NUMERICAL METHODS FOR TWO SECOND ORDER ELLIPTIC EQUATIONS by Brittany Dawn Froese B.Sc., Trinity Western University, 2007 a Thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in the Department of Mathematics c Brittany Dawn Froese 2009 SIMON FRASER UNIVERSITY Summer 2009 All rights reserved. However, in accordance with the Copyright Act of Canada, this work may be reproduced, without authorization, under the conditions for Fair Dealing. Therefore, limited reproduction of this work for the purposes of private study, research, criticism, review, and news reporting is likely to be in accordance with the law, particularly if cited appropriately.
All rights reserved. However, in accordance with the Copyright Act of Canada,
this work may be reproduced, without authorization, under the conditions for
Fair Dealing. Therefore, limited reproduction of this work for the purposes of
private study, research, criticism, review, and news reporting is likely to be
in accordance with the law, particularly if cited appropriately.
Name:
Degree:
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Examining Comrnittee: Dr. Ralf \\ '-ittenberg
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Date Approved: J u I y L J , 2 0 0 9
Last revision: Spring 09
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Abstract
Among the second order elliptic equations that arise frequently in science and engineering are
the Monge-Ampere equation and the non-divergence structure linear equation with rapidly
oscillating coefficients. Both of these PDEs are challenging from a numerical viewpoint: the
first due to its nonlinearity and the second due to the impracticality of properly resolving the
coefficients. In the first part of this thesis we construct finite difference schemes for the two-
dimensional Monge-Ampere equation. Numerical investigation indicates that these schemes
converge even in situations where smooth solutions do not exist. Secondly, we use formal
asymptotic techniques to recover an appropriate average of the non-divergence structure
operator. For several special cases we construct this operator in closed form. When this is
not possible, the averaged operator is obtained numerically. Numerical investigations are
consistent with the assertion that solutions of the original equation converge to solutions of
Second order elliptic equations arise frequently in areas such as science and engineering.
This thesis focuses on two such equations. The first is the Monge-Ampere equation, a fully
nonlinear PDE with important applications such as optimal mass transportation, classical
mechanics and meteorology [6, 16]. The second is the non-divergence structure linear equa-
tion with rapidly varying coefficients, which arises in the study of heterogeneous materials
and stochastic processes [30]. Both of these problems are challenging from a computa-
tional viewpoint—the Monge-Ampere equation because it is fully nonlinear and the highly-
oscillatory equations due to the impracticality of sufficiently resolving coefficients that vary
on a very small scale. The aim of this thesis is to construct practical numerical methods for
solving these equations. The main results presented in this thesis are also available in [3, 20].
1.1 The Monge-Ampere Equation
The Monge-Ampere equation is a fully nonlinear elliptic PDE which is closely related to im-
portant applications such as optimal mass transportation, classical mechanics and meteorol-
ogy [6, 16]. In this thesis we focus on the numerical solution of a 2-D Monge-Ampere equation
with a gentler non-linearity than in the above mentioned applications and simple Dirichlet
boundary conditions. The problem formulation isdet(D2u) = f x ∈ Ω
u = g x ∈ ∂Ω
u is convex.
(MA)
1
CHAPTER 1. INTRODUCTION 2
or in 2D
uxxuyy − u2xy = f in Ω. (1.1)
Here Ω ⊂ Rn is a bounded domain, ∂Ω is the boundary of Ω, D2u is the Hessian of u,
and f : Ω → R is a non-negative function. The solution u must be convex in order for
the equation to be elliptic. Without the convexity constraint, this equation does not have
a unique solution. (For example, in two dimensions with g = 0, if u is a solution then −uis also a solution.) Generally, without this convexity constraint, (MA) has two solutions
in two dimensions. The solution can also be understood in the weak viscosity sense. In
this case the solution is still convex but the gradient may be discontinuous across kinks
or creases or simply blow up at the boundary (this is when one of the eigenvalues of the
Hessian vanishes).
The nonlinearity and non-uniqueness of the solutions of Monge-Ampere make the solu-
tion of this equation a challenging problem. Until recently, little work had been done on
the numerical solution of the Monge-Ampere equation and even less on the convergence of
these methods.
1.1.1 Related Work
Oliker and Prussner [33] proposed a method that uses a discretisation based on the geometric
interpretation of the solutions [2]. In two dimensions this method converges to the geometric
solution.
Oberman [31] presented monotone wide-stencil finite difference methods for this problem.
This method converges to the viscosity solution of the equation with an error depending
on the angle discretisation of the sphere achieved by the stencil. Both of these latter two
methods involve techniques outside of the standard scientific computing methodology.
Dean and Glowinski [9, 10, 11, 12] have investigated Lagrangian and least squares meth-
ods for the numerical solution of Monge-Ampere. There are known examples where these
methods converge to something other than the solution. In fact, what is computed is the
projection of the solution into a space with more derivatives than the solution will usually
have.
Feng and Neilan [18, 17] proposed a vanishing moment method that uses a higher order
quasilinear PDE to approximate second order fully nonlinear PDEs such as the Monge-
Ampere equation.
CHAPTER 1. INTRODUCTION 3
Fulton [21] suggested the possibility of using multigrid methods to speed up the solution
of Monge-Ampere.
1.2 The Rapidly Varying Non-Divergence Structure Opera-
tor
Second order linear elliptic PDEs—in both divergence and non-divergence form—are another
class of equations that is important in science and engineering [30]. These equations carry
an additional challenge when the coefficients vary rapidly on a small spatial scale since it
is usually not practical or possible to resolve the computational domain enough to obtain
accurate solutions. One resolution to this challenge is to replace the original operator
with a different, slowly varying operator with solutions that are close to the solutions of the
original problem. When the equations contain singular terms (as in the divergence structure
case), this approach is known as homogenisation and the solution is obtained via the cell
problem. In the non-divergence stucture case studied herein, the macroscopic coefficients
are simply averages of the original terms against the invariant measure and the problem is
called averaging. (There is some inconsistency in the literature concerning terminology; we
follow [38].)
Let Lε denote the following non-degenerate elliptic differential operator
Lε = −n∑
i,j=1
aij(x/ε)∂2
∂xixj+
n∑i=1
bi(x/ε)∂
∂xi+ c(x/ε) (1.2)
Here we assume that the coefficients A = (a)ij , b = (b)i, and c are periodic of period one in
all their arguments with A ∈ C1(Rn) and b, c ∈ L∞(Rn). We study the problemLεuε = f(x), x ∈ Ω
uε = g(x), x ∈ ∂Ω(PDE)ε
in a bounded domain Ω ⊂ Rn with smooth boundary ∂Ω. Here for simplicity, f(x), g(x)
are continuous functions on Ω, ∂Ω, respectively.
Notation. We again use the notation
(D2)ij =∂2
∂xixj, ∇ = (∂x1, . . . , ∂xn)
CHAPTER 1. INTRODUCTION 4
for the Hessian operator and the gradient operator, respectively. We also use
M : N = tr(MTN) =n∑
i,j=1
MijNij
for the inner product between matrices and b · d =∑n
i=1 bidi for the dot product of vectors.
With this notation we can rewrite Lε defined in (1.2) as
Lε = −A(x/ε) : D2 + b(x/ε) · ∇+ c(x/ε)
1.2.1 Related Work
Homogenisation has received much attention in recent years. Introductions to homogenisa-
tion are found in the books [23, 38, 7] and the review paper [14].
The non-divergence structure equation has a probabilistic interpretation. Early work
by Freidlin [19] and Bensoussan, Lions and Papanicolaou [5] averaged the inhomogeneous,
non divergence-structure equation (PDE)ε by considering the limiting distribution of the
underlying stochastic differential equation.
There has been considerably more work on divergence structure equations, either in the
variational form
−∇ ·(A(x/ε)∇u
)= f (1.3)
or expanded into the singular form
−A(x/ε) : D2u+1εb(x/ε) · ∇u = f. (1.4)
We note that averaging the non-divergence structure equation (PDE)ε and the expanded
divergence structure equation (1.4) are qualitatively different problems due to the coefficient
ε−1 of the drift term b in the divergence structure problem.
Bensoussan, Lions and Papanicolaou [4], Papanicolaou [35], and Pardoux [37] have used
probabilistic techniques to homogenise the expanded form of the operator.
Keller [25] described a smoothing method, which involves applying a smoothing operator
to the equation in order to project the solution into some subspace. In the same paper, Keller
describes a two-space method that introduces an additional spatial variable that varies on
the small scale.
Knapek [26] presented a multigrid homogenisation method. This approach uses a Galerkin
approximation to construct the homogenised operator.
CHAPTER 1. INTRODUCTION 5
Hughes, Fiejoo, Mazzei, and Quincy [24] described the variational multiscale method.
In this approach, the desired solution is decomposed into a sum of two functions that vary
on different scales and a variational equation is obtained for the homogenised (“coarse”)
solution.
Owhadi and Zhang [34] presented a numerical approach that simplifies the solution
of (1.3) when this equation needs to be solved many times (with different right-hand sides).
This approach involves solving the homogeneous problem n times with different boundary
conditions and using the results to extract the most important small-scale information.
Desbrun, Donaldson, and Owhadi [13] suggested a geometric interpretation of the ho-
mogenisation problem. This approach relates the rapidly-varying coefficients to convex
functions in order to re-express the problem in terms of linear interpolation.
In many situations, the operators of interest vary randomly. Stochastic techniques can
be used to study the properties of such operators. This has been done by Kozlov [27],
Papanicolaou and Varadhan [36], and Dal Maso and Modica [8].
1.3 Overview of this Thesis
The first part of this thesis concentrates on the Monge-Ampere equation (MA). In Chapter 2
we motivate this problem with a brief review of the theory of optimal transport and its
relation to the Monge-Ampere equation. In Chapter 3 we use centred finite differences,
together with simple algebraic manipulations and the convex solution assumption, to build
iterative numerical methods for solving Monge-Ampere. In Chapter 4 we present a numerical
investigation of these methods; these computations indicate (though we do not prove this)
that the methods converge even for viscosity solutions where second derivatives do not exist.
The second part of this thesis focuses on the second order, non-divergence structure,
linear elliptic equation. The goal here is approximation of (PDE)ε for small ε using an
appropriate average of the rapidly varying operator (1.2). In Chapter 5 we provide a brief
background in stochastic theory in order to motivate the study of this problem and provide
intuition into the averaging formula. In Chapter 6 we use formal asymptotic calculations to
recover an averaged operator (first obtained by Freidlin [19] using probabilistic methods);
this formula involves averaging the operator against the invariant distribution for the under-
lying dynamics. In Chapter 7 we construct explicit representations of the averaged operator
for several special cases that require additional structure on the coefficients. In Chapter 8
CHAPTER 1. INTRODUCTION 6
we implement a finite difference method for computing the invariant distribution (and thus
the averaged operator); we compute several numerical examples that validate the method.
Chapter 2
Background in Optimal Transport
The elliptic Monge-Ampere equation arises in the context of optimal mass transportation. In
this chapter we briefly describe the problem of optimal transport and show how the Monge-
Ampere equation describes the solution of this problem with a quadratic cost function [1,
16, 40].
2.1 Monge-Kantorovich Mass Transport
The problem originally considered by Monge is how to transport a given pile of sand into
a hole with minimum cost, where the original cost is simply the magnitude of the distance
the sand is transported (Figure 2.1). That is, the problem is to find a mapping s(x) from
the original space X to the space Y that minimises the cost functional
I[s] =∫X
∣∣x− s(x)∣∣ dx. (2.1)
The more general Monge-Kantorovich problem describes the transportation of mass
densities using more general cost functions. That is, we want to find a mapping that takes
the density f(x) in the space X into the density g(y) in space Y . We denote the set of such
functions as the admissible set A. We are also given a cost function c(x, y), which gives the
cost of transporting a unit of mass from location x to location y. The problem is then to
find a mapping s(x) ∈ A that minimises the cost functional
I[s] =∫Xc(x, s(x))f(x) dx. (2.2)
7
CHAPTER 2. BACKGROUND IN OPTIMAL TRANSPORT 8
f(x)
X
g(y)
Y
y = s(x)
Figure 2.1: The mass transport problem.
CHAPTER 2. BACKGROUND IN OPTIMAL TRANSPORT 9
2.2 Conservation of Mass
It is useful to consider in more detail the requirement that the minimiser of the cost (2.2)
must push the density f(x) in X entirely onto the density g(y) in Y . Since this conserves
mass, the following equality must hold for any continuous function h(y).∫Xh(s(x))f(x) dx =
∫Yh(y)g(y) dy
By introducing the change of variables y = s(x) into the right-hand side of this equation we
obtain ∫Xh(s(x))f(x) dx =
∫Xh(s(x))g(s(x)) det(∇s(x)) dx.
