≤ 4
27K2|q|.
Indeed, in this case, Aδ(p, q) = Kp and so
d
dp
(
A(p, q) − Aδ(p, q))
= K − 2
3
(
q
p
)1/3
= 0 ⇔ 8
27q = K3p ⇔ p =
8
27K3q
where α = 2/3. Hence
maxp
∣
∣
∣Aδ(p, q) − A(p, q)∣
∣
∣ ≤∣
∣
∣
∣
Aδ(
8
27K3q, q
)
− A(
8
27K3q, q
)∣
∣
∣
∣
=4
27K2|q|.
If |q| ≤ L−3/2|p| then∣
∣
∣Aδ(p, q) − A(p, q)∣
∣
∣ ≤ 2
3√
3L|p|.
Indeed, in this case, Aδ(p, q) = Lq and so
d
dq
(
A(p, q) − Aδ(p, q))
= L − 1
3
(
p
q
)2/3
= 0 ⇔ q = (3L)−3/2p.
Hence
maxq
∣
∣
∣Aδ(p, q) − A(p, q)∣
∣
∣ ≤∣
∣
∣Aδ(
p, (3L)−3/2p)
− A(
p, (3L)−3/2p)∣
∣
∣ =2
3√
3L|p|.
The result now follows straightforwardly.
Now, we show that Aff 1D,e,δ[u] is Lipschitz continuous.
Lemma 6.3.8. Aff 1D,e,δ[u] is Lipschitz continuous with constant
Ch =K
h+
2L
h2.
Proof. Using the derivative estimates proved in Lemma 6.3.7 and the triangle inequality,
we get
∣
∣
∣Aδ(p1, q1) − Aδ(p2, q2)∣
∣
∣ ≤∣
∣
∣Aδ(p1, q1) − Aδ(p2, q1)∣
∣
∣+∣
∣
∣Aδ(p2, q1) − Aδ(p2, q2)∣
∣
∣
≤∣
∣
∣
∣
∣
d
dpAδ(p, q1)
∣
∣
∣
∣
∣
|p1 − p2| +
∣
∣
∣
∣
∣
d
dqAδ(p2, q)
∣
∣
∣
∣
∣
|q1 − q2|
≤ K|p1 − p2| + L|q1 − q2|.
for some p, q ∈ R. Here, the schemes |uhx|± take the place of p1, p2 and have Lipschitz
constant of 1/h. On the other hand, q1, q2 are replaced by (−uhxx)± which has Lipschitz
Chapter 6. Affine curvature 111
constant 2/h2. The result then follows easily.
Finally, we study how K and L can be chosen to ensure that Aff 1D,e,δ[u] is consistent
with Aff 1D[u].
Lemma 6.3.9. Let x ∈ Ω be a reference point on the grid and φ be a C4 function that is defined in a
neighborhood of the grid. Assume K = O(h−α) and L = O(h−β) such that K√
L ≥ 1, α ∈ (0, 1)
and β ∈ (0, 2). Then the scheme Aff 1D,e,δ defined by (AC)1D,e,δ is consistent with Aff 1D and has
accuracy
Aff 1D,e,δ[φ](x) − Aff 1D[φ](x) = O(
h1−α + h2−β + hmin(2α,β/2))
.
Moreover, the optimal choice of α and β is given by α = 1/3 and β = 4/3, in which case the
accuracy is O(h2/3).
Proof. We showed in the previous lemma that
∣
∣
∣Aδ(p1, q1) − Aδ(p2, q2)∣
∣
∣ ≤ K|p1 − p2| + L|q1 − q2|.
Here,∣
∣
∣uhx
∣
∣
∣
±take the place of p1, while p2 is replaced by |ux|. On the other hand, q1 is
replaced by −uhxx, while q2 is replaced by −uxx. The result follows from the consistent of
the finite difference operators
∣
∣
∣uhx
∣
∣
∣
±= |ux| + O(h),
(
−uhxx
)±= (−uxx)± + O(h2).
Hence
Aff 1D,e,δ[φ](x) − Aff 1D,δ[φ](x) = O(
h1−α + h2−β)
.
A direct application of Lemma 6.3.7, where ux and uxx take the place of p and q
respectively, leads to the estimate
∣
∣
∣Aff 1D,δ[φ] − Aff 1D[φ]∣
∣
∣ ≤ max
(
4
27K2|uxx|, 2
3√
3L|ux|
)
.
Therefore, since K = O(h−α) and L = O(h−β)
Aff 1D,e[φ](x) − Aff 1D[φ](x) = O(
hmin(2α,β/2))
.
The accuracy of Aff 1D,e,δ then follows from the equality
Aff 1D,e,δ[φ](x) − Aff 1D[φ](x) = Aff 1D,e,δ[φ](x) − Aff 1D,δ[φ] + Aff 1D,e[φ](x) − Aff 1D[φ](x)
112 6.3. NUMERICAL METHODS FOR THE MODEL EQUATION
Finally, we observe that
max(min(1 − α, 2α)) = max(min(2 − β, β/2)) =2
3,
with the maximums being attained at α = 1/3 and β = 4/3, thus justifying the optimal
choice of α and β.
Remark 6.5. As a result of the proof, we show as well that Aff 1D,e,δ is consistent with Aff 1D,δ
with accuracy
Aff 1D,e,δ[φ](x) − Aff 1D,δ[φ](x) = O(
h1−α + h2−β)
.
Remark 6.6. With the optimal choice of α and β, the overall accuracy of Aff 1D,e,δ and Aff 1D,e
with respect to Aff 1D is the same (see Remark 6.4), meaning that no accuracy is lost due to
the regularization.
6.3.3 Convergence theorems for the one-dimensional model
Having proved the ellipticity and consistency of the schemes, the uniform convergence
follows as discussed in chapter 3.
The first convergence result is for the elliptic problem, where there is no need for the
regularized scheme.
Theorem 6.3.10. Let u(x) be the unique viscosity solution of Aff 1D[u] = f in Ω, along with
suitable boundary conditions, which we assume to satisfy a strong comparison principle (see
Definition 2.3.7). For each h > 0, let u1D,e,h be the uniformly bounded solution of Aff 1D,e[u] = f .
Then u1D,e,h → u locally uniformly, as h → 0.
Proof. Convergence for the elliptic discretization −Aff 1D,e follows from Theorem 3.1.5.
Unlike in the elliptic problem, in the parabolic problem we need to use the regularized
scheme, with the time discretization being given by a forward Euler step. This leads us to
the scheme
u(·, t + dt) = u(·, t) + dt Aff 1D,e,δ[u(·, t)]. (6.9)
Theorem 6.3.11. Let u(x, t) be the unique viscosity solution of ut = Aff 1D[u] in Ω × [0, ∞),
along with u(x, 0) = u0(x) and suitable boundary conditions. Assume as well that K = h−1/3
and L = h−4/3. For each h > 0, let u1D,e,h be the uniformly bounded solution of the monotone
time discretization (6.9) with dt ≤ 1/Ch given by (6.8). Then u1D,e,h → u locally uniformly, as
dt, h → 0.
Chapter 6. Affine curvature 113
Proof. The elliptic scheme Aff 1D,e,δ leads to the monotone time discretization (6.9) provided
dt ≤ 1/Ch where Ch is the Lipschitz constant of Aff 1D,e,δ. The convergence then follows
from Theorem 3.1.5.
Remark 6.7. It is also true that, for fixed values of K, L, there is a unique viscosity solution,
uδ, of the regularized PDE. Then, fixing K, L and using the discretization above with
dt = O(h2), the forward Euler method converges uniformly to uδ as h → 0.
6.4 NONC ONVERG ENC E OF S TANDAR D FINITE DIFFERENC ES
In this section we show that standard finite differences are unstable for both the one
dimensional and the two dimensional operators that we study. This instability can be
understood from the one dimensional model, which, if we take |ux| = 1, reduces to the
operator u1/3xx . It is certainly plausible that the singularity near uxx = 0 could lead to
instabilities. This motivates the regularization introduced in the previous section, which
replaces the singularity of the cube root with a linear function with large slope. It also
motives the convergent finite difference schemes, which have an explicit time step that
ensures the convergence of the time dependent schemes.
We begin with an example where the standard finite scheme does not converge for the
one-dimensional model. Next we consider the time dependent equation and the associated
scheme obtained with the (unregularized) elliptic scheme in space and a forward Euler
step in time. A scaling argument suggest that the choice dt = O(h4/3) should provide a
stable scheme. In fact, this scaling argument can be improved to a proof of the maximum
principle for the homogeneous equation with the same time step. However, the maximum
principle is not enough for convergence (we need the comparison principle) and we
demonstrate divergence with that time step. Using a smaller time step dt = O(h2) appears
to fix the problem. (The standard non-elliptic finite difference scheme diverges for the
example we present no matter how small the time step). The example is then generalized
to the two-dimensional operator.
6.4.1 Nonconvergence of standard finite differences in one dimension
Consider the discretization given by inserting the standard centered differences, given by
(3.7),
Aff 1D,a[u] =(
(
uhx
)2 (
uhxx
)
)1/3
.
We considered an exact solution u0(x) = sin(2πx) of (AC-1D). Then we solved the time
dependent problem, using the forward Euler time discretization, with initial data given
114 6.4. NONCONVERGENCE OF STANDARD FINITE DIFFERENCES
by the solution u(x, 0) = u0(x) We found that this solution was unstable for the standard
finite difference scheme. The results are displayed in Figure 6.1, which illustrates that the
numerical solution diverges from the exact solution. This effect persists over different grid
sizes, and over smaller time steps.
-1 -0.5 0 0.5 1
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
-1 -0.5 0 0.5 1
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
-1 -0.5 0 0.5 1
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
-1 -0.5 0 0.5 1
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
Figure 6.1: Solution of the one-dimensional model equation using standard finite differences at timest ∈ 0, 1, 2, 5. Here dt = h2/2 on a 256-point grid.
Next we consider the discretization of the time-dependent equation
ut = Aff 1D[u] − f. (6.10)
Using a forward Euler method in time, and the elliptic method in space leads to
u(·, t + dt) = Φ1D,et (u) := u(·, t) + dt(Aff 1D,e[u(·, t)] − f). (6.11)
A scaling argument suggests dt = O(h43 ) might be stable, since the operator Aff 1D is
homogeneous to this order in h. In fact, we are able to prove the following.
Lemma 6.4.1. When f = 0 in (6.10), the solution map Φ1D,e satisfies the maximum principle
min Φ1D,et (u) ≤ Φ1D,e
t+dt(u) ≤ max Φ1D,et (u),
provided dt ≤ (h4/2)1/3.
Proof. We have
−2
h
∣
∣
∣uhx
∣
∣
∣
− ≤ −uhxx ≤ 2
h
∣
∣
∣uhx
∣
∣
∣
+.
Thus, since A+ ∈ ND+(R2) and A− ∈ ND−(R2),
0 ≤ A+(
∣
∣
∣uhx
∣
∣
∣
+, −uh
xx
)
≤ A+(
∣
∣
∣uhx
∣
∣
∣
+,
2
h
∣
∣
∣uhx
∣
∣
∣
+)
=(
2
h
)1/3∣
∣
∣uhx
∣
∣
∣
+
and
0 ≥ A+(
∣
∣
∣uhx
∣
∣
∣
−, −uh
xx
)
≥ A+(
∣
∣
∣uhx
∣
∣
∣
−, −2
h
∣
∣
∣uhx
∣
∣
∣
−)= −
(
2
h
)1/3∣
∣
∣uhx
∣
∣
∣
−
and so
−(
2
h
)1/3∣
∣
∣uhx
∣
∣
∣
− ≤ −Aff 1D,e[u] ≤(
2
h
)1/3∣
∣
∣uhx
∣
∣
∣
+
Chapter 6. Affine curvature 115
Hence
u(x) − dt(
2
h
)1/3∣
∣
∣uhx
∣
∣
∣
+ ≤ u(x) + dtAff 1D,e[u] ≤ u(x) + dt(
2
h
)1/3∣
∣
∣uhx
∣
∣
∣
−
The proof now follows due to the choice of dt and from observing that
u(x) + h∣
∣
∣uhx
∣
∣
∣
−= maxu(x + h), u(x − h), u(x)
and
u(x) − h∣
∣
∣uhx
∣
∣
∣
+= minu(x + h), u(x − h), u(x).
However, as the following example shows, the maximum principle by itself is not
enough to guarantee convergence and so the above choice for the time step is not guaran-
teed to produce a convergent scheme. In practice, the above choice for the time step leads
to a divergent but bounded scheme with nonlinear instabilities.
Example 6.1. We solved (6.10) using (6.11). We took u(x, 0) = u0 where u0(x) = Cx4/3 and
f = Aff 1D[u0]. In Figure 6.2, we plot the numerical solution obtained at different times
with dt = (h4/4)1/3 (The conservative choice of the time step is to account for the fact the
equation is not homogeneous). The exact solution grows in time. For larger times, the
solution has the same shape but with small high frequency oscillations in time. On the
other hand, choosing dt = h2/2 leads to convergence. (The data is not presented to save
space.)
-1 -0.5 0 0.5 1
-5
-4
-3
-2
-1
0
1
-1 -0.5 0 0.5 1
-5
-4
-3
-2
-1
0
1
-1 -0.5 0 0.5 1
-5
-4
-3
-2
-1
0
1
-1 -0.5 0 0.5 1
-5
-4
-3
-2
-1
0
1
Figure 6.2: Plot of the solution obtained described in Example 6.1 at time t ∈ 0, 1, 5, 20 on a 128-pointgrid.
6.4.2 Two dimensions
In this section we define and study the standard finite difference scheme for Aff [u]. We
show that the accuracy degenerates near points where the affine curvature is zero. We give
an example of a steady solution where the standard finite difference scheme breaks down.
116 6.4. NONCONVERGENCE OF STANDARD FINITE DIFFERENCES
Using (6.1), we can write
Aff [u] =(
uxxu2y − 2uxuyuxy + uyyu2
x
)1/3.
Definition 6.4.2. Let u : R2 → R. We define the scheme
Aff a[u] =(
uhxx(uh
y)2 − 2uhxuh
yuhxy + uh
yy(uhx)2)1/3
(AC)a
that approximates Aff [u].
Remark 6.8. The uhxy term is not elliptic, and consequently −Aff a is not elliptic.
Lemma 6.4.3. Let (x, y) ∈ Ω be a reference point on the grid and φ be a smooth function that is
defined in a neighborhood of the grid. Then the scheme Aff a[φ] defined by (AC)a approximates
Aff [φ] with accuracy
Aff a[φ](x, y) − Aff [φ](x, y) =
O(h2), Aff [φ](x, y) 6= 0,
O(h23 ), Aff [φ](x, y) = 0.