Rearranged, this becomes∫X
(f(x)− g(s(x)) det(∇s(x))
)h(s(x)) dx
Again, this holds for every continuous function h(x). Consequently, we obtain the equation
det(∇s(x)) = f(x)/g(s(x)).
2.3 Cyclical Monotonicity
The simplest and most widely studied cost function is the quadratic cost function
c(x, y) =12|x− y|2 .
With this cost, the Monge-Kantorovich problem becomes
minimise∫X
12
∣∣x− s(x)∣∣2 f(x) dx
subject to det(∇s(x)
)= f(x)/g(s(x))
(2.3)
It turns out that a solution of this problem will be cyclically monotone. Intuitively, this
means that mass is not being “twisted.” To see why, we assume that a minimiser s(x) exists
and choose any finite number N ∈ N of distinct points xk ∈ X. Then we denote by Ek the
ball of radius rk centred at xk. Here the rk are chosen so that all of the balls are disjoint
and contain the same total mass ε. That is, for every 1 ≤ k ≤ N∫Ek
f(x) dx = ε. (2.4)
CHAPTER 2. BACKGROUND IN OPTIMAL TRANSPORT 10
We also define the points and regions that the xk, Ek are mapped onto by
yk = s(xk), Fk = s(Ek).
We observe that the new regions Fk also contain mass ε since the mapping s(x) conserves
mass. ∫Fk
g(y) dy =∫Ek
f(x) dx
= ε
We can now define a new mapping s′(x) by cyclically permuting the images of Ek and
leaving the remainder of the mapping s(x) unchanged (Figure 2.2).
s′(x) =
s(x+ xk+1 − xk) x ∈ Ek, 1 ≤ k < N
s(x+ x1 − xN ) x ∈ EN
s(x) x ∈ X\N⋃k=1
Ek
By design, this new mapping will also push the density f(x) entirely onto g(y).
We recall that s(x) is a minimiser of the cost functional in (2.3). This means that
I[s] ≤ I[s′].
Substituting in the quadratic cost we see that∫X
∣∣x− s(x)∣∣2 f(x) dx ≤
∫X
∣∣x− s′(x)∣∣2 f(x) dx.
Expanding the quadratic term , we obtain∫X
(∣∣s(x)∣∣2 − 2x · s(x)
)f(x) dx ≤
∫X
(∣∣s′(x)∣∣2 − 2x · s′(x)
)f(x) dx.
Since both s(x) and s′(x) push the density f(x) onto g(y) and since these two mappings are
identical over much of the domain, this simplifies to
N∑k=1
∫Ek
x · (s′(x)− s(x))f(x) dx ≤ 0.
CHAPTER 2. BACKGROUND IN OPTIMAL TRANSPORT 11
E3 F3
E2 F2
E1 F1
(a)
E3 F3
E2 F2
E1 F1
(b)
Figure 2.2: (a) A mapping that minimises (2.3). (b) A cyclical permutation of the minimiser.
CHAPTER 2. BACKGROUND IN OPTIMAL TRANSPORT 12
Dividing both sides by ε (that is, replacing the integrals by averages over the balls Ek) we
obtainN∑k=1
1ε
∫Ek
x · (s′(x)− s(x))f(x) dx ≤ 0.
In the limit as ε→ 0 this becomes
N∑k=1
xk · (yk+1 − yk) ≤ 0.
This is exactly the statement that the mapping s(x) is cyclically monotone.
2.4 Monge-Ampere Equation
The Monge-Ampere equation emerges from the Monge-Kantorovich mass transport problem
with quadratic cost function via a result proved by Rockafellar [39].
Theorem 2.1. Every cyclically monotone subset of Rn ×Rn lies in the subdifferential of a
convex mapping of Rn → R.
This means that the solution to the transport problem (2.3) can almost everywhere be
expressed as
s(x) = ∇u(x)
where u is a convex function [29]. Given the constraints on s(x) in (2.3), this convex function
must satisfy the Monge-Ampere equationdet(D2u(x)
)= f(x)/g(∇u(x)) x ∈ X
∇u : X → Y
u is convex.
(2.5)
Chapter 3
Numerical Methods for
Monge-Ampere
In this chapter we present two methods for numerically solving the two-dimensional Monge-
Ampere equation. The first is an explicit Gauss-Seidel iteration that is obtained by solving
a quadratic equation at each grid point and choosing the smaller root to ensure selection of
the convex solution. This method gives the value of the solution at each grid point in terms
of the solution values at the neighbouring points.
The second method is an algebraic method that involves the solution of a sequence of
linear equations. Here the positive square root is selected to enforce convexity.
Notation. We use∣∣D2u
∣∣ to denote the Frobenius norm of the Hessian of a function u∣∣∣D2u∣∣∣ =
√√√√ n∑i,j=1
u2xixj .
We use ∆u for the Laplacian of a function u
∆u =n∑i=1
uxixi .
3.1 An Explicit Finite Difference Method
The simplest idea is to begin by discretising the second derivatives in (1.1) using standard
centred differences on a uniform Cartesian grid:(ui+1,j + ui−1,j − 2uij
h2
)(ui,j+1 + ui,j−1 − 2uij
h2
)
13
CHAPTER 3. NUMERICAL METHODS FOR MONGE-AMPERE 14
−(ui+1,j+1 + ui−1,j−1 − ui−1,j+1 − ui+1,j−1
4h2
)2
= fij
Solving this quadratic equation for uij and choosing the smaller root in order to select
the convex solution, we obtain:
uij =d1 + d2
2−
√(d1 − d2
2
)2
+(d3 − d4
4
)2
+14fijh4 (Method 1)
where we introduce the notation
d1 = ui+1,j+ui−1,j
2 d2 = ui,j+1+ui,j−1
2
d3 = ui+1,j+1+ui−1,j−1
2 d4 = ui−1,j+1+ui+1,j−1
2 .
(3.1)
We can now use Gauss-Seidel iteration to find the fixed point of (Method 1). Dirichlet
boundary conditions are enforced at boundary grid points.
Remark. In the computations of Chapter 4, we perform the Gauss-Seidel iteration using a
lexicographical ordering. Other orderings are possible and may improve convergence and
allow for parallelisation of the method.
3.2 Improving Convexity
As explained in the convergence proof of the wide stencil schemes [31], the main obstacle
to monotonicity of the discrete scheme is the lack of convexity along directions other than
grid lines. Because we are looking for the convex solution of the Monge-Ampere equation,
the solution should satisfy
u(x) ≤ u(x+ h) + u(x− h)2
(3.2)
for all grid directions h. We check that this holds in some of the grid directions. This
convexity is partially built in to Method 1.
Lemma 3.1. The fixed point of Method 1 satisfies the inequalities (3.2) for the grid directions
h = (1, 0), (0, 1).
Proof. We assume without loss of generality that
ui,j+1 + ui,j−1
2≤ ui+1,j + ui−1,j
2.
CHAPTER 3. NUMERICAL METHODS FOR MONGE-AMPERE 15
In the notation of Equation (3.1) this reads
d2 ≤ d1.
Since f is non-negative,
uij ≤d1 + d2
2− d1 − d2
2= d2
=ui,j+1 + ui,j−1
2
≤ ui+1,j + ui−1,j
2.
From this lemma, we observe that solutions of Method 1 are neccessarily “convex” in
the x and y directions. Along the lines of [31] (where convexity along several directions
ensures convergence), we can also build more convexity requirements into our method. This
is accomplished by modifying Method 1 slightly:
uij = min
d1 + d2
2−
√(d1 − d2
2
)2
+(d3 − d4
4
)2
+14fijh4, d3, d4
. (Method 2)
Lemma 3.2. The fixed point of Method 2 satisfies the inequality (3.2) for the grid directions
h = (1, 0), (0, 1), (−1, 1), (1, 1).
Proof. The proof of the first part of this lemma is the same as the proof of the first part of
Lemma 3.1. The second half of this lemma is built directly into Method 2.
3.3 A Linear Iterative Method
The next method is based on a reformulation of the Monge-Ampere equation. We again use
the convexity requirement to select the correct square root.
Definition 3.1. First define the operator
T [u] = ∆−1(√
(∆u)2 + 2(f − det(D2u))).
This operator can be used to reformulate (MA) due to the following lemma.
CHAPTER 3. NUMERICAL METHODS FOR MONGE-AMPERE 16
Lemma 3.3. The convex solution of (MA) satisfies
u = T [u]. (Method 3)
Proof. Let v be the convex solution of (MA), which satisfies
f − det(D2v) = 0.
Inserting this into Definition 3.1 we obtain
T [v] = ∆−1(√
(∆v)2)
= ∆−1(|∆v|).
Since v is convex,
∆v > 0.
As a result,
T [v] = ∆−1(|∆v|)
= ∆−1(∆v)
= v.
Therefore, v is a fixed point of Method 3.
In this thesis we focus on the two dimensional case. With this in mind we rewrite the
operator T [u] in two dimensions.
Lemma 3.4. In R2, the operator T [u] defined in Definition 3.1 is equivalent to
T [u] = ∆−1
(√u2xx + u2
yy + 2u2xy + 2f
)= ∆−1
(√∣∣D2u∣∣2 + 2f
).
(3.3)
CHAPTER 3. NUMERICAL METHODS FOR MONGE-AMPERE 17
Proof. In R2, T [u] takes the form
T [u] = ∆−1(√
(∆u)2 + 2(f − det(D2u)))
= ∆−1
(√(uxx + uyy)2 + 2f − 2(uxxuyy − u2
xy))
= ∆−1
(√u2xx + u2
yy + 2uxxuyy + 2f − 2uxxuyy + 2u2xy))
= ∆−1
(√u2xx + u2
yy + 2u2xy + 2f
)= ∆−1
(√∣∣D2u∣∣2 + 2f
).
Method 3 consists in iterating un → un+1 = T [un] by solving
∆un+1 =√u2n,xx + u2
n,yy + 2u2n,xy + 2f.
with the prescribed Dirichlet boundary conditions.
We implemented Method 3 using a simple finite difference method. This involves dis-
cretising (3.3) using central differences (as with the first method) and iterating to find the
fixed point. In the computations of Chapter 4 we solved the resulting Poisson equation
using the MATLAB backslash operator.
3.4 Towards Contractivity of the Linear Iterative Map
In the numerical experiments of Chapter 4 we observe that Method 3 (the linear iteration)
converges very quickly when the solutions are smooth and the function f is strictly positive,
but is fairly slow when solutions are not smooth or f is very close to 0. In this section we
consider a one-dimensional version of Method 3 and prove that this mapping is a contraction
with a rate of convergence depending on how far f is from zero. We provide a similar result
for the two-dimensional case on a rectangle, although we do not have a complete proof
that Method 3 is a contraction mapping on a general domain. We begin with an observation
about the contractivity of the real valued function h(x) =√a2 + x2.
Lemma 3.5. The function
h(x) =√a2 + x2
is a strict contraction on the domain
|x| ≤ ka.
CHAPTER 3. NUMERICAL METHODS FOR MONGE-AMPERE 18
In other words, there exists a constant µk < 1 such that∣∣h(x1)− h(x2)∣∣ ≤ µk|x1 − x2|
for any x1, x2 in |x| ≤ ka.
Proof. ∣∣h′(x)∣∣ =
|x|√a2 + x2
≤ ka√a2 + k2a2
=k√
1 + k2
= µk < 1.
It follows that ∣∣h(x1)− h(x2)∣∣ ≤ µk|x1 − x2| .
Lemma 3.6. Let v be an exact, smooth solution of (1.1) and u a smooth function. Further
suppose that
f ≥ α > 0
is a strictly positive function. Then at every point in the domain∣∣∆(T [u]− T [v])∣∣ =
∣∣∣∣√2f +∣∣D2u
∣∣2 −√2f +∣∣D2v
∣∣2∣∣∣∣≤ µ
∣∣∣D2(u− v)∣∣∣
for some constant µ < 1.
Proof. Since u, v are smooth and f is strictly positive, there exists a constant k so that∣∣∣D2u∣∣∣ ,∣∣∣D2v
∣∣∣ ≤ k√2f.
It follows from Lemma 3.5 that∣∣∣∣√2f +∣∣D2u
∣∣2 −√2f +∣∣D2v
∣∣2∣∣∣∣ ≤ µk
∣∣∣∣ ∣∣∣D2u∣∣∣−∣∣∣D2v
∣∣∣ ∣∣∣∣≤ µk
∣∣∣D2(u− v)∣∣∣ ,
which completes the proof.
CHAPTER 3. NUMERICAL METHODS FOR MONGE-AMPERE 19
Remark. It is worth noting that as f → 0 or u becomes more and more non-smooth, the
constant k will increase so that µk increases and approaches 1.
Now we define the semi-norm
‖u‖L =∫
Ω(∆u)2 dx dy. (3.4)
Lemma 3.7. Let u(x, y) be a C2 function that vanishes on the boundary of a rectangle Ω.
Then ∫Ω
(∆u)2 dx dy =∫
Ω|D2u|2 dx dy.
Proof. Using repeated integration by parts we find that∫Ω
(∆u)2 dx dy =∫
Ω
(u2xx + u2
yy + 2uxxuyy)dx dy
=∫
Ω
(u2xx + u2
yy − 2uxuxyy)dx dy
=∫
Ω
(u2xx + u2
yy + 2uxyuxy)dx dy
=∫
Ω|D2u|2 dx dy.
Throughout this computation the boundary terms vanish since u is constant along the sides
of the rectangle (and thus at any point on the boundary either ux, uxx or uy, uyy vanish).
Theorem 3.1 (Contractivity on a Rectangle). The mapping T on a rectangular domain Ω
is a contraction in the semi-norm ‖u‖L.
Proof. Let u, v be any C2 functions that satisfy the Dirichlet boundary conditions associated
with (1.1). Compute
‖T (u)− T (v)‖L =∫
Ω[∆(T (u)− T (v))]2 dx dy
≤∫
Ωµ2∣∣∣D2(u− v)
∣∣∣2 dx dy,where the last step follows from Lemma 3.6. Since u and v are identical on ∂Ω, we can
apply Lemma 3.7 to obtain
‖T (u)− T (v)‖L ≤ µ2
∫Ω
[∆(u− v)]2 dx dy
= µ2‖u− v‖L.
Since µ2 < 1, this completes the proof.
CHAPTER 3. NUMERICAL METHODS FOR MONGE-AMPERE 20
Remark. Although we can only prove contractivity of the linear iterative map on a rectangle,
it may be possible to extend this result to a more general domain by include appropriate
boundary terms in the seminorm defined by (3.4).
We have already noted that for f close to zero or u with large second derivatives, the
constant µ will be close to 1. This suggests that the mapping T [u] will converge more slowly
in these situations, which is exactly what we observe in the computations of Chapter 4.
Chapter 4
Numerical Solutions of
Monge-Ampere
In this chapter we demonstrate the capabilities of our finite difference methods with a
number of numerical examples. The solutions are computed on an N ×N grid and iteration
is continued until the maximum difference between two subsequent iterates is less than
10−14. By convention, h is the spatial discretisation parameter h = L/N where L is the
length of one side of the square domain.
4.1 An Exact Radial Solution
An exact radial solution is
u(x, y) = exp
(x2 + y2
2
), f(x, y) = (1 + x2 + y2) exp(x2 + y2).
All finite difference methods converge to the same numerical solution, which is accurate to
second order in h (i.e. O(h2), see Figure 4.1(a)). Because of the radial symmetry of this
example, there is no difference between Method 1 and Method 2—both require exactly
the same number of iterations to converge. The extra constraint of Method 2 is, therefore,
unnecessary for this example. For this smooth example, Method 3 is by far the fastest
method; see Tables 4.1 and 4.2 and Figure 4.1(b).
Remark. In Figure 4.1(b), the CPU time required by Method 3 appears to increase sharply
for the largest values of N . This is most likely a byproduct of memory limitations and is
21
CHAPTER 4. NUMERICAL SOLUTIONS OF MONGE-AMPERE 22
not representative of the actual behaviour of Method 3.
(a) (b)
Figure 4.1: Convergence results for u(x, y) = exp(x2+y2
2
)on an N × N grid. (a) Error
versus N . (b) Total CPU time (seconds).
4.2 Noisy Data
We continue this first example, with exact solution
u(x, y) = exp
(x2 + y2
2
), f(x, y) = (1 + x2 + y2) exp(x2 + y2),
and study the effects of adding noise to the data (f and the boundary conditions). Results
are shown in Figure 4.2. Although the data is not convex, all methods find a solution that
is convex except at the boundary. The noise does not seem to have any effect on the rate
of convergence for Method 1 and Method 2. However, Method 3 becomes much slower
when dealing with the noisy, non-convex data; compare Tables 4.2 and 4.3.
CHAPTER 4. NUMERICAL SOLUTIONS OF MONGE-AMPERE 23
N∥∥∥u− u(N)
∥∥∥∞
21 2.1× 10−3
41 5.4× 10−4
61 2.4× 10−4
81 1.4× 10−4
101 8.6× 10−5
121 6.0× 10−5
141 4.4× 10−5
161 3.4× 10−5
181 2.7× 10−5
201 2.2× 10−5
221 1.8× 10−5
241 1.5× 10−5
Table 4.1: Errors for the exact solution u(x, y) = exp(x2+y2
2
)on an N ×N grid. Results
are the same for all methods.
N Iterations CPU Time (seconds)Method 1 Method 3 Method 1 Method 3
Next we consider the solution of Monge-Ampere when f = 1 with constant boundary values
1 on the square [−1, 1]× [−1, 1]. (MA) does not have smooth solutions for this data (since
uxx = 0 on part of the boundary, uyy → ∞). Method 1 and Method 3 converge to
a solution that is convex except near the corners of the domain (Figure 4.8(b)). This is
comparable to the solutions presented in [10]. Method 2 converges to a slightly different,
more convex, solution.
As no exact solution is available for comparison, we focus on the minimum value of the
solution produced by each method. Naturally, the solution obtained with Method 2 (which
enforces more convexity by computing the minimum of Method 1 and second directional
derivatives in some directions) is lower than the solution obtained with the other methods.
We also consider the minimum values obtained using the monotone scheme in [31]; see
Table 4.11. The solutions obtained with our methods are lower than the solutions obtained
with the monotone, wide-stencil scheme. The monotone scheme is known to converge to a
supersolution of (MA), so we do expect these values to be higher than the true values. This
is evident in the numerics as the solution obtained with the monotone method becomes lower
(and closer to the solutions obtained with our methods) as the stencil width is increased.
Although the finite difference implementation of Method 3 was quite fast for the smooth
example, it becomes much slower for this non-smooth example. It is also much slower than
Method 1 and Method 2, which require about the same number of iterations to converge
as they did for the smooth example. See Figure 4.9 and Table 4.12.
Remark. The time per iteration for the finite difference implementation of Method 3 is
the time to do a linear solve using the MATLAB backslash. We could instead save the LU
decomposition of the matrix or use an FFT based method to speed up each iteration.
CHAPTER 4. NUMERICAL SOLUTIONS OF MONGE-AMPERE 34
(a) (b)
Figure 4.8: Solutions when f = 1 with constant boundary values 1. (a) Surface plot ofsolution. (b) Method 1 and Method 3 produce a solution with slight negative curvaturealong the line y = x.
(a) (b)
Figure 4.9: Convergence results when f = 1 with constant boundary values 1 on an N ×Ngrid. (a) Total number of iterations required verus N . (b) Total CPU time (secons) versusN .
Table 4.11: Minimum value of u when f = 1 with constant boundary values 1 on an N ×Ngrid. We include results from the wide stencil schemes of [31] on nine, seventeen, andthirty-three point stencils.
N Iterations CPU Time (seconds)Method 1 Method 2 Method 3 Method 1 Method 2 Method 3
Table 4.12: Computation times when f = 1 with constant boundary values 1 on an N ×Ngrid.
Chapter 5
Background in Stochastic Theory
The non-divergence structure elliptic operator arises in stochastic processes. Understanding
the stochastic interpretation of the operator can provide insight into the PDE. As it turns
out, the invariant distribution that arises in this context plays an important role in the
method we use to average the non-divergence structure operator.
In this chapter we briefly describe the probabilistic context of the non-divergence struc-
ture operator in order to provide intuition into the use of the invariant distribution to
average the rapidly varying operator. The intent of this chapter is not to give rigorous ar-
guments, but rather to provide some motivation and intuition into the averaging problem.
However, the arguments used here can be made rigorous using semigroup theory and Ito’s
calculus [32, 38].
5.1 Finite State, Discrete Time Processes
For simplicity, we begin by considering finite state, discrete time Markov processes. These
are stochastic processes that “forget” the past. To be concrete, we suppose that there are
n possible states associated with the process z. In this case, the Markov process can be
represented by an n× n transition matrix P with entries giving the probabilities of ending
up in each state j at a later time t+ ∆t given the situation at the current time t.
pij(∆t) = P(z(t+ ∆t) = j|z(t) = i
)(5.1)
As an example of a finite state Markov process, we can consider a symmetric random
36
CHAPTER 5. BACKGROUND IN STOCHASTIC THEORY 37
walk on a lattice (Figure 5.1(a)). The associated transition matrix has entries
pij(∆t) =
14 if i, j are neighbours
0 otherwise(5.2)
Another example is an asymmetric random walk that prefers vertical movement to horizontal
movement (Figure 5.1(b)). For example:
pij(∆t) =
16 if i, j are horizontal neighbours
13 if i, j are vertical neighbours
0 otherwise
(5.3)
The transition matrix satisfies a few important properties:
pij ≥ 0 ∀ij∑jpij = 1 ∀i
P1 = 1
P (t+ ∆t) = P (t)P (∆t)
(5.4)
Here we denote the vector whose entries are all equal to 1 by 1 =(
1 1 . . . 1)T
.
We also define the generator of a Markov chain:
L(∆t) =P (∆t)− I
∆t(5.5)
At this point, a relationship between finite-state Markov process and second order elliptic
PDEs is already emerging. We note that the generator associated with the symmetric
random walk (5.2) is identical to the finite difference matrix for the laplacian operator while
the generator describing the asymmetric walk (5.3) is the finite difference matrix for the
operator ∂xx + 2∂yy.
From the required properties of a transition matrix (5.4), we can deduce several prop-
erties of the generator.
lij ≥ 0 ∀i 6= j
li,i < 0 ∀i∑jlij = 0 ∀i
L1 = 0
CHAPTER 5. BACKGROUND IN STOCHASTIC THEORY 38
(a)
(b)
Figure 5.1: (a) Three symmetric random walks. (b) Three asymmetric random walks.
CHAPTER 5. BACKGROUND IN STOCHASTIC THEORY 39
5.2 Finite State, Continuous Time Processes
It is now straightforward to consider the continuous time limit of the finite state processes
of §5.1. In this case, the generator defined by (5.5) becomes
L = lim∆t→0
L(∆t)
= lim∆t→0
P (∆t)− I∆t
= lim∆t→0
P (∆t)− P (0)∆t
=dP
dt
∣∣∣∣t=0
.
(5.6)
We can also derive a useful rule for the time evolution of the transition matrix, which is
known as the master equation of the Markov chain.
Lemma 5.1. In the continuous time limit, the transition matrix defined by (5.1) evolves
according to the ODE dPdt = PL = LP
P (0) = I(5.7)
Proof. Equation (5.7) really states three things.
1. First we show thatdP
dt= PL.
We begin by considering component (i, j) of the transition matrix.
dpijdt
= lim∆t→0
pij(t+ ∆t)− pij(t)∆t
= lim∆t→0
1∆t
∑k
pik(t)pkj(∆t)− pij(t)
= lim
∆t→0
1∆t
∑k 6=j
pik(t)pkj(∆t) + pij(t)(pjj(∆t)− 1)
= lim
∆t→0
∑k 6=j
pik(t)lkj(∆t) + pij(t)ljj(∆t)
=∑k
pik(t)lkj
CHAPTER 5. BACKGROUND IN STOCHASTIC THEORY 40
It follows immediately thatdP
dt= PL.
2. We similarily show thatdP
dt= LP
by demonstrating that P and L commute.
PL = lim∆t→0
P (∆t)L(∆t)
= lim∆t→0
1∆t
P (∆t)(P (∆t)− I
)= lim
∆t→0
1∆t(P (∆t)− I
)P (∆t)
= LP
3. Finally, we require the initial condition P (0) = I. This follows immediately from the
definition of the transition matrix in (5.1).
The master equation is useful because it enables us to derive equations for the evolution
of expectation values and probability densities on the state space. We begin by considering
the expectation value of a function φ on the state space. We define the n-vector v as the
expected value of φ at time t given that the process starts in the ith state.
vi(t) = E(φ(z(t))|z(0) = i
)(5.8)
Lemma 5.2 (Backward Kolmogorov Equation). The vector v of expectation values evolves
according to the backward Kolmogorov equation.dvdt = Lv
v(0) = φ(5.9)
Proof. The ith component of the expectation vector can be expressed as
vi(t) =∑j
pij(t)φj .
In other words,
v(t) = P (t)φ.
CHAPTER 5. BACKGROUND IN STOCHASTIC THEORY 41
Taking the time derivative and making use of the master equation (5.7), we obtain
dv
dt=dP
dtφ
= LPφ
= Lv.