Proof. All the standard finite differences used are second order accurate. Therefore
φhxx
(
φhy
)2 − 2φhxφh
yφhxy + φh
yy
(
φhx
)2= φxxφ2
y − 2φxφyφxy + φyyφ2x + O(h2),
in other words, (Aff a[φ])3 = (Aff [φ])3 + O(h2). In addition, the Taylor expansion tells us
that
(r + t)1/3 = r1/3 +t
3r2/3+ O(t2)
when r 6= 0. Hence, when Aff [φ](x, y) 6= 0, it follows that Aff a[φ] = Aff [φ] + O(h2).
Otherwise, when Aff [φ](x, y) = 0, we can only conclude that Aff a[φ] = Aff [φ] + O(h23 ).
Now we present an example that shows that the scheme Aff a does not converge. We
choose a level set function and a right hand side so that the equation is a steady state (see
Example 6.6(d) for more details). Starting from initial data corresponding to the exact
solution, and evolving in time with a forward Euler step, the solution changes to order
one. Indeed, the solution does not appear to reach a steady state, even after running for
a long time. See Figure 6.3 for snapshots in time of the solution. Similar behaviour was
observed on finer grids (although it took a longer time to reach a comparable change in
the solution).
Chapter 6. Affine curvature 117
5 10 15 20 25 30
5
10
15
20
25
30
5 10 15 20 25 30
5
10
15
20
25
30
5 10 15 20 25 30
5
10
15
20
25
30
5 10 15 20 25 30
5
10
15
20
25
30
5 10 15 20 25 30
5
10
15
20
25
30
5 10 15 20 25 30
5
10
15
20
25
30
Figure 6.3: Lack of convergence when using a standard finite difference scheme: example 6.6 (d) with thestandard finite difference solver: level sets of the solution at times t = 0 (upper left), t = 15 (upper center),t = 17 (upper right), t = 20 (lower left), t = 40 (lower center) and t = 50 (lower right) with dt = h2/2 ona 32 × 32 grid.
6.5 CONVER GENT FINITE DIFFERENC E METHODS
Based on the convergence theory discussed above, and on the elliptic discretization of the
one dimensional model equation, we now build a discretization of the affine curvature
operator. This discretization is based on the median scheme for the mean curvature
operator from [Obe04]. We could also use the morphological operator [CDK95], which
results in a very similar discretization of the operator Aff [u]. We establish the accuracy of
the discretization, and show that it is elliptic. The scheme is augmented to a more accurate,
but still convergent, filtered scheme which interpolates between the standard scheme and
the elliptic scheme. We regularize the operator, which allows us to build a convergent
monotone time discretization.
6.5.1 Median for ∆1u
In [Obe04] an elliptic scheme for ∆1u(x) is presented based on taking the median of the
values of u sampled in a small approximately circular neighborhood of x. The motivation
follows from observing, using (6.1), that
∆1u = utt, t =(−uy, ux)
(u2x + u2
y)1/2,
118 6.5. CONVERGENT FINITE DIFFERENCE METHODS
where t is the (Euclidean) unit tangent. The median captures an approximation to utt,
the second tangential derivative of u, since the larger values point in the direction of the
gradient and the smaller values point in the opposite directions.
We outline the scheme here. We will define it at the reference point (x, y). Let
(x1, y1), . . . , (xnS, ynS
) be the nS neighbours and set dθ = 2πnS
. We refer to dθ as the directional
resolution. Denote the value of the solution at the point (xi, yi) by ui. Set vi = (xi, yi)−(x, y)
and (x−i, y−i) = (x, y) − vi. We choose the neighbours such that
(xi+1, yi+1) = h nθ (cos(idθ), sin(idθ)) + (ei, fi)
where |ei|, |fi| ≤ h. Thus h is the spatial resolution and nθ denotes the width of the stencil.
In fact, for nθ ≤ 5, the neighbours in the first quadrant are given by
(xi+1, yi+1) = h (bnθ cos(idθ)e, bnθ sin(idθ)e)
where i = 0, . . . , nS
4− 1, with the points on the remaining three quadrants being obtained
by π2, π and 3π
2rotations. Here bxe denotes the integer closest to x.
Figure 6.4: Neighbour points of the stencil for nθ = 3 (smaller circle) and nθ = 7 (larger circle).
nθ ns neighbours in the first quadrant1 8 (1, 0), (1, 1)
2 12 (2, 0), (2, 1), (1, 2)
3 16 (3, 0), (3, 1), (2, 2), (1, 3)
4 32 (4, 0), (4, 1), (4, 2), (3, 2), (3, 3), (2, 3), (2, 4), (1, 4)
5 40 (5, 0), (5, 1), (5, 2), (4, 2), (4, 3), (4, 4), (3, 4), (2, 4), (2, 5), (1, 5)
Table 6.1: Coordinates of the neighbours in the first quadrant of a stencil with width nθ.
Chapter 6. Affine curvature 119
Definition 6.5.1. Let u : R2 → R. Define the scheme
∆e1u(x, y) = 2
mediani=1,...,nS
ui − u(x, y)
(h nθ)2(MC)e
that approximates ∆1u.
In general, consistency of finite difference schemes follows by Taylor expansions.
However, additional care is needed when the PDE is singular, as when ∇u(x) = 0 in (MC).
We then recall here the definition of consistency for the mean curvature operator.
Definition 6.5.2. The numerical scheme F h,dθ is consistent with ∆1 if for every smooth function
φ and every (x, y) ∈ R2
limh,dθ→0
F h,dθ[φ] = ∆1φ
at (x, t) if ∇φ(x, y) 6= 0 and
λ ≤ lim infh,dθ→0
F h,dθ[φ] ≤ lim suph,dθ→0
F h,dθ[φ] ≤ Λ
at (x, t) where λ, Λ are the least and greatest eigenvalues of D2φ(x, y), otherwise.
Lemma 6.5.3. The finite difference scheme −∆e1u, given by (MC)e, is elliptic.
Proof. We can rewrite the scheme as
−∆e1u = 2
mediani=1,...,nS
(u(x, y) − ui)
(h nθ)2.
Since median is a nondecreasing function, we can immediately conclude that ∆e1u is elliptic.
Lemma 6.5.4. Let (x, y) ∈ Ω be a reference point on the grid and φ be a smooth function that
is defined in a neighborhood of the grid. Then the scheme ∆e1φ defined by (MC)e is consistent.
Further, it approximates ∆1 with accuracy
∆e1φ(x, y) = ∆1φ(x, y) + O
(
(h nθ)2 + dθ
)
,
when |∇u(x, y)| 6= 0.
Remark 6.9. Since dθ = O( 1nθ
), the “optimal” choice is h = O(n−3/2θ ), which results in
accuracy of O(h2/3). However this also requires a dense stencil. In practice, we use a fairly
narrow stencil, and combine with a filtered scheme for more accuracy.
The proof can be found in [Obe04].
120 6.5. CONVERGENT FINITE DIFFERENCE METHODS
6.5.2 Elliptic scheme for −Aff [u]
We now construct an elliptic scheme for −Aff [u] following the same procedure as in
subsection 6.3.1. This is accomplished by writing Aff [u] in terms of |∇u| and ∆1u as
Aff [u] = |∇u| k[u]1/3 = (|∇u|2 ∆1u)1/3 = A(|∇u| , ∆1u)
and using the elliptic schemes |∇uh|+ and −|∇uh|− presented in section 3.3 and −∆e1u
described in subsection 6.5.1.