We are also interested in the probability density vector ρ(t).
Lemma 5.3 (Fokker-Planck Equation). The probability density vector evolves according to
the Fokker-Planck equation. dρdt = LTρ
ρ(0) = ρ0(5.10)
Proof. We consider the ith component of the probability density vector at time t:
ρi(t) =∑j
ρ0jpji(t).
The equation for the full vector is then
ρ(t) = P (t)Tρ0.
Differentiating with respect to time and applying the master equation (5.7), we find that
dρ
dt=dP T
dtρ0
= (PL)Tρ0
= LT(P Tρ0
)= LTρ.
We have already observed that the vector 1 is in the nullspace of the generator L and is
an eigenvector of the transition matrix P (with corresponding eigenvalue 1). A consequence
of this is that the nullspace of LT is also non-empty. That is, there is a non-zero vector ρ∞
satisfying
LTρ∞ = 0.
From the definition of L in (5.5), we deduce that ρ∞ is also an eigenvector of P T .
P Tρ∞ = ρ∞
CHAPTER 5. BACKGROUND IN STOCHASTIC THEORY 42
In other words, ρ∞ is a fixed point of the Fokker-Planck equation (5.10). This fixed point
is known as the invariant distribution.
At this point, the concept of ergodicity is important in ensuring the uniqueness of the
(normalised) invariant distribution. A Markov process is said to be ergodic if the transition
matrix P has a simple eigenvalue at 1, with the remainder of the spectrum lying strictly
inside the unit circle. This means that the probability distribution will always approach the
invariant distribution ρ∞ after a long time. That is, the process “forgets” its initial state.
In this case, the long time average of a function of the state space is simply given by the
average of the function against the invariant distribution [38, Theorem 5.15].
limT→∞
1T
∫ T
0φ(z(t)) dt = (ρ∞, φ) (5.11)
We have already observed that finite difference matrices for certain elliptic operators
can also play the role of the generator L of a Markov chain. In this case, the homogeneous
problem for the elliptic operator can be solved numerically by computing the steady states
of the backward Kolmogorov equation (5.9).
dv
dt= Lv = 0
For the rapidly varying operator, we can imagine that the system has a large number of
states and that the matrix L is dependent on the state of the system. For example we could
divide the domain Ω into two regions (Ω1,Ω2), define L1 and L2 as the finite difference
matrix of two constant coefficient elliptic operators, and define the generator by
L(x) =
L1 x ∈ Ω1
L2 x ∈ Ω2.
We would like to be able to replace this rapidly variable operator with some kind of
averaged operator. We recall that the time average of a function of the state space is simply
the average against the invariant distribution. Since we are interested in the long-time
behaviour of the generator (i.e. the steady states of Equation (5.9)), the idea is to compute
the time average of the generator matrix L in the same way to obtain
L = (L, ρ∞) . (5.12)
In fact, this is exactly how we will numerically average the elliptic operators in Chapter 8.
CHAPTER 5. BACKGROUND IN STOCHASTIC THEORY 43
5.3 Continuous Processes
The above analysis can also be extended to the infinite-state problem using Ito’s calculus.
In this section we provide an outline of the important results. These ideas can be made
rigorous using more sophisticated analysis [32, 38].
5.3.1 The Stochastic Process
In this case the relevant stochastic process takes the formdX(t) = b(X(t))dt+ γ(X(t))dW (t)
X(0) = x(5.13)
where b is known as the drift, γ is the diffusion coefficient, and W is Brownian motion. This
is known as an Ito process and can also be written in the form
X(t) =∫ t
0b(X(s)) ds+
∫ t
0γ(X(s)) dW (s).
The key to relating this stochastic process to the non-divergence structure PDE is Ito’s
Formula [32, Theorem 4.2.1].
Theorem 5.1 (Ito’s Formula). Let
dX(t) = b(X(t))dt+ γ(X(t))dW (t)
be an n-dimensional Ito process and let g ∈ C2(Rn × [0,∞]→ Rp
). Then dg(X(t), t) is also
an Ito process satsifying
dgk =∂gk∂t
+∑i
∂gk∂xi
dXi +12
∑ij
∂2gk∂xi∂xj
dXidXj
where dWidWj = δijdt
dWidt = dtdWi = 0.
CHAPTER 5. BACKGROUND IN STOCHASTIC THEORY 44
5.3.2 The Generator
Just as in the finite state case, we can define a generator of the stochastic process. Now the
generator is defined as the initial time derivative of the expectation of a function:
Lφ = limt→0
E(φ(X(t)) : X(0) = x)− φ(x)t
=dE(φ(X(t)) : X(0) = x)
dt
∣∣∣∣t=0
.(5.14)
At this point, the link between Ito processes and the non-divergence structure elliptic oper-
ator begins to appear, as the following theorem explains.
Theorem 5.2 (The Generator). The generator of the SDE (5.13) is given by
L = b · ∇+12
(γγT
): D2. (5.15)
Proof. We begin by applying Ito’s Formula (Theorem 5.1) to the stochastic process φ(X(t))
where X(t) is governed by the SDE (5.13).
dφ =∑i
∂φ
∂xidXi +
12
∑i,j
∂2φ
∂xi∂xjdXidXj
=∑i
∂φ
∂xi
(bidt+ (γdW )i
)+
12
∑i,j
∂2φ
∂xixj(γdW )i(γdW )j
Again using Ito’s Formula, we can see that
(γdW )i(γdW )j =
∑k
γik dWk
∑k
γjk dWk
=∑k
γikγjk dt
=(γγT
)ijdt.
We combine these results to obtain
dφ =
∑i
bi∂φ
∂xi+
12
∑i,j
(γγT
)ij
∂2φ
∂xixj
dt+∑i,k
γik∂φ
dxidWk.
This can be rewritten in the integral form
φ(X(t)) = φ(x) +∫ t
0
∑i
bi∂φ
∂xi+
12
∑i,j
(γγT
)ij
∂2φ
∂xixj
ds+∫ t
0
∑i,k
γik∂φ
dxidWk.
CHAPTER 5. BACKGROUND IN STOCHASTIC THEORY 45
Next we compute the expected value of both sides of this equation.
E(φ(X(t)) : X(0) = x) =
φ(x) + E
∫ t
0
∑i
bi∂φ
∂xi+
12
∑i,j
(γγT
)ij
∂2φ
∂xixj
ds : X(0) = x
+∑i,k
E
(∫ t
0γik
∂φ
∂xidWk : X(0) = x
)
It is possible to show [32, §7.3] using more sophisticated analysis that if g(X) is a bounded
Borel function then
E
(∫ t
0g(X(s)) dW (s) : X(0) = x
)= 0.
Combining this with the previous results, we find that
E(φ(X(t)) : X(0) = x) =
φ(x) + E
∫ t
0
∑i
bi∂φ
∂xi+
12
∑i,j
(γγT
)ij
∂2φ
∂xixj
ds : X(0) = x
.
CHAPTER 5. BACKGROUND IN STOCHASTIC THEORY 46
We can substitute this result into the definition of the generator (5.14) to obtain:
Lφ(x) = limt→0
E(φ(X(t)) : X(0) = x)− φ(x)t
= limt→0
1tE
∫ t
0
∑i
bi(X(s))∂φ
∂xi(X(s))
+12
∑i,j
(γγT
)ij
(X(s))∂2φ
∂xixj(X(s))
ds : X(0) = x
= E
limt→0
1t
∫ t
0
∑i
bi(X(s))∂φ
∂xi(X(s))
+12
∑i,j
(γγT
)ij
(X(s))∂2φ
∂xixj(X(s))
ds : X(0) = x
= E
∑i
bi(X(0))∂φ
∂xi(X(0)) +
12
∑i,j
(γγT
)ij
(X(0))∂2φ
∂xixj(X(0)) : X(0) = x
=∑i
bi(x)∂φ
∂xi(x) +
12
∑i,j
(γγT
)ij
(x)∂2φ
∂xi∂xj(x)
= b · ∇φ+12
(γγT
): D2φ.
5.3.3 Backward Kolmogorov Equation
As in the finite state case, we can define the expected value of a function φ on the state
space.
v(x, t) = E(φ(X(t)) : X(0) = x
)(5.16)
Using Ito’s formula, it is possible to prove a continuous analogue of Lemma 5.2.
Lemma 5.4 (Backward Kolmogorov Equation). The expected value of a function φ on the
state space evolves according to the backward Kolmogorov equation:dvdt = Lv
v(0) = φ.(5.17)
CHAPTER 5. BACKGROUND IN STOCHASTIC THEORY 47
Proof. Fix t > 0 and use the definitions of the generator (5.14) and the expectation function
v (5.16) to compute
Lv(x, t) = lims→0
E(v(X(s), t) : X(0) = x)− v(x, t)s
= lims→0
E(E(φ(X(t)) : X(0) = X(s)
): X(0) = x
)− E
(φ(X(t)) : X(0) = x
)s
= lims→0
E(φ(X(t+ s)) : X(0) = x
)− E
(φ(X(t)) : X(0) = x
)s
= limx→0
v(x, t+ s)− v(x, t)s
=∂v(x, t)∂t
The initial condition holds because, from the definition of v,
v(x, 0) = E(φ(X(0)) : X(0) = x
)= φ(x).
We can express the solution of Equation (5.17) in terms of the exponential of the gen-
erator L:
v(z, t) =(eLtφ
)(z). (5.18)
5.3.4 Fokker-Planck Equation
We are also interested in the probability distribution function ρ(t), which evolves in a similar
way as in the finite-state case.
Lemma 5.5 (Fokker-Planck Equation). The probability distribution function of the stochas-
tic process described by (5.13) evolves according to the Fokker-Planck equation.dρdt = L∗ρ
ρ(0) = ρ0.(5.19)
Proof. We begin by recalling Equation (5.16) for the expected value of a function φ for a
process beginning in the state x.
v(x, t) = E(φ(X(t)) : X(0) = x
)
CHAPTER 5. BACKGROUND IN STOCHASTIC THEORY 48
We now compute the expected value of this function over all possible initial states x.
E(φ(X(t))
)=∫
Ωv(X, t)ρ0(X) dX
=∫
Ω
(eLtφ
)(x)ρ0(x) dx
=∫
Ω
(eL
∗tρ0)
(x)φ(x) dx
Alternatively, we can compute this expected value as
E(φ(X(t))
)=∫
Ωφ(x)ρ(x, t) dx.
Subtracting these two expressions for the expected value of φ we find that∫Ω
(eL
∗tρ0 − ρ(t))φdx
This expression holds for every smooth function φ on the state space Ω. As a result, we
conclude that
ρ(t) =(eL
∗t)ρ0.
This is equivalent to saying that ρ satisfies the adjoint (Fokker-Planck) equation.
As before, the concept of ergodicity is important. We say that an SDE is ergodic if
the kernel of its generator L consists only of constants. This is in fact the case for the
SDE (5.13), whose generator is the non-divergence structure elliptic operator. In other
words, the generator L has a one-dimensional nullspace. By the Fredholm Alternative (Ap-
pendix A), the adjoint operator L∗ must also have a one-dimensional nullspace. This means
that the Fokker-Planck equation (5.19) has a unique (normalised) steady state solution ρ∞,
which we again define as the invariant distribution.
As in the finite-state process, the probability distribution of the ergodic process “for-
gets” its initial state and long-time averages can again be replaced by averages against the
invariant distribution:
limT→∞
1T
∫ T
0φ(X(t)) dt = 〈ρ∞φ〉 . (5.20)
For the homogeneous elliptic problem, we are really interested in the long-time behaviour
of the generator L. If L varies rapidly on the state space, we would like to replace L with
its long-time average
L = 〈Lρ∞〉 (5.21)
CHAPTER 5. BACKGROUND IN STOCHASTIC THEORY 49
in the hope that this will not significantly affect the solutions of the steady-state backward
Kolmogorov equation. That is, the hope is that solutions of
Lu = 0
will be close to solutions of
Lu = 0.
This intuitive idea for “averaging” elliptic operators will be justified in the formal asymp-
totic calculations of Chapter 6. As it turns out, we can use this approach to average a
non-divergence structure operator on a periodic domain and use this averaged operator to
(approximately) solve inhomogeneous equations with Dirichlet boundary conditions.
Chapter 6
The Averaging Formula
In this chapter we use formal asymptotic calculations and PDE methods to average the
operator (1.2). When the solutions are sufficiently smooth, we can prove that solutions
of the non-divergence structure equation converge to solutions of the averaged equation.
Another option for making the formal calculation rigorous is to use the two-scale method
as in [38, Chapter 19].
Notation. Use angle brackets to denote the average over one periodic domain U = [0, 1]n,
〈v〉 =
∫U v
|U |
Denote the kernel of an operator L by N(L).
Remark. Due to the periodicity assumption on the domain U = [0, 1]n, we can also think of
this domain as the unit torus.
6.1 The Solvability Condition
Before we proceed with a derivation of the averaged coefficients, we require a solvability
condition for the periodic elliptic equations we are considering.
Consider the operator
L0 = −A : D2 (6.1)
where the coefficient matrix is periodic, C1, and uniformly positive definite. We will show
that the adjoint of L0 is given by
L∗0v = −D2 : (Av), (6.2)
50
CHAPTER 6. THE AVERAGING FORMULA 51
also equipped with periodic boundary conditions. We define the invariant distribution ρ∞
as the kernel of the adjoint operator L∗0. This can be made unique by enforcing the normal-
isation condition 〈ρ∞〉 = 1. L∗0ρ
∞ = −D2y : (Aρ∞) = 0 in U
ρ∞ is periodic
〈ρ∞〉 = 1
(6.3)
In this case we can restate the solvability condition of Theorem A.1 as follows.
Theorem 6.1 (The Solvability Condition). Consider the operator (6.1) with a coefficient
matrix that is C1, periodic, and uniformly positive definite. Then the inhomogeneous equa-
tion
L0u = f
has a solution if and only if the invariant distribution ρ∞ corresponding to L0 is orthogonal
to the forcing f :
〈ρ∞f〉 = 0. (6.4)
6.1.1 The Adjoint Operator
It is a simple matter to verify that L∗0 is in fact the adjoint of the operator L0. We do this
using integration by parts.
(L0u, v) = −∫U
(A : D2yu)T v dy
= −∫U
∑i,j
aijuyiyjv dy
=∫U
∑i,j
(aijv
)yiuyj dy
= −∫U
∑i,j
(ai,v)yiyj
u dy
= −∫UuT(D2y : (Av)
)dy
= (u, L∗0v)
Throughout this calculation the boundary terms vanish due to periodicity.
CHAPTER 6. THE AVERAGING FORMULA 52
6.1.2 Uniqueness of the Invariant Distribution
In order to properly apply the solvability condition of Theorem A.1 we need to know the
dimension of the nullspace of the adjoint operator (6.2). We begin by considering the kernel
of the original operator (6.1). Clearly, constants are included in the kernel of this operator.
In fact, these are the only elements in the kernel, as the following lemma asserts.
Lemma 6.1 (One-Dimensional Kernel). Consider the non-divergence structure elliptic oper-
ator (6.1) on the periodic domain U and let u ∈ C2 belong to the nullspace of this operator.
Then u is a constant.
Proof. Let u ∈ C2 be any element of the kernel of (6.1), choose any y0 ∈ U and for
sufficiently small δ > 0 define the domain
Uδ = U\B(y0, δ).
Then u solves the Dirichlet problemL0v = 0 in Uδ
v = u on ∂B(y0, δ).
By the weak maximum principle [15, §6.4.1] we see that
min∂B(y0,δ)
u = minUδ
u
≤ maxUδ
u
= max∂B(y0,δ)
u.
Since u is continuous, in the limit as δ → 0 we have
u(y0) = minUu
≤ maxU
u
= u(y0).
Therefore u is constant in U .
Since the operator (6.1) has a one-dimensional nullspace, the adjoint operator (6.2) will
also have a one-dimensional nullspace by the Fredholm alternative (Theorem A.4). Thus the
normalised invariant distribution will be uniquely prescribed. Combined with Theorem A.1,
this completes the proof of Theorem 6.1.
CHAPTER 6. THE AVERAGING FORMULA 53
6.2 The Averaged Coefficients
We now have the tools required to formally derive the averaging formula for the non-
divergence structure equation (PDE)ε, which we recall here.−A(x/ε) : D2uε(x) + b(x/ε) · ∇uε(x) + c(x/ε)uε(x) = f(x) x ∈ Ω
uε(x) = g(x) x ∈ ∂Ω
Theorem 6.2. Let the operator Lε be as in (1.2). Let ρ∞ be the invariant distribution,
given by the unique solution of (6.3). Formally, in the limit ε → 0, solutions of the equa-
tion (PDE)ε converge to solutions of the averaged equationLu = f(x), x ∈ Ω
u = g(x) x ∈ ∂Ω(PDE)
where
L = −A : D2 + b · ∇+ c
has coefficients given by averaging the original coefficients against the invariant distribution
A = 〈ρ∞A〉, b = 〈ρ∞b〉, c = 〈ρ∞c〉. (6.5)
In addition, formally,
uε(x) = u0(x) +O(ε2). (6.6)
Proof. We begin the analysis by introducing a new variable
y = x/ε
where ε is the size of the small cells on which the coefficients vary. We assume the two scales
x and y are independent and write the coefficients as
A(x/ε) = A(y), b(x/ε) = b(y), c(x/ε) = c(y).
Since we are treating the two spatial scales as independent, we rewrite the partial derivative
∂xi as
∂xi +1ε∂yi
CHAPTER 6. THE AVERAGING FORMULA 54
The equation (PDE)ε then becomes
− 1ε2A : D2
yuε
+1ε
(−2A : ∇x∇Ty uε + b · ∇yuε
)+ (−A : D2
xuε + b · ∇xuε + cuε) = f.
(6.7)
Now we look for a solution of the form
uε(x) = u0(x, x/ε) + ε2u2(x, x/ε) + . . .
= u0(x, y) + ε2u2(x, y) + . . .(6.8)
Substituting this solution form into the PDE (6.7) we obtain
1ε2L0u0 +
1εL1u0 + (L0u2 + L2u0) +O(ε) = f (6.9)
Here we have defined the operators
L0 = −A : D2y
L1 = −2A : ∇x∇Ty + b · ∇y
L2 = −A : D2x + b · ∇x + c.
To ensure that uε satisfies the required boundary conditions we also enforce the appropriate
boundary conditions on the terms in the asymptotic expansion (6.8). For the leading order
term we require u0 = g(x) on ∂Ω
u0 periodic in y.
For k > 0 we enforce uk = 0 on ∂Ω
uk periodic in y.
We first look at the leading order, O( 1ε2
), terms in (6.9) to obtain the equation−A : D2yu0 = 0
u0 is periodic in y.
CHAPTER 6. THE AVERAGING FORMULA 55
Because of the periodic boundary conditions (in y), the solution of this equation will be
independent of the y variable:
u0 = u0(x).
Next we consider the O(1ε ) terms in (6.9) to obtain
−2A : ∇x∇Ty u0 + b · ∇yu0 = 0.
This is automatically satisfied since u0 is independent of y.
Finally, we consider the O(1) terms in (6.9) to obtain the equation
−A : D2yu2 = A : D2
xu0 − b · ∇xu0 − cu0 + f. (6.10)
We can now use the Theorem 6.1 to obtain a solvability condition for (6.10):∫U
ρ∞(A : D2
xu0 − b · ∇xu0 − cu0 + f)dy = 0.
Because u0 and f are independent of the y variable and the solution of the adjoint problem
is normalised, we can simplify the last equation to obtain
− A : D2xu0 + b · ∇xu0 + cu0 = f (6.11)
where the averaged coefficients are given by (6.5)
A = 〈ρ∞A〉, b = 〈ρ∞b〉, c = 〈ρ∞c〉.
Returning now to the asymptotic expansion for uε (6.8), we see that (6.6) holds, where u0
satisfies the averaged equation (6.11).
6.3 Ellipticity of the Averaged Operator
Since the original operator Lε is uniformly elliptic, the homogenised operator L will also be
uniformly elliptic. We first note that the uniform ellipticity of Lε means that the coefficient
matrix A is uniformly positive definite. That is, there exists a constant α so that for any
ξ ∈ Rn
ξTAξ ≥ α|ξ|2 .
CHAPTER 6. THE AVERAGING FORMULA 56
A simple calculation shows that the same property hold for the homogenised coefficient
matrix.
ξT Aξ =∑i,j
aijξiξj
=∑i,j
(∫Uρ∞aij dy
)ξiξj
=∫Uρ∞
∑i,j
aijξiξj
dy
≥∫Uρ∞α|ξ|2 dy
= α|ξ|2
Here we have used the fact that ρ∞ is a positive, normalised function and that the original
coefficient matrix A is uniformly positive definite.
6.4 Convergence of Smooth Solutions
When the rapidly varying equation (PDE)ε, the averaged equation (PDE), and the first
correction equation (6.10) have smooth (C2) solutions, we can use the maximum principle
to provide a simple proof that the solutions of the rapidly varying equation converge to
solutions of the averaged equation as ε→ 0. This proof is along the lines of the proof given
by Bensoussan, Lions and Papanicolaou [5, §2.4] for the divergence structure problem.
Theorem 6.3 (Convergence of Smooth Solutions). Let A be C1, periodic and uniformly
elliptic. Let b, c belong to L∞ and suppose that at least one of the following conditions holds:
1. c ≤ 0 in Ω.
2. The domain Ω is sufficiently narrow (see [22, Corollary 3.8] for details).
Let uε(x) be a C2 solution of (PDE)εLεuε = f(x), x ∈ Ω
uε = g(x), x ∈ ∂Ω
CHAPTER 6. THE AVERAGING FORMULA 57
and u0(x) a C2 solution of the averaged equation (PDE)Lu = f(x), x ∈ Ω
u = g(x) x ∈ ∂Ω.
Recalling that the definition of u0 guarantees (via the Fredholm alternative) the existence of
a y-periodic solution u2(x, y) to the equation for the first correction term (6.10)
−A : D2yu2 = −Lεu0 + f,
we further assume that this equation has a C2 solution (boundary conditions do not matter).
Then there exists a constant K such that for any ε > 0 and any point x ∈ Ω,∣∣uε(x)− u0(x)∣∣ ≤ Kε.
Proof. We begin by defining the C2 function
vε(x) = uε(x)− u0(x)− ε2u2(x, x/ε).
Now we make use of the definition of Lε (1.2) to compute
Lεvε(x) = Lεuε(x)− Lεu0(x)− ε2Lεu2(x, x/ε)
= f(x)− Lεu0(x)− ε2(−A : D2
xu2(x, x/ε) + b · ∇xu2(x, x/ε) + cu2(x, x/ε))
= f(x)− Lεu0(x)
− ε2(−A : D2
xu2(x, y)− 2εA : ∇x∇Ty u2(x, y)− 1
ε2A : D2
yu2(x, y)
+ b · ∇xu2(x, y) +1εb · ∇yu2(x, y) + cu2(x, y)
)∣∣∣∣∣y=x/ε
.
From the definition of u2(x, y), we can simplify this expression as follows.
Lεvε(x) = f(x)− Lεu0(x)
− ε2(−A : D2
xu2(x, y)− 2εA : ∇x∇Ty u2(x, y) +
1ε2(−Lεu0(x) + f(x)
)+ b · ∇xu2(x, y) +
1εb · ∇yu2(x, y) + cu2(x, y)
)∣∣∣∣∣y=x/ε
= ε(
2A : ∇x∇Ty u2(x, y)− b · ∇yu2(x/ε))∣∣∣∣y=x/ε
+ ε2(A : D2
xu2(x, y)− b · ∇xu2(x, y)− cu2(x, y))∣∣∣∣y=x/ε
CHAPTER 6. THE AVERAGING FORMULA 58
Since u2 is C2 and A, b, c are bounded, we can find a constant C1 (independent of ε) so that∣∣Lεvε(x)∣∣ ≤ C1ε x ∈ Ω.
Additionally, since uε and u0 are identical on the boundary ∂Ω, we have∣∣vε(x)∣∣ = ε2
∣∣u2(x, x/ε)∣∣ ≤ ε2‖u2(x, y)‖∞ x ∈ ∂Ω.
From the maximum principle (see [22, Corollary 3.8]), we conclude that there is a constant
C such that
|vε| ≤ Cε x ∈ Ω.
Recalling the definition of vε, we see that for x ∈ Ω,∣∣uε(x)− u0(x)∣∣ =∣∣∣vε(x) + ε2u2(x, x/ε)
∣∣∣≤ ε
(C + ‖u2(x, y)‖∞
)Remark. In addition to proving convergence of smooth solutions, this proof indicates a rate
of convergence of (at least) O(ε). In the numerical experiments of Chapter 8 we actually
observe faster, O(ε2), convergence.
Chapter 7
Explicit Solutions
In this chapter we consider a number of special cases for which we can solve the adjoint
problem (6.3) exactly. In these cases we can generate explicit formulas for the averaged
operators.
We provide a visual representation of several of these operators by plotting the ellipses
given by the equation yTAy = 1. The shading of these pictures is such that lighter colouring
corresponds to larger entries in the coefficient matrix.
Remark. Since the lower order terms average in the same way as the coefficient matrix,
but do not affect the formula for the invariant distribution, we omit them to shorten the
exposition.