Remark 6.10. We chose to discretize ∆1u with the median scheme (MC)e. However, other
schemes could have been used, for instance the morphological scheme in [CDK95].
Definition 6.5.5. Let u : R2 → R. We define the scheme
−Aff e[u] = A+(
∣
∣
∣∇uh∣
∣
∣
+, −∆e
1u)
+ A−(
−∣
∣
∣∇uh∣
∣
∣
−, −∆e
1u)
(AC)e
that approximates −Aff [u].
Remark 6.11. The above approach can be generalized to obtain an elliptic scheme for the
elliptic operator − |∇u| G(k[u]), where G is nondecreasing and homogeneous of order
α ≤ 1, G(tr) = tαG(r). (This PDE represents curves evolving with normal speed G(k)). In
such cases, write
|∇u| G(k[u]) = |∇u|1−α G(|∇u| k[u]) = |∇u|1−α G(∆1u).
The elliptic scheme is then given by
(
|∇uh|+)1−α
G(
(−∆e1u)+
)
+(
|∇uh|−)1−α
G(
(−∆e1u)−) .
Now we prove the ellipticity and consistency of the scheme −Aff e[u].
Lemma 6.5.6. The finite difference scheme −Aff e[u], given by (AC)e, is elliptic.
Proof. The proof is similar to Lemma 6.3.5.
Lemma 6.5.7. Let (x, y) ∈ Ω be a reference point on the grid and φ be a smooth function that is
defined in a neighborhood of the grid. Then the scheme Aff e[φ] defined by (AC)e is consistent with
Aff [φ] and has accuracy
Aff e[φ](x, y)−Aff [φ](x, y) =
O (h + (h nθ)2 + h dθ + dθ) , if Aff [φ](x, y) 6= 0,
O(
((h nθ)2 + h dθ + dθ)1/3
)
, if Aff [φ](x, y) = 0 and |∇φ| (x, y) 6= 0,
O(
h2/3)
if |∇φ| (x, y) = 0.
Chapter 6. Affine curvature 121
Proof. Suppose first that |∇φ| (x, y) 6= 0. We have
|∇φ(x, y)|2 =(
|∇φh(x, y)|±)2
+ O(h) and ∆e1φ(x, y) = ∆1φ(x, y) + O
(
dθ + (h nθ)2)
.
(See Lemma 6.5.4). Hence, when Aff [φ](x, y) 6= 0,
Aff e[φ]3 = Aff [φ]3 + O(
h + (h nθ)2 + h dθ + dθ
)
and, by the Taylor expansion like in Lemma 6.4.3, we have Aff e[φ](x, y) = Aff [φ](x, y) +
O (h + (h nθ)2 + h dθ + dθ). Otherwise, when Aff [φ](x, y) = 0, we necessarily have ∆1φ = 0
and so
Aff e[φ]3 = Aff [φ]3 + O(
(h nθ)2 + h dθ + dθ
)
.
Therefore, Aff e[φ](x, y) = Aff [φ](x, y) + O(
((h nθ)2 + h dθ + dθ)1/3
)
.
When |∇φ| (x, y) = 0,
|∇φ(x, y)|2 =(
|∇φh(x, y)|±)2
+ O(h2)
with ∆e1u being bounded. Hence, Aff e[φ] = Aff [φ] + O(h2/3).
Remark 6.12. Since dθ = O( 1nθ
), the “optimal” choice is h = O(n−3/2θ ), which results in
accuracy of O(h2/9) in the worst case. However this also requires a dense stencil. In
practice, we use a fairly narrow stencil, and combine with a filtered scheme for more
accuracy.
6.5.3 Filtered scheme for Aff [u]
Filtered schemes were discussed in section 3.4. Here, we define filtered schemes in a
slightly different form: they are a continuous linear interpolation between the accurate and
the elliptic scheme, which equals the accurate scheme when the two schemes are within ε
of each other. In order to so, we make use of the auxiliary function Sε : R2 → R, defined to
be a continuous function which for (a, b) ∈ R2 is equal to a near the diagonal and b off the
diagonal.
Definition 6.5.8. Define for ε > 0, Aε = (a, b) ∈ R2 | |a − b| < ε. Set ρ = 10ε. Define
d = dist ((a, b), Aε).
Sε(a, b) =
a if (a, b) ∈ Aε,
ρ−dρ
a + dρb if d ≤ ρ,
b otherwise.
122 6.5. CONVERGENT FINITE DIFFERENCE METHODS
Define the filtered scheme
Aff f [u] = Sε (Aff a[u], Aff e[u]) (AC)f
where ε = ε(h, dθ).
While theoretically, the only requirement on ε to ensure the convergence of the filtered
schemes is that ε → 0 as h, dθ → 0, in practice ε must be chosen carefully. Intuitively, it
should be large enough to permit the accurate scheme to be active where the solution
is smooth (see Proposition 3.4.9), and small enough to force the monotone scheme to be
active whenever needed for convergence (for instance, when the solution is singular) .
Remark 6.13. Proposition 3.4.9 tells us that heuristically we could choose ε =√
Acc[Aff e],
where Acc[Aff e] denotes the accuracy of Aff e (which corresponds to the choice α = βM/2
in the Proposition). In the numerical results presented here, we defined ε based on the
accuracy of the scheme away from the singularities of Aff [u].
In practice, the filtered scheme allows us to neglect the error coming from the wide
stencil, while in theory we still need to send dθ → 0 for convergence of the filtered scheme.
In our numerical examples, we obtain the accuracy of the accurate scheme in most cases.
6.5.4 Regularization and Forward Euler method
Similarly to the one dimensional model equation (see subsection 6.3.2), in order to build a
provably convergent scheme for the time dependent equation (AC) we need to regularize
the operator.
Write Aff [u] = A(|∇u| , ∆1u), where A(p, q) = (p2q)1/3, and regularize the cube root
function as before. This leads to
Aff δ[u] = Aδ(|∇u| , ∆1u),
where Aδ(p, q) = sgn(q) min(|A(p, q)|, K|p|, L|q|).Remark 6.14. The regularized operator, Aff δ[u], is still a level set operator. To see this, it is
enough to take K = L = 1/δ:
Aff δ[u] = |∇u| sgn(k[u]) min
(
|k[u]|1/3 ,|k[u]|
δ,1
δ
)
,
The operator reduces to either a multiple of the mean curvature operator, or the Eikonal
equation and otherwise we obtain Aff [u].
Similar results regarding ellipticity and consistency hold as for the one-dimensional
model, which we present without proof.
Chapter 6. Affine curvature 123
Lemma 6.5.9. For K = K(δ), L = L(δ) such that K√
L ≥ 1, define the finite difference scheme
−Aff e,δ[u] = Aδ,+(
|∇uh|+, −∆e1u)
+ Aδ,−(
−|∇uh|−, −∆e1u)
(AC)e,δ
Then, −Aff e,δ[u] is elliptic and consistent with −Aff δ[u].
As with the one-dimensional case, K and L need to be chosen appropriately in order
for −Aff e,δ[u] to be consistent with −Aff [u]. Assuming K = O(h−α), L = O(h−β) and
dθ = O(1/nθ) = O(hγ), full consistency is obtained when α ∈ (0, 1), β ∈ (0, 23) and
γ ∈ (β, 2−β2
). The optimal choices are α ∈ [1/9, 7/9], β = 4/9, γ = 2/3 and, in such case, the
accuracy is O(h2/9). The proof is similar to Lemma 6.3.9. As in the one dimensional model,
no accuracy is lost with the regularization (see Remark 6.12).