7.1 Multiples of a Constant Coefficient Matrix
One of the simplest cases involves a coefficient matrix that is a (variable) scalar multiple of
a constant matrix
A(x/ε) = a(x/ε)B,
which yields the harmonic mean
A =⟨a(y)−1
⟩−1B. (7.1)
A typical example is illustrated in Figure 7.1, which is the piecewise constant periodic
checkerboard.
59
CHAPTER 7. EXPLICIT SOLUTIONS 60
As a special case we recover the well-known one-dimensional result which follows. The
operator
−a(x/ε)uεxx = f
averages to the harmonic mean
a =⟨a(y)−1
⟩−1. (7.2)
Calculation. The adjoint problem (6.3) becomes
−B : D2y(a(y)ρ∞(y)) = 0
with solution ρ∞(y) = a(y)−1⟨a(y)−1
⟩−1 yielding (7.1). In particular (since B is constant)
we see that each element in the coefficient matrix averages to its harmonic mean:
aij =⟨a(y)−1b−1
ij
⟩−1=⟨aij(y)−1
⟩−1. (7.3)
7.2 A Special Case: Layered Material
Next we consider the special case of a layered material, where the coefficient matrix is a
function of only one variable:
A(y) = A(y1).
See Figure 7.2 for a visualisation of the operator.
The adjoint equation is now
−D2y(A(y1)ρ∞(y)) = 0,
which has the solution
ρ∞(y) = a11(y1)−1⟨a11(y1)−1
⟩−1.
Thus the averaged coefficient matrix from (6.5) is
A =⟨a−1
11
⟩−1 ⟨a−1
11 A⟩. (7.4)
CHAPTER 7. EXPLICIT SOLUTIONS 61
(a) (b)
(c) (d)
Figure 7.1: A visual representation of a piecewise constant multiple of the laplacian on aperiodic checkerboard. (a) The original grid. (b) The homogenised grid. (c) The originaloperator on one small cell. (d) The homogenised operator on one small cell.
CHAPTER 7. EXPLICIT SOLUTIONS 62
(a) (b)
(c) (d)
Figure 7.2: A visual representation of a piecewise constant multiple of the laplacian on alayered material. (a) The original grid. (b) The homogenised grid. (c) The original operatoron one small cell. (d) The homogenised operator on one small cell.
CHAPTER 7. EXPLICIT SOLUTIONS 63
7.3 A Special Case: Separable, Diagonal Coefficient Matrix
We also consider the case where the coefficient matrix is diagonal and has separable entries.
That is, the matrix can be expressed as a product of diagonal matrices of the form
A(y) =n∏j=1
Aj(yj)
and each entry on the diagonal of the resulting matrix can be written as
Aii(y) =n∏i=1
Ajii(yj).
See Figure 7.3 for a visualisation of such an operator. The averaged coefficients are
A =
⟨1
n∏j=1
Ajjj(yj)A
⟩⟨1
n∏j=1
Ajjj(yj)
⟩−1
, (7.5)
which is verified by the following calculation.
Calculation. In this case, the adjoint equation (6.3) takes the form
−n∑i=1
n∏j=1
Ajii(yj)ρ∞(y)
yiyi
= 0.
Since the coefficients are separable, the invariant distribution will also be separable:
ρ∞ =1
n∏j=1
Ajjj(yj)
⟨1
n∏j=1
Ajjj(yj)
⟩−1
and the result follows by averaging using (6.5).
7.4 Comparison with Divergence Structure Results
A given coefficient matrix, A, can be used to generate either the non-divergence structure
operator (1.2)
Lεu = −A(x/ε) : D2u,
CHAPTER 7. EXPLICIT SOLUTIONS 64
(a) (b)
(c) (d)
(e) (f)
Figure 7.3: A visual representation of a separable operator. (a) The original grid. (b) Thehomogenised grid. (c) The vertical structure of the operator on one small cell. (d) Thehorisontal structure of the operator on one small cell. (e) The original operator on onesmall cell. (f) The averaged operator on one small cell.
CHAPTER 7. EXPLICIT SOLUTIONS 65
or the divergence structure operator (1.3)
Lεdivu = −∇ ·(A(x/ε)∇u
).
Only in very simple cases will the coefficient matrix A homogenise to the same operator for
both the divergence and non-divergence case. The main example where the results are the
same is the one-dimensional case, where the coefficient matrix homogenises to the harmonic
mean for both problems [38], (7.2). However, in general the two operators will result in
different homogenised coefficients.
Here we compare the explicit solutions of this chapter to known results for the divergence
structure problem.
7.4.1 A Layered Material
First we consider a two-dimensional layered material. The divergence structure result is
given in [38]. Although most elements of the coefficient matrix average in the same way
for both problems, one element is different. In the non-divergence structure case we obtain
from (7.4)
a22 =⟨a−1
11
⟩−1⟨a22
a11
⟩.
The divergence case, on the other hand, yields
adiv22 =⟨a21
a11
⟩⟨a12
a11
⟩⟨a−1
11
⟩−1+⟨a22 −
a12a21
a11
⟩.
7.4.2 A Separable Material
We can also consider a two-dimensional separable operator that is a scalar multiple of the
identity.
A(x/ε) = a1(x1/ε)a2(x2/ε)I
This is a special case of both the constant operator of §7.1 and the separable case of §7.3.
In either case, the coefficients for the non-divergence structure problem homogenises to the
constant multiple of the identity, given by a generalised harmonic mean,
A = 〈a−11 a−1
2 〉−1I,
where I denotes the identity matrix. In the divergence problem, which is considered in [23],
the homogenised coefficients are not all equal and are obtained from a combination of the
CHAPTER 7. EXPLICIT SOLUTIONS 66
harmonic and arithmetic means of the coefficients.
Adiv11 = 〈a2〉⟨a−1
1
⟩−1
Adiv22 = 〈a1〉⟨a−1
2
⟩−1.
Chapter 8
Numerical Averaging
Although there are a number of situations in which we can write down an explicit repre-
sentation of the averaged coefficients, this is not always possible. However, the averaging
result can be implemented numerically. To accomplish this we need only solve one linear
equation with normalisation constraints (6.3). The computational effort is comparable (ar-
guably less) to the effort for the divergence structure problem, where n equations must be
solved for the cell problem.
The finite difference schemes we use have a natural interpretation as a discrete space
Markov chain approximation of the underlying diffusion process. Consequently, one ap-
proach to interpreting the numerical method would be to use Markov chain approximations
along the lines of [28, 42, 41]. Here we will focus on the PDE aspects in order to discretise
the problem. To present the ideas in the simplest setting, we perform the discretisation in
two dimensions; the generalisation to higher dimensions is straightforward.
In this chapter we numerically average several test problems and solve the averaged
equations. We demonstrate that the approach presented in this thesis does correctly average
the non-divergence structure elliptic operator and compute the operators is some cases where
the exact formula is not known. We begin by validating the approach: we demonstrate
numerically that the solutions of the original equation
−A(x/ε) : D2uε = f
converge pointwise to the solutions of the averaged equation
−A : D2u = f
67
CHAPTER 8. NUMERICAL AVERAGING 68
in the limit as ε→ 0.
A further goal of this chapter is to investigate the relationship between cell resolution
and accuracy of both the computed averaged operators and the solutions of the averaged
equations.
Notation. Below we denote by diag(a) the diagonal matrix whose jth diagonal element
is given by the jth element of the vector a. We use 0,1 for vectors whose elements are
identically zero or one respectively. The identity matrix is represented by I.
8.1 Discretisation and Numerical Solution
We will be solving the non-divergence form equation (PDE)ε. This is discretised using
centred differences and the linear systems was solved directly (in this case using the default
Matlab backslash solver). In all cases, the computational domain is the unit square [0, 1]×[0, 1].
We also want to solve Equation (6.3) for the invariant distribution. Let ρ = (ρij) be a
grid vector indexed by the grid coefficients. The fact that the grid vector ρ approximates
the solution ρ∞ at the corresponding grid points,
ρij = ρ∞(hi, hj) +O(h2)
will follow from from the standard linear finite difference analysis. After discretising the
two-dimensional adjoint equation (6.3) by standard centered finite differences, with spatial
resolution h, we obtain the finite dimensional linear equation
Mρ = 0 (8.1)
where M is a linear operator defined below, along with the discretisation of the constraint∑i,j
ρijh2 = 1. (8.2)
The matrix M is obtained by combining the finite difference operators for each term as
follows. To shorten the formulas, we write the grid matrix A = Aij as a collection of scalar
functions for each component
A = Aij =
a11 a12
a21 a22
ij
=
qij sij
sij rij
.
CHAPTER 8. NUMERICAL AVERAGING 69
Then we define the linear operators
Q = diag(qij), R = diag(rij), S = diag(sij),
which correspond to scalar multiplication. Next we define Dy1y1 , Dy1y2 , Dy2y2 to be the
linear operators corresponding to the finite difference operators for the second derivatives.
The equation (8.1) then takes the form(Dy1y1Q+Dy2y2R+ 2Dy1y2S
)ρ = 0. (8.3)
Writing out the operator more explicitly, term by term, results in
(Dy1y1Qρ)ij =1h2
(qi+1,jρi+1,j + qi−1,jρi−1,j − 2qi,jρi,j
)(Dy2y2Rρ)ij =
1h2
(ri,j+1ρi,j+1 + ri,j−1ρi,j−1 − 2ri,jρi,j
)2(Dy1y2Sρ)ij =
2h2
(si+1,j+1ρi+1,j+1 + si−1,j−1ρi−1,j−1
−si+1,j−1ρi+1,j−1 − si−1,j+1ρi−1,j+1
).
(8.4)
The equation (8.3) can be solved by a direct method, enforcing the single constraint (8.2).
For simplicity, we solve the equation iteratively by a Gauss-Seidel method. Since the oper-
ator preserves mass, even at the discrete level, the constraint (8.2) is satisfied by choosing
initial data which satisfies the constraint.
8.2 Multiples of a Constant Coefficient Matrix
We begin with a simple example, where the coefficient matrix is a (variable) scalar multiple
of a constant matrix. That is
A = p(y)K
where K is a constant matrix.
We have already showed that the solution (7.1) of the adjoint equation is simply
ρ∞(y) = p(y)−1⟨p(y)−1
⟩−1.
It is easy to see that this will also be a steady state solution of the discretised equation no
matter what cell resolution we use. To demonstrate this, we make the following replacements
in the discretised equation (8.4):
Q = k11P, R = k22P, S = k12P.
CHAPTER 8. NUMERICAL AVERAGING 70
Here P = diag(p) is a diagonal matrix and k11, k22, k12 are constant. The proposed steady
state solution in discrete form is
ρ∞ = P−11〈P−11〉−1.
Substituting these expressions into the discrete equation (8.4) we find that
Dy1y1Qρ∞ = Dy1y1k11PP
−11〈P−11〉−1
= k11〈P−11〉−1Dy1y11
= 0
where the last step holds since the rows of Dy1y1 sum to one. The other terms vanish
similarily, showing that (7.1) is in fact the steady state solution of the discrete iterative
scheme (8.4) regardless of how well or poorly the cell is resolved.
We now choose a specific example where the coefficient matrix is piecewise constant in
a checkerboard pattern. The coefficient matrix will vary between
A1 =
2 1
1 4
A2 =
4 2
2 8
= 2A1.
See Figure 8.1(a) for a surface plot of the resulting solution. Using the formula (7.3), we
expect that the coefficient matrix should average to
A =
8/3 4/3
4/3 16/3
.
We set forcing f = 50, enforce zero Dirichlet boundary conditions, and run through
different values of ε to investigate the convergence of solutions of the original equation to
solutions of the averaged equation as ε→ 0.
Convergence results are presented in Table 8.1 and Figure 8.1(b). These computations
are consistent with the assertion that the solutions of the original equation (PDE)ε do, in
fact, converge to the solutions of the averaged equation (6.5) as ε → 0. Furthermore, the
convergence appears to be second order in ε. That is, the exact solution of (PDE)ε is given
by
uε = u+O(ε2),
consistent with (6.6).
CHAPTER 8. NUMERICAL AVERAGING 71
(a)
(b)
Figure 8.1: Results for a coefficient that is a multiple of a constant matrix. (a)A surface plotof the solution on the fully resolved grid (25 × 25 cells of 24 × 24 grid points). (b)Log-logplot of L2 difference between solutions of averaged equation and original equation.
CHAPTER 8. NUMERICAL AVERAGING 72
ε ‖u− u‖21/3 5.474× 10−3
1/5 2.003× 10−3
1/7 1.029× 10−3
1/9 6.250× 10−4
1/11 4.194× 10−4
1/13 3.008× 10−4
1/15 2.262× 10−4
1/17 1.763× 10−4
1/19 1.412× 10−4
1/21 1.157× 10−4
1/23 9.648× 10−5
1/25 8.169× 10−5
Table 8.1: Results for a coefficient that is a multiple of a constant matrix. The error givesthe L2 difference between the solution of the averaged equation and the solution of theoriginal equation for a given value of ε.
8.3 A Layered Material
Next we consider a layered material, where the coefficient matrix varies in vertical stripes.
That is, A(x, y) = A(x). For this example we allow the coefficients to vary between the
following two matrices:
A1 =
2 1
1 4
A2 =
20 0
0 1
.
Again we set the forcing to f = 50 and enforce zero Dirichlet boundary conditions. The
resulting solution is plotted in Figure 8.2.
According to (7.4) the homogenised coefficient matrix should be
A =
40/11 10/11
10/11 41/11
.
The difference between the solutions of the original equation (PDE)ε and the homogenised
equation (6.5) (measured in L2) are presented in Table 8.2 and Figure 8.2. As in the first
example, we observe quadratic convergence as ε→ 0.
CHAPTER 8. NUMERICAL AVERAGING 73
(a)
(b)
Figure 8.2: Results for a layered material. (a)A surface plot of the solution on the fullyresolved grid (25× 25 cells of 24× 24 grid points). (b)Log-log plot of L2 difference betweensolutions of the averaged equation and original equation.
CHAPTER 8. NUMERICAL AVERAGING 74
ε ‖u− u‖21/3 1.565× 10−2
1/5 5.609× 10−3
1/7 2.867× 10−3
1/9 1.740× 10−3
1/11 1.168× 10−3
1/13 8.384× 10−4
1/15 6.312× 10−4
1/17 4.924× 10−4
1/19 3.948× 10−4
1/21 3.237× 10−4
1/23 2.702× 10−4
1/25 2.289× 10−4
Table 8.2: Results for a layered material. The error gives the L2 difference between thesolution of the averaged equation and the solution of the original equation for a given valueof ε.
8.4 A Separable, Diagonal Coefficient Matrix
Next we consider a separable, diagonal coefficient. We again choose a piecewise constant
coefficient matrix. In this case, each cell is divided into quarters with the following coeffi-
cients:
A1 =
1 0
0 1
1 0
0 1
A2 =
2 0
0 4
1 0
0 1
A3 =
1 0
0 1
20 0
0 1
A4 =
2 0
0 4
20 0
0 1
.
As before we set f = 50 with u = 0 on the boundary. See Figure 8.3(a) for a surface plot of
the solution.
According to (7.5), the homogenised coefficients should be
A =
14 0
0 2
.
CHAPTER 8. NUMERICAL AVERAGING 75
(a)
(b)
Figure 8.3: Results for a coefficient matrix with separable entries. (a)A surface plot of thesolution on the fully resolved grid (25× 25 cells of 24× 24 grid points). (b)Log-log plot ofL2 difference between solutions of the averaged equation and original equation.
CHAPTER 8. NUMERICAL AVERAGING 76
ε ‖u− u‖21/3 6.105× 10−2
1/5 2.337× 10−2
1/7 1.214× 10−2
1/9 7.405× 10−3
1/11 4.984× 10−3
1/13 3.582× 10−3
1/15 2.698× 10−3
1/17 2.106× 10−3
1/19 1.689× 10−3
1/21 1.385× 10−3
1/23 1.156× 10−3
1/25 9.794× 10−4
Table 8.3: Results for a coefficient matrix with separable entries. The error gives the L2
difference between the solution of the averaged equation and the solution of the originalequation for a given value of ε.
As in the previous examples, we observeO(ε2) convergence to the solution of the averaged
equation; see Table 8.3 and Figure 8.3(b).
8.5 A More General Linear Operator
In all the examples we have considered so far we have assumed that the coefficients of the
drift and source terms (b, c in (1.2)) are equal to zero. Here we consider a more general
operator where these terms are present in the equation. We choose the same coefficient
matrix used in Section 8.2, which is a scalar multiple of a constant matrix. The remaining
coefficients do not need to have this property (i.e., b need not be a multiple of a constant
vector) in order for the analysis in Section 7.1 to apply. In particular we consider the
coefficients:
A1 =
2 1
1 4
b1 =
1
1
c1 = 1
A2 =
4 2
2 8
b2 =
2
4
c1 = 2.
As before we let f = 50 and enforce zero Dirichlet boundary conditions. The solution is
plotted in Figure 8.4(a).
CHAPTER 8. NUMERICAL AVERAGING 77
The analysis in Section 7.1 is easily extended to this situation, and predicts the following
averaged coefficients:
A1 =
8/3 4/3
4/3 16/3
b1 =
4/3
2
c1 = 4/3.
Convergence results are contained in Table 8.4 and Figure 8.4. As before, we observe
quadratic convergence as ε→ 0.
ε ‖u− u‖21/3 5.486× 10−3
1/5 2.008× 10−3
1/7 1.031× 10−3
1/9 6.263× 10−4
1/11 4.203× 10−4
1/13 3.014× 10−4
1/15 2.267× 10−4
1/17 1.767× 10−4
1/19 1.415× 10−4
1/21 1.159× 10−4
1/23 9.669× 10−5
1/25 8.187× 10−5
Table 8.4: Results for a general elliptic operator. The error gives the L2 difference betweenthe solution of the averaged equation and the solution of the original equation for a givenvalue of ε.
8.6 Non-Zero Boundary Conditions
In all the examples we have studied so far we have enforced zero Dirichlet boundary con-
ditions. Now we demonstrate numerically that the approach presented in this thesis also
works for non-constant Dirichlet boundary conditions.
CHAPTER 8. NUMERICAL AVERAGING 78
(a)
(b)
Figure 8.4: Results for a general elliptic operator. (a)A surface plot of the solution on thefully resolved grid (25 × 25 cells of 24 × 24 grid points). (b)Log-log plot of L2 differencebetween solutions of the averaged equation and original equation.
CHAPTER 8. NUMERICAL AVERAGING 79
In this example we use the y-periodic coefficient matrix with entries
aε11 =1
cos(
2πx1ε −
πε
)[cos(πε
)− cos
(2πx2ε −
πε
)]+ 4
aε22 =1
cos(
2πx2ε −
πε
)[cos(πε
)− cos
(2πx1ε −
πε
)]+ 4
a12 = 0.
We set forcing
f = −2
and enforce the boundary conditions
u = 2x21 + 2x2
2 x1 = 0, x1 = 1, x2 = 0, x2 = 1.
This problem has the exact solution
uε =ε2
4π2
[cos(π
ε
)− cos
(2πx1
ε− π
ε
)][cos(π
ε
)− cos
(2πx2
ε− π
ε
)]+ 2x2
1 + 2x22.
We fix ε = 1/499 and look at the relationship between cell resolution and solution
accuracy. This solution is plotted in Figure 8.5(a).
The convergence results are presented in Table 8.5 and Figure 8.5(b). It is evident from
the log-log plot of error that for fixed (small) ε, the error is linearly dependent on the spatial
step size used to solve the cell problem (6.3).
8.7 Random Coefficients
So far we have considered operators that vary periodically. However, the averaging method
discussed in this article can be applied to average random operators. While we avoid the
analysis, we can compute the invariant distribution in this case.
Next we consider the case where the coefficient matrix at each point is randomly chosen
(with equal probability) to be one of two constant operators. Since the operator no longer
varies periodically, it is now neccessary to solve for the invariant distribution in the entire
domain. As before, we enforce periodic boundary conditions. In order to compute the
CHAPTER 8. NUMERICAL AVERAGING 80
(a)
(b)
Figure 8.5: Results for a problem with non-constant boundary data averaged with spatialresolution 1/h. (a)A surface plot of the exact solution (with ε = 1/499). (b)Log-log plot ofL2 difference between solutions of the averaged equation and original equation.
CHAPTER 8. NUMERICAL AVERAGING 81
1/h a11 a22 ‖u− u‖210 0.24514 0.24514 6.527× 10−3
50 0.24891 0.24891 1.437× 10−3
90 0.24939 0.24939 8.103× 10−4
130 0.24958 0.24958 5.607× 10−4
170 0.24967 0.24967 4.290× 10−4
210 0.24973 0.24973 3.499× 10−4
250 0.24978 0.24978 2.907× 10−4
290 0.24981 0.24981 2.531× 10−4
330 0.24983 0.24983 2.239× 10−4
370 0.24985 0.24985 1.989× 10−4
410 0.24986 0.24986 1.792× 10−4
450 0.24988 0.24988 1.630× 10−4
490 0.24989 0.24989 1.494× 10−4
Table 8.5: Results for a problem with non-constant boundary conditions homogenised withspatial resolution 1/h. The averaged coefficients are a11, a22, a12. The error gives the L2
difference between the solution of the homogenised equation and the solution of the originalequation.
averaged coefficients, we solve for the invariant distribution with K different randomly
chosen coefficients Ai and compute the averaged coefficients
Ai =⟨ρ∞Ai
⟩.
The results of these trials are averaged to compute the averaged coefficient matrix for the
general random problem.
A =1K
K∑i=1
Ai.
In the following computations, we average the coefficients over K = 10 trials. To demon-
strate that this approach correctly averages the random operator we will solve for twenty
different random coefficient matrices and compute the mean difference between the solutions
of the random and averaged problems.
In particular, we randomly vary between the following two coefficient matrices:
A1 =
2 1
1 4
A2 =
20 0
0 1
.
We enforce zero Dirichlet boundary conditions and set forcing f = 30.
CHAPTER 8. NUMERICAL AVERAGING 82
Results are shown in Table 8.6 and Figure 8.6. We see that as the number of grid points
is increased, the solution of the random equation approaches (on average) the solution of
the averaged equation. Moreover, the average difference between the two solutions is O(h).
N a11 a22 a12 ‖u− u‖2⟨‖u− u‖2
⟩8 6.834 3.194 0.731 2.747× 10−2 3.635× 10−2
16 6.745 3.209 0.736 2.622× 10−2 2.146× 10−2
24 6.657 3.224 0.741 9.839× 10−3 1.699× 10−2
32 6.628 3.229 0.743 9.934× 10−3 9.940× 10−3
40 6.672 3.221 0.740 1.923× 10−2 8.358× 10−3
48 6.683 3.220 0.740 5.064× 10−3 6.583× 10−3
56 6.538 3.244 0.748 4.283× 10−3 6.029× 10−3
64 6.496 3.251 0.750 5.144× 10−3 6.624× 10−3
72 6.624 3.229 0.743 5.976× 10−3 5.100× 10−3
80 6.636 3.227 0.742 3.008× 10−3 4.009× 10−3
88 6.594 3.234 0.745 3.239× 10−3 3.454× 10−3
96 6.592 3.235 0.745 3.821× 10−3 3.902× 10−3
104 6.564 3.239 0.746 2.369× 10−3 3.145× 10−3
112 6.574 3.238 0.746 2.870× 10−3 2.979× 10−3
120 6.572 3.238 0.746 3.141× 10−3 2.986× 10−3
Table 8.6: Results for a problem with random coefficients on an N ×N grid. The averagedcoefficients are a11, a22, a12. The errors include the L2 difference between the solution ofthe homogenised equation and the solution of a random equation as well as the averagedifference over 20 trials.
CHAPTER 8. NUMERICAL AVERAGING 83
(a)
(b)
Figure 8.6: Results for a problem with random coefficients. (a)Log-log plot of the L2
difference between the homogenised solution and a random solution. (b)Log-log plot ofthe average L2 difference between the homogenised solution and a random solution over 20trials.
Chapter 9
Conclusions
9.1 Summary
In this thesis we have focused on numerical techniques for solving two important second-
order elliptic equations.
9.1.1 The Monge-Ampere Equation
Firstly, we have investigated several finite difference methods for the two-dimensional el-
liptic Monge-Ampere equation. For examples with smooth, convex data and solutions, the
methods converge to the exact solution and are accurate to second order in h (O(h2)). In
situations with non-smooth or noisy (i.e. non-convex) data or solutions, the methods still
converge and agree with the exact solution when one is available.
The first method we considered is an explicit finite difference method. A slight modifica-
tion of this method gives us a method that ensures more convexity. Numerical investigation
indicates that this modified method is sometimes more accurate when solutions are not
smooth or strictly convex. The computation time required by these methods seems to be
independent of the regularity of the solution.
The other method we presented involves repeatedly solving Poisson’s equation. Nu-
merical investigation indicates that this method is very fast when smooth, strictly convex
solutions exist. The method seems to converge, though much more slowly, even when so-
lutions are non-smooth or not strictly convex. In the special case where the domain is a
rectangle we show that this iteration is a contraction mapping. The proof of this suggests a
84
CHAPTER 9. CONCLUSIONS 85
rate of convergence that will be much faster when solutions are smooth and strictly convex,
just as we observed numerically.