The Lipschitz constant of Aff e,δ[u] is given by
Ch =K
h+
2L
(h nθ)2, (6.12)
with the proof similar to Lemma 6.3.8. In practice, we will choose K = cKh−1/9 and
L = cLh−4/9, which leads to
Ch = cKh−10/9 + 2cL
nθ2h−22/9.
We can finally define and prove the convergence of the discretizations for the time
dependent equation (AC). The time derivative is discretized with an explicit forward
Euler step, while Aff [u] is discretized using either the regularized elliptic scheme Aff e,δ
or the regularized filtered scheme Aff f,δ[u] (this is easily defined by replacing the elliptic
scheme in Aff f by its regularized version). This leads to the monotone (resp. filtered) time
discretization with solution map given by
u(·, t + dt) = u(·, t) + dtAff e,δ[u(·, t)],(
resp. = u(·, t) + dtAff f,δ[u(·, t))
. (6.13)
6.5.5 Convergence Theorems
Having proved the ellipticity and consistency of the schemes, the uniform convergence
follows as discussed in chapter 3, provided there exists unique viscosity solutions to the
PDEs (AC) and Aff [u] = f along with the homogeneous Neumman boundary conditions,
which is assured by the theory in [Gig06], as explained above.
Existence of stable solutions to the schemes is also required. Here, we will make this an
assumption of our result for readability. In rigour, it follows from adding a small multiple
124 6.5. CONVERGENT FINITE DIFFERENCE METHODS
of u to the scheme which makes it proper (see the discussion in section 3.2, in particular
Remark 3.7).
There are two convergence results. The first is for the elliptic problem, where there is
no need for regularization. The second is for the parabolic problem, where we need to
regularize and use an explicit Euler time step.
Theorem 6.5.10. Let Ω be a bounded convex domain with a C2 boundary and u denote the unique
viscosity solution of Aff [u] = f in Ω, along with homogeneous Neumann boundary conditions
on ∂Ω. For each ε = ε(h, dθ) > 0, let ue,ε, (resp. uf,ε) be the uniformly bounded solution of
Aff e[u] = f , (resp. Aff f [u] = f ). Then
ue,ε → u and uf,ε → u, locally uniformly, as ε → 0.
Proof. The assumptions on Ω guarantee that the PDE satisfies a comparison principle.
Convergence for the elliptic discretization −Aff e then follows from Theorem 3.1.5. As for
the filtered schemes, it follows from Theorem 3.4.3.
Theorem 6.5.11. Let Ω be a bounded convex domain with a C2 boundary. Assume that u(x, t)
is the unique viscosity solution of ut = Aff [u] in Ω × [0, ∞), along with u(x, 0) = u0(x) and
homogeneous Neumman boundary conditions. Assume as well that K and L are picked so that
Aff e,δ is consistent with Aff . For each ε = ε(h, dθ) > 0, let ue,ε (resp. uf,ε) be the uniformly
bounded solution of the monotone (resp. filtered) time discretization (6.13) with dt ≤ 1/Ch where
Ch is the Lipschitz constant of Aff e,δ[u] (6.12). Then
ue,ε → u and uf,ε → u
locally uniformly, as dt, ε → 0.
Proof. The assumptions on Ω guarantee that the PDE satisfies a comparison principle. The
elliptic scheme leads to the monotone time discretization (6.13) provided dt ≤ 1/Ch where
Ch is the Lipschitz constant of Aff e,δ[u] (6.12). The convergence then follows from Theorem
3.1.5. For the time discretization of the filtered scheme, the convergence of filtered schemes
follows from Theorem 3.4.3.
Remark 6.15. Just like in the one-dimensional model, for fixed values of K, L, there is a
unique viscosity solution, uδ, of the regularized PDE. Then, fixing K, L and using the
discretization above with dt = O((h nθ)2), the forward Euler method converges uniformly
to uδ as h → 0.
Chapter 6. Affine curvature 125
6.6 NUMERIC AL RES ULTS
In this section, we start with a simple example to compare the affine invariant curvature
motion and the regularized model. We present different examples on the evolution of
a single curve in subsection 6.6.1 under the affine curvature motion and compare it to
the mean curvature motion. We test the accuracy of the schemes by computing solutions
to the time-independent PDE in subsection 6.6.2 and to the time dependent equation
in subsection 6.6.3. We mostly considered stationary problems because it was easier to
generate exact solutions by applying the operator Aff to a given function u, and including
f = Aff [u] as a source function. For the time dependent problem, we took advantage of
the exact solution from Lemma 6.2.6 to compare the accuracy of the solutions after a short
time, T = 0.1. We test as well in subsection 6.6.4 if the schemes satisfy numerically the
morphology and affine invariance properties that (AC) satisfies (see Theorem 6.2.5).
Definition 6.6.1 (Parameters for the discretization). We used the regularized schemes with
K = 20h−1/9 and L = 20h−4/9. For the elliptic discretization, we used two different stencils:
the narrow elliptic scheme used nθ = 3 and the wider elliptic scheme used nθ = 7. The filtered
discretization used the wider elliptic scheme with
ε(h, dθ) =√
h + dθ/10.
For the forward Euler method with the elliptic and filtered schemes, we used a constant time step
of dt = 1/Ch where Ch is given by (6.12). For the forward Euler method with the standard finite
difference scheme, we used the same time step as the filtered scheme, except when computing the
steady state solutions in Example 6.6 where we used dt = h2/2 (this choice of time step proved
to be enough in practice). The stopping criteria was simply that the l∞ norm of the residual∣
∣
∣Aff f,δ[un] − f∣
∣
∣ (and similarly for Aff e,δ, Aff a) was below tol = 10−5.
Example 6.2. [Comparing the regularized to the unregularized operators] We compare
the evolution of an ellipse by the affine invariant curvature motion and by the regularized
model. The ellipse should remain an ellipse of fixed eccentricity. The results were obtained
by numerically solving (AC) and ut = Aff δ[u] respectively, with the initial condition
u0(x, y) = min
(
x
2
)2
+ y2 − 1, 1
and homogeneous Neumann boundary conditions. We took [−4, 4]2 as the computational
domain on a 128 × 128 grid. The narrow elliptic scheme was used for the spatial discretiza-
tion. The difference between the level sets of the solution of the two equations is visually
126 6.6. NUMERICAL RESULTS
indistinguishable. Measured in the l∞ norm ranges from 10−6 for early times steps and
10−7 for the later time steps. The level sets of the numerical solution obtained by solving
the regularized PDE are depicted in Figure 6.5.
Remark 6.16. While the regularized scheme with the more stringent time step is needed for
the convergence proof, in practise we found, as reported in the previous example, using
the (unregularized) elliptic discretization with the time step dt = h2/2 gave nearly identical
results to within two or more significant digits. For the remaining examples, we present
results using the regularized schemes (with K = 20h−1/9 and L = 20h−4/9).
6.6.1 Numerical examples showing curve evolution
In this section we present some numerical examples illustrating the geometric properties
of the PDEs.
Example 6.3 (Ellipse). We compare the evolution of an ellipse by the affine invariant
curvature motion and by mean curvature. For the former, the ellipse should remain an
ellipse of fixed eccentricity. For the latter, the ellipse asymptotically approaches a circle
instead. The results were obtained by numerically solving (AC) and (MC), respectively,
with the initial condition
u0(x, y) = min
(
x
2
)2
+ y2 − 1, 1
and homogeneous Neumann boundary conditions. We took [−4, 4]2 as the computational
domain on a 128 × 128 grid. As for the scheme used, we chose the narrow elliptic schemes
for both. In Figure 6.5, we plot the zero level sets obtained at t ∈ 0, 0.1, 0.3, 0.5, 0.7, 0.9.
Example 6.4 (Diamond). We also compute the solution of (AC) with initial condition
(a) u0(x, y) = min |x| + |y| − 1, 1 , (b) u0(x, y) = min |x| + 2|y| − 1, 1
and homogeneous Neumann boundary conditions. The exact solutions are not known.
However, we do know that smooth convex curves evolving under affine curvature con-
verge to ellipses until collapsing to a point and that is the behaviour we observed here (see
Figure 6.5). We took [−2, 2]2 as the computational domain on a 128 × 128 grid. As for the
scheme, we chose again the narrow elliptic scheme. In Figure 6.5, we plot the zero level
sets of the numerical solution from time t = 0 to t = 0.5 in increments of 0.1 for example
(a) and from t = 0 to t = 0.3 in increments of 0.05 for example (b).
Chapter 6. Affine curvature 127
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
-1 -0.5 0 0.5 1
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
-1 -0.5 0 0.5 1
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Figure 6.5: (Top:) Evolution of an ellipse by (left) affine curvature, (right) mean curvature. (Bottom:)Evolution by affine curvature of (left) a diamond and (right) a flatter diamond.
-1 -0.5 0 0.5 1
-1
-0.5
0
0.5
1
-1 -0.5 0 0.5 1
-1
-0.5
0
0.5
1
-1 -0.5 0 0.5 1
-1
-0.5
0
0.5
1
-1 -0.5 0 0.5 1
-1
-0.5
0
0.5
1
-1 -0.5 0 0.5 1
-1
-0.5
0
0.5
1
-1 -0.5 0 0.5 1
-1
-0.5
0
0.5
1
-1 -0.5 0 0.5 1
-1
-0.5
0
0.5
1
-1 -0.5 0 0.5 1
-1
-0.5
0
0.5
1
Figure 6.6: Evolution of a fan-shape like curve under affine curvature motion (top) and mean curvature(bottom) at time t ∈ 0, 0.05, 0.1, .2 (see Example 6.5 for more details).
Example 6.5 (Fan-shape curve). We also compute the solution of (AC) and (MC) with
initial condition
u0(x, y) = min
c(1)+ (x, y), c
(1)− (x, y), c
(2)+ (x, y), c
(2)− (x, y), 1
,
where
c(1)± (x, y) =
(
x ± 1
2
)2
+ 5(
y ± 1
4
)2
− 1
2and c
(2)± (x, y) = 5
(
x ± 1
4
)2
+(
y ∓ 1
4
)2
− 1
2
and homogeneous Neumann boundary conditions. The exact solution is not known. As in
the previous example, we took [−2, 2]2 as the computational domain on a 128 × 128 grid
128 6.6. NUMERICAL RESULTS
together with the narrow elliptic scheme.
Under the both curvature motions, the curve should initially become convex. At later
times, under the affine curvature motion, the curve should evolve to an ellipse as opposed
to a circle in the mean curvature case. In Figure 6.6, we plot the zero level set of the
numerical solutions at time t ∈ 0, 0.05, 0.1, .2 and observe the exact behaviour described.
6.6.2 Accuracy of stationary solutions
We test the accuracy for the following Dirichlet problem
Aff [u] = f, in Ω,
u = g, on ∂Ω.
Solutions are obtained by computing the steady state solution of ut = Aff [u] − f , with
u(·, t) = g on ∂Ω and u(x, 0) = u0(x). We set dt = 1/Ch where Ch is given by (6.12) for
the elliptic and filtered schemes and dt = h2/2 for the standard finite difference scheme
scheme. These examples also demonstrate stability of the time dependent problem for
elliptic and filtered scheme, as well as convergence to the steady solution, since we used
the time dependent problem to obtain the solution.
Remark 6.17. The unregularized schemes were also used. For these we set dt = h2/2 for
all examples, except for the filtered scheme in example (d) where we set dt = h2/8. The
results obtained were virtually the same.
We set u0(x) to be the exact solution in a layer of seven grid points adjacent to the
boundary. As a result, each discretization is initialized at the same set of grid points and
therefore we can make a fair comparison of their accuracy. On the coarsest grid, we set
u0(x) = 0 on the interior grid points. To speed up calculations, on finer grids we set u0(x)
to be interpolated solutions from the coarser grids at interior grid points.
Example 6.6. We consider the following exact solutions
(a) u(x, y) = x2 + y2, f(x) = 2(
x2 + y2)
13 ,
(b) u(x, y) = ex2+y2
, f(x, y) = 2(
e3(x2+y2)(x2 + y2))
13 ,
(c) u(x, y) =(
x2 + y2)
13 , f(x) =
4
3,
(d) u(x, y) =sin(2πx) sin(2πy)
4, f(x) =
π43
2(− (2 + cos(4πx) + cos(4πy)) sin(2πx) sin(2πy))
13 ,
Chapter 6. Affine curvature 129
with Ω = [−1, 1]2. The solutions in (a),(b) and (d) are smooth, but the functions f are only
Holder continuous, C0,2/3, with singularities at the origin for (a) and (b), and at several
points in example (d). The solution in (c) is only C0,2/3 with a singularity at the origin, but
in this case the function f is constant.
The results are presented in Table 6.2. In Example 6.6(a) the solution is a quadratic
polynomial. The accurate scheme Aff a gives essentially machine precision, which is not
surprising, since this scheme is second order accurate. The filtered scheme Aff f gives
nearly the same accuracy (with a small discrepancy which could be eliminated by tuning
the parameter). On the other hand, both the narrow and wider elliptic scheme are much
less accurate. In contrast, in Example 6.6(b) the solution is smooth but not quadratic. In this
case we see that the accurate scheme appears to be converging to O(h). The elliptic schemes
are less accurate and the filtered scheme is in between. In Example 6.6(c) the solution is
only Holder continuous. In this case, the elliptic schemes have almost constant error near
0.01 over the range of parameters used. Despite the singular solution, the accurate scheme
gives accuracy O(h), and the filtered scheme does just as well. Example 6.6(d) shows
that the standard finite difference scheme does not converge as discussed in section 6.4
(see Figure 6.3). The narrow elliptic scheme has almost constant error 0.03 and the wider
elliptic scheme has error 0.1 for the smallest grid, decreasing by a factor of two as the grid
is refined. The filtered scheme has the best accuracy, achieving an error of 0.001 at the
finest grid.
When comparing the different schemes, we have to account for the width of the stencil
since for the elliptic schemes the wider schemes also have a larger spatial discretization
error. In general, the accuracy improved with the use of the wider stencil. Moreover,
the filtered scheme performed as expected by providing better accuracy than the elliptic
scheme and almost as good accuracy as the accurate scheme. The final example shows
that the standard finite difference scheme may not converge. This may be due to the fact
that, despite the solution being smooth, there were multiple points where f was singular.
Geometrically, this solution has several points where curves shrunk to zero.
We also considered for Example 6.6 the elliptic schemes with nθ = 1 and 2. These
provided poor accuracy with the directional resolution error easily dominating the spatial
resolution error. The errors do not decrease to zero as we decrease the grid size. This
property is consistent with the theoretical results since convergence is only proved as both
h, dθ → 0, which is indeed observed in the numerical results. On the other hand, using the
narrow and wider schemes, the accuracy of the elliptic scheme was good enough for the
filtered scheme to give accuracy comparable to the accurate scheme in many examples. In
principle we still need to send dθ → 0, but in practice, the rate at which it needs to go to
130 6.6. NUMERICAL RESULTS
zero is much slower than h when the filtered scheme is used.
Finally, we point that the choice of ε is not an easy one and it is hard to pick an ε that
yields optimal results in each example presented. By choosing ε larger, it is possible to
achieve with the filtered schemes the accuracy of the standard schemes in Examples 6.6
(a), (b), (c). However, such choice is too permissive for Example 6.6 (d). As pointed out in
subsection 6.5.3, ε needs to be chosen small enough in order for the monotone scheme to be
active to ensure convergence. (This is comparable to the CFL condition in time dependent
equations: methods are convergent as dt, h → 0, with dt satisfying the CFL condition.)
Errors and order, Example 6.6 (a)
N Narrow Elliptic (nθ = 3) Wide Elliptic (nθ = 7) Standard Filter32 3.565 × 10−2 3.978 × 10−2 2.922 × 10−7 5.104 × 10−7
64 2.555 × 10−2 2.696 × 10−2 3.019 × 10−7 2.827 × 10−7
128 1.623 × 10−2 1.678 × 10−2 1.982 × 10−7 1.984 × 10−7
256 1.050 × 10−2 1.055 × 10−2 9.293 × 10−8 5.678 × 10−5
Errors and order, Example 6.6 (b)
N Narrow Elliptic (nθ = 3) Wide Elliptic (nθ = 7) Standard Filter32 5.462 × 10−2 7.666 × 10−2 1.922 × 10−3 1.985 × 10−3
64 5.207 × 10−2 5.872 × 10−2 9.875 × 10−4 8.904 × 10−3
128 4.105 × 10−2 3.725 × 10−2 3.385 × 10−4 7.262 × 10−3
256 3.173 × 10−2 2.240 × 10−2 9.798 × 10−5 8.065 × 10−3
Errors and order, Example 6.6 (c)
N Narrow Elliptic (nθ = 3) Wide Elliptic (nθ = 7) Standard Filter32 1.387 × 10−2 2.386 × 10−2 1.129 × 10−2 1.129 × 10−2
64 1.310 × 10−2 7.683 × 10−3 4.625 × 10−3 4.625 × 10−3
128 9.302 × 10−3 8.202 × 10−3 1.859 × 10−3 1.872 × 10−3
256 6.445 × 10−3 7.156 × 10−3 7.426 × 10−4 7.570 × 10−4
Errors and order, Example 6.6 (d)
N Narrow Elliptic (nθ = 3) Wide Elliptic (nθ = 7) Standard Filter32 3.964 × 10−2 9.813 × 10−2 - 1.926 × 10−2
64 3.788 × 10−2 4.679 × 10−2 - 8.309 × 10−3
128 3.688 × 10−2 2.367 × 10−2 - 2.482 × 10−3
256 3.003 × 10−2 1.798 × 10−2 - 9.697 × 10−4
Table 6.2: Accuracy in the l∞ norm and order of convergence of the schemes for Example 6.6 with regularizedschemes.
6.6.3 Accuracy for the time dependent problem
Recall here that for general boundary conditions, the time dependent PDE (AC) requires
Neumann boundary conditions in order for uniqueness of viscosity solutions to hold. We
Chapter 6. Affine curvature 131
have already established (in the previous section) the stability of the numerical method. In
the following examples, we test the accuracy of solutions, comparing two different wide
stencil discretizations, along with regularized filtered discretization and the (generally
unstable) standard finite difference method. Consequently we test as well the accuracy
using Dirichlet boundary conditions coming from the exact solution of Lemma 6.2.6.
Example 6.7 (Neumann BC). We consider the exact solution
u(x, y, t) = min
t +3
4
(
b
ax2 +
a
by2
)2/3
− 1, 0
.
with a = 2 and b = 1 (see Lemma 6.2.6). By taking the minimum with 0, we are imposing
homogeneous Neumann boundary conditions, thus avoiding having to deal with boundary
of the computational domain. We set Ω = [−3, 3]2.
We display the numerical error in the l∞ norm at time T = 0.1 in Table 6.3.
Example 6.8 (Dirichlet BC). We consider the exact solution
u(x, y, t) = t +3
4
(
b
ax2 +
a
by2
)2/3
.
with a = 2 and b = 1 (see Lemma 6.2.6). This is the same example as in Example 6.7, but we
consider Dirichlet boundary conditions instead. Therefore, we prescribe the exact solution
at a seven point layer at the boundary for all time t. This way all schemes are initialized at
the same set of grid points and thus we can compare their accuracy.
The error in the l∞ norm at time T = 0.1 is presented in Table 6.3.
When using Neumman boundary conditions, we observed slow convergence, with
errors near 0.01, slowly decreasing as the grid size improved. The accuracy improved as
we went from the narrow to the wider elliptic scheme, and further improved as we went
to the accurate and then the filtered scheme. In the case of Dirichlet boundary conditions,
the accuracy is better overall and the error decrease is slightly faster. The difference is
explained by the cap off done in the Neumman boundary conditions that introduces an
additional error in the solution. However, this error does not propagate to the whole
domain as the level sets of the solution shrink to its interior and so when we look at the
error away from the cap off, we recover results very similar to the Dirichlet boundary
conditions.
132 6.6. NUMERICAL RESULTS
Errors and order, Example 6.7N Narrow Elliptic (nθ = 3) Wide Elliptic (nθ = 7) Standard Filter32 4.845 × 10−2 6.691 × 10−2 4.894 × 10−2 4.888 × 10−2
64 4.432 × 10−2 4.607 × 10−2 2.977 × 10−2 2.975 × 10−2
128 3.544 × 10−2 2.823 × 10−2 2.457 × 10−2 2.438 × 10−2
256 2.971 × 10−2 2.080 × 10−2 1.747 × 10−2 1.724 × 10−2
512 2.764 × 10−2 1.652 × 10−2 1.205 × 10−2 1.182 × 10−2
Errors and order, Example 6.8N Narrow Elliptic (nθ = 3) Wide Elliptic (nθ = 7) Standard Filter32 2.182 × 10−2 1.449 × 10−2 1.985 × 10−2 1.985 × 10−2
64 1.435 × 10−2 1.160 × 10−2 1.279 × 10−2 1.279 × 10−2
128 9.580 × 10−3 7.517 × 10−3 5.566 × 10−3 5.567 × 10−3
256 6.404 × 10−3 4.854 × 10−3 2.442 × 10−3 2.409 × 10−3
512 6.090 × 10−3 4.288 × 10−3 1.036 × 10−3 1.002 × 10−3
Table 6.3: Error in the l∞ norm of the whole computational domain at time t = 0.1 for the time dependentExample 6.7 and 6.8.
6.6.4 Numerical study of the morphology and affine invariance properties
In this section, we test if our proposed schemes satisfy numerically the morphology and
affine invariance properties that (AC) satisfies (see Theorem 6.2.5).
Example 6.9. In this example, we test numerically if the schemes presented here satisfy the
morphology property of the affine curvature evolution ((ii) in Theorem 6.2.5). We consider
two examples: (a) g1(x) = ex, (b) g2(x) = x3. We take
u0(x, y) = min
(
x
2
)2
+ y2 − 1, 0
and compare Φt(gv u0) with gv Φt(u0) at t = 1 for v = 1, 2. We took [−3, 3]2 as the
computational domain with homogeneous Neumann boundary conditions. The results
are displayed in Table 6.4. The difference in the l∞ norm is one order of magnitude smaller
than the observed accuracy for the schemes in subsection 6.6.2. Based on these examples,
the morphology property seems to hold numerically.
Example 6.10. In this example we do a qualitative test of the affine invariance property,
which in practice is what one needs for applications in image analysis. In order to do so
Chapter 6. Affine curvature 133
Difference in the l∞ norm, Example 6.9 (a)N Narrow Elliptic (nθ = 3) Wide Elliptic (nθ = 7) Standard Filter32 8.943 × 10−3 1.037 × 10−2 3.419 × 10−3 3.446 × 10−3
64 5.709 × 10−3 6.111 × 10−3 1.954 × 10−3 1.970 × 10−3
128 4.061 × 10−3 3.257 × 10−3 1.061 × 10−3 1.109 × 10−3
256 3.115 × 10−3 1.871 × 10−3 5.907 × 10−4 6.109 × 10−4
512 2.604 × 10−3 1.161 × 10−3 3.207 × 10−4 3.397 × 10−4
Difference in the l∞ norm, Example 6.9 (b)N Narrow Elliptic (nθ = 3) Wide Elliptic (nθ = 7) Standard Filter32 3.450 × 10−2 7.023 × 10−2 6.947 × 10−3 6.947 × 10−3
64 1.730 × 10−2 2.842 × 10−2 1.881 × 10−3 1.877 × 10−3
128 1.032 × 10−2 9.355 × 10−3 6.254 × 10−4 6.333 × 10−4
256 6.894 × 10−3 4.307 × 10−3 2.283 × 10−4 2.307 × 10−4
512 5.372 × 10−3 2.316 × 10−3 8.343 × 10−5 8.506 × 10−5
Table 6.4: Difference in the l∞ norm between Φt(gv u0) and gv Φt(u0) for v = 1, 2 for Example 6.9.
we plot the level sets of the affine invariant motion by curvature (AC) with
u(x, y) =(
x
2
)2
+ y2 − 1
and u φ as the initial solutions. For the affine transformations φ(x) = Ax, we consider
(a) (rotation by π/4) A =
√2
2
√2
2
−√
22
√2
2
, (b) A =
12
1
1 12
.
We take [−5, 5]2 as the computational domain on a 256 × 256 grid.
In Figure 6.7 we plot the zero level set of Φt(u φ) and(
Φt(det φ)2/3(u))
φ at t = 1. The
filtered scheme provided the best results, being indistinguishable to the naked eye. For
long time, the elliptic scheme did not provide as good results, a consequence of its lower
accuracy (see subsection 6.6.2). As for the standard finite difference scheme only in (a) the
difference is indistinguishable to the naked eye like the filtered scheme. For (b), where A
is not a special affine transformation, the difference is significant, but can be removed by
taking time steps of half the size (For shorter times all the curves were very close).
We point out that when the affine transformation is a rotation by a multiple of π/2 or
a reflection over a line L that makes an angle multiple of π/4 with the x-axis, we obtain
essentially machine precision (The difference in the l∞ norm was of the order 10−10.). This
is expected since our stencil is invariant under these transformations.
134 6.7. CONCLUSIONS
-0.5 0 0.5
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
-0.5 0 0.5
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
-0.5 0 0.5
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
-1 -0.5 0 0.5 1
-1
-0.5
0
0.5
1
-1 -0.5 0 0.5 1
-1
-0.5
0
0.5
1
-1 -0.5 0 0.5 1
-1
-0.5
0
0.5
1
Figure 6.7: Plot of the zero level sets in Example 6.10 of Φt(u φ) and(
Φt(det φ)2/3(u))
φ for regular
elliptic scheme (left), standard scheme (center) and regular filtered scheme (right) at time t = 1 with φ givenby (a) (top) and (b) (bottom).
6.7 CONC LUS IONS
We presented a convergent finite difference discretization of the PDE for motion of level
sets by affine curvature in two dimensions. Computational examples demonstrated that the
standard finite difference method is unstable, which motivates the need for a convergent
method.
The foundation of the scheme used an existing wide stencil discretization of the mean
curvature operator, combined with an elliptic discretization of the positive and negative
eikonal operators, ±|∇u|. However, explicit time discretizations require Lipschitz continu-
ous operators, which the affine curvature operator fails to be. Thus, we approximated it by
a Lipschitz continuous regularization. In theory, the explicit Euler discretization is stable
using a time step dt ≤ O(h22/9) , with the constant determined by the width of the stencil.
In practice, we achieved numerically equivalent results using dt = h2/2 and without the
regularization, although there is no proof of stability with the less restrictive time step.
A careful choice of the regularization parameters allowed for the regularized elliptic
scheme to maintain the same order of accuracy as the unregularized scheme, while being
provably convergent. The lower accuracy of both schemes, which results from the singular-
ity of the operator, is overcomed by the use of the filtered schemes, which essentially attain
the accuracy of the standard finite difference schemes, while being provably convergent.
Chapter 6. Affine curvature 135
Simulations demonstrated the geometric properties of the PDE were nearly preserved
by the numerical solutions, including affine invariance, morphological properties, and the
accurate representation of the shrinking ellipses. Simulations validated the convergence of
the elliptic scheme, and the improved accuracy of the filtered scheme.
CHAPTER 7
CONCLUSIONS
In this thesis, we tackled three different elliptic partial differential equations (PDEs).
Filtered schemes for Hamilton-Jacobi equations were proposed, which allow us to construct
provably convergent, high order accurate finite difference schemes. For the 2-Hessian
equation in the three-dimensional case, we gave two different discretizations: a naive
one obtained by simply using standard finite differences and a monotone discretization.
The monotone discretization is provably convergent but less accurate due to the use of a
wide stencil that introduces a directional resolution error. As for the PDE that governs the
planar motion of level sets by affine curvature, we presented a convergent finite difference
discretization. A standard finite difference method was also considered, but computational
examples demonstrated its lack of stability.
Further extensions of the work can be explored. In the case of Hamilton-Jacobi equa-
tions, two fast solvers are available for monotone schemes: fast sweeping and fast marching.
In this thesis, fast sweeping solvers for the filtered schemes were proposed. A natural
extension is to develop a fast marching algorithm for filtered schemes for Hamilton-Jacobi
equations. As for the 2-Hessian equation, the schemes proposed in this thesis can be used
to build schemes for the prescribed scalar curvature of a three dimensional graph. The
reason for this is that the 2-Hessian equation is related to the scalar curvature, they are
equal up to a constant when the gradient of the function vanishes. Moreover, an extension
of the scheme for 2-Hessian equation in arbitrary dimensions can also be explored.
Finally, it is worth emphasizing that the filtered schemes are simple to implement, and
allow for an unrestricted choice of accurate schemes. It should also be clear that they
can be used for different PDEs and frameworks (e.g. discontinuous Garlekin), while still
retaining the advantages of accuracy, stability, and convergence to the viscosity solution
of the monotone schemes. However, there are still some challenges to be addressed. The
choice of the filter parameter has to be made carefully: currently, there is only a good
heuristic available. Moreover, the choice of the filter function is also worth exploring. In
the case of the Monge-Ampère equation and Hamilton-Jacobi equations, the choice was
motivated by the type of solver used, while for the affine curvature evolution a different
formulation was proposed, with the same underlying principle, which exhibits better
accuracy numerically.
137
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