9.1.2 The Rapidly Varying Non-Divergence Structure Operator
Secondly, we have investigated the non-divergence structure elliptic operator with rapidly
varying, periodic coefficients. Using formal asymptotic calculations, we recovered a sim-
ple formula for the averaged operator. This formula was available in the literature using
probabilistic techniques; we gave a formal derivation and (in the case of smooth data and
solutions) a proof using Partial Differential Equations techniques. We also presented sev-
eral special cases where the adjoint problem could be solved in closed form, leading to a
closed form result for the averaged operator. This could be accomplished for a wider set
of examples than in the divergence structure case. We also showed that the homogenised
coefficients are generally different for the two cases.
In addition, we presented numerical computations for this problem. The adjoint equation
was discretised using finite differences and solved using a Gauss-Seidel method. The effort
involved in solving this problem is comparable (or less) that the corresponding problem
in the divergence structure case. The numerics were validated for finite ε against resolved
numerical solutions. The formal asymptotic convergence rate O(ε2) was also validated.
9.2 Future Work
Much work remains to be done for both of the problems we have considered in this thesis.
The first part of this thesis has aimed at solving the Monge-Ampere equation (MA).det(D2u) = f in Ω
u = g on ∂Ω
u is convex.
However, the equation that arises most naturally in the theory of optimal transport is
Equation (2.5). det(D2u(x)
)= f(x)/g(∇u(x)) x ∈ X
∇u : X → Y
u is convex.
CHAPTER 9. CONCLUSIONS 86
This equation presents a greater numerical challenge for two reasons.
1. The equation includes a (possibly) non-linear function of the gradient.
2. Dirichlet boundary conditions are replaced with the requirement that the gradient of
the solution map the space X onto a specified space Y .
The second part of this thesis has considered the solution of the second order, non-
divergence structure linear elliptic equation with periodic coefficients. However, in the
study of heterogeneous materials it is natural to expect the coefficients to vary randomly
rather than periodically. This presents a more challenging problem as the equation for the
invariant distribution (6.3) must be solved on the entire domain rather than just one cell.
Although we have done some preliminary computations in the random setting, further work
is needed to explore practical numerical techniques for handling the random problem.
Finally, we would like to take the techniques used here for a linear equation and ex-
tend them to develop methods for solving non-linear problems that involve rapidly varying
operators.
Appendix A
The Fredholm Alternative
A key to using asymptotics to homogenise the operators considered in this paper is the use
of the Fredholm alternative to derive solvability conditions for certain inhomogeneous linear
equations. In particular we will want to develop solvability conditions for the equation
Lu = f
where L is a second order, periodic, non-divergence structure, linear elliptic operator of the
form
L = −A : D2. (A.1)
The coefficient matrix A associated with this operator is periodic and the operator is
equipped with periodic boundary conditions on the bounded domain Ω. The chief goal
of this chapter is to prove that the Fredholm alternative holds for this operator.
In order to carry out the analysis, we will require the operator (A.1) to be uniformly
elliptic with sufficiently smooth, periodic coefficients. This results in the following conditions
on the coefficient matrix A with entries aij :∃α > 0 s.t. ξTAξ ≥ α|ξ|2 ∀ξ ∈ Rn
A ∈ C1(Ω; Rn×n)
A is periodic on Ω
(A.2)
A consequence of A being continuously differentiable on a closed domain is the additional
87
APPENDIX A. THE FREDHOLM ALTERNATIVE 88
conditions Axi ∈ L∞ i = 1, . . . , n
A ∈ L∞.(A.3)
We can now state the specific result we wish to prove.
Theorem A.1. Consider the operator (A.1) with a coefficient matrix that satisfies the
conditions (A.2) and (A.3). Then the inhomogeneous equation
Lu = f
has a solution if and only if the kernel of the adjoint of L is orthogonal to the forcing f :
(f, v) = 0 ∀v ∈ N(L∗). (A.4)
A.1 Background Results
Before proceeding with the proof, we recall two important theorems. The first is the classical
Fredholm alternative [15].
Theorem A.2 (The Fredholm Alternative). Let K : H → H be a compact linear operator
on a Hilbert space H. Then the following hold:
(i) dim(N(I −K)) <∞
(ii) R(I −K) is closed
(iii) R(I −K) = N(I −K∗)⊥
(iv) N(I −K) = 0 iff R(I −K) = H
(v) dim(N(I −K)) = dim(N(I −K∗))
We will also use the Lax-Milgram Theorem [15].
Theorem A.3 (Lax-Milgram Theorem). Let H be a Hilbert space and assume that
B : H ×H → R
is a bilinear mapping, for which there exist constant β, γ > 0 such that
APPENDIX A. THE FREDHOLM ALTERNATIVE 89
(i)∣∣B[u, v]
∣∣ ≤ β‖u‖ ‖v‖ (u, v ∈ H)
(ii) γ‖u‖2 ≤ B[u, u] (u ∈ H)
Furthermore, let f : H → R be a bounded linear functional on H.
Then there exists a unique element u ∈ H such that
B[u, v] =< f, v >
for all v ∈ H.
A.2 The Proof
We will use the Fredholm Alternative (Theorem A.2) to prove the solvability condition.
However, for this theorem to apply we require a compact operator. With this in mind, we
proceed as in the divergence structure problem [38] and consider the resolvent operator
K : L2per → L2
per defined by
K = µ(L+ µI)−1 (A.5)
where µ is some positive constant. We also define the function
h =1µKf. (A.6)
We divide the proof into four main steps.
1. Prove that the operator K is well-defined.
2. Prove that the operator K is compact.
3. Prove that the Fredholm alternative for I −K leads to the Fredholm alternative for
L.
4. Prove that the Fredholm alternative for L gives the desired solvability condition.
A.2.1 Well-Defined
The first step is to prove that the operator K defined by (A.5) is in fact well-defined. That
is, we need to show that the equation
1µ
(L+ µI)u = f (A.7)
APPENDIX A. THE FREDHOLM ALTERNATIVE 90
has a unique solution u ∈ H1per ⊂ L2
per for every f ∈ L2per.
With the goal of developing a weak formulation of the problem, we begin by taking the
inner product of (A.7) with a test function v and multiplying through by the constant µ.
∫Ω
−∑i,j
aijuxixj + µu
v dx =∫
Ωµfv dx
Integration by parts leads to
∫Ω
∑i,j
(aijv
)xiuxj + µuv
dx =∫
Ωµfv dx.
Here the boundary terms vanish due to periodicity. Thus we can rewrite Equation (A.7) in
the weak formulation
Bµ[u, v] = (µf, v) ∀v ∈ L2per (A.8)
where the bilinear operator is defined by
Bµ[u, v] =∫
Ω
∑i,j
(aijv
)xiuxj + µuv
dx. (A.9)
Next we want to show that this bilinear form has the bounds required by Theorem A.3.
We first show that this function is bounded.
Lemma A.1 (Bounded). There exists β > 0 such that for every µ > 0∣∣Bµ[u, v]∣∣ ≤ β‖u‖H1‖v‖H1 ∀u, v ∈ H1
per(Ω). (A.10)
Proof. We begin by using the product rule to rewrite (A.9).
∣∣Bµ[u, v]∣∣ ≤ ∫
Ω
∣∣∣∣∣∣∣∑
i,j
(aijv
)xiuxj + µuv
∣∣∣∣∣∣∣ dx
=∫
Ω
∣∣∣∣∣∣∑i,j
(aijvxiuxj + (aij)xivuxj
)+ µuv
∣∣∣∣∣∣ dx≤∫
Ω
∑i,j
(∣∣∣aijvxiuxj ∣∣∣+∣∣∣(aij)xivuxj ∣∣∣)+ µ|uv|
dx
APPENDIX A. THE FREDHOLM ALTERNATIVE 91
Since by (A.3) both A and its derivatives are bounded in L∞, we obtain the bounds
∣∣Bµ[u, v]∣∣ ≤ C ∫
Ω
∑i,j
(∣∣∣vxiuxj ∣∣∣+∣∣∣vuxj ∣∣∣)+|uv|
dx
≤ C∫
Ω
(|∇u||∇v|+|v||∇u|+|u||v|
)dx.
Making use of the Cauchy-Schwarz inequality leads to∣∣Bµ[u, v]∣∣ ≤ C (‖∇v‖L2 ‖∇u‖L2 + ‖v‖L2 ‖∇u‖L2 + ‖u‖L2 ‖v‖L2
)≤ C ‖u‖H1 ‖v‖H1 .
This is exactly the bound we required.
Next we show that the bilinear function is coercive for some value of µ > 0.
Lemma A.2 (Coercive). There exist γ, µ > 0 such that
γ‖u‖2 ≤ Bµ[u, u] ∀u ∈ H1per.
Proof. We begin by using the uniform ellipticity bounds in (A.2).
α ‖∇u‖2L2 ≤∫
Ω
∑ij
aijuxiuxj dx
=∫
Ω
∑ij
(aijuxiuxj + (aij)xiuuxj
)dx−
∫Ω
∑ij
(aij)xiuuxj dx
= B0[u, u]−∫
Ω
∑ij
(aij)xiuuxj dx
≤ B0[u, u] + C
∫Ω|u||∇u| dx
The last step is possible because the derivatives of A are bounded (A.3). For the next step,
we recall the inequality
u|∇u| ≤ δu2 +14δ|∇u|2 ∀δ > 0.
Choosing δ so thatC
4δ≤ α
2we obtain
α ‖∇u‖2L2 ≤ B0[u, u] + C
∫Ω
(δu2 +
14δ|∇u|2
)dx
≤ B0[u, u] + Cδ ‖u‖2L2 +α
2‖∇u‖2L2 .
APPENDIX A. THE FREDHOLM ALTERNATIVE 92
Rearranging this inequality we find that
α
2‖∇u‖2L2 ≤ B0[u, u] + Cδ ‖u‖2L2 .
Now we add α2 ‖u‖L2 to both sides of the inequality.
α
2‖∇u‖2L2 +
α
2‖u‖L2 ≤ B0[u, u] +
(Cδ +
α
2
)‖u‖2L2
This simplifies to the desired inequality:
BCδ+α2≥ α
2‖u‖2H1 .
Together Lemmas A.1 and A.2 satisfy the hypotheses of the Lax-Milgram Theorem.
Since for f ∈ L2per, (µf, v) defines a bounded linear functional on H1
per, we can use Lax-
Milgram to conclude that the weak formulation (A.8)
Bµ[u, v] = (µf, v) ∀v ∈ L2per
has a unique solution u ∈ H1per for every f ∈ L2
per.
We conclude that Kf is uniquely defined for every f ∈ L2per and the operator K is thus
well-defined.
A.2.2 Compact
In order to use the Fredholm alternative we must show that the operator K : L2per → L2
per
is compact. We begin by assigning µ the value required by Lemma A.2, fixing any f ∈ L2per,
and defining
uf = Kf. (A.11)
Making use of Lemma A.2 and the Cauchy-Schwarz inequality we find that
γ‖uf‖2H1 ≤ Bµ[uf , uf ]
= (f, uf )
≤ ‖f‖L2‖uf‖L2
≤ ‖f‖L2‖uf‖H1 .
A consequence of this result is
‖Kf‖H1 = ‖uf‖H1
≤ 1γ‖f‖L2 .
APPENDIX A. THE FREDHOLM ALTERNATIVE 93
Since H1 is compactly embedded in L2, we see that K maps sets that are bounded in L2
into sets that are compact in in L2. Therefore K is a compact operator.
A.2.3 The Fredholm alternative
Since we have shown that K is a well-defined, compact operator, the Fredholm alternative
(Theorem A.2) holds for this resolvent operator. We want to show that the Fredholm
alternative must also hold for the original operator L. That is, we want the following
theorem.
Theorem A.4 (The Fredholm Alternative for a Periodic Linear Elliptic Operator). Let L be
the operator defined by (A.1) satisfying the conditions (A.2). Then the following conditions
hold:
(i) dim(N(L)) <∞
(ii) R(L) is closed
(iii) R(L) = N(L∗)⊥
(iv) N(L) = 0 iff R(L) = H
(v) dim(N(L)) = dim(N(L∗))
Proving this theorem requires three main results:
1. N(L) = N(I −K)
2. N(L∗) = N(I −K∗)
3. R(L) = R(I −K)
With these results, we can simply replace I − K with L in the Fredholm alternative to
obtain Theorem A.4.
The nullspace of the operators
We begin by showing that L and I −K have identical kernels. To do this, we choose any
element v in the nullspace of L. This is equivalent to saying that
Lv = 0.
APPENDIX A. THE FREDHOLM ALTERNATIVE 94
Adding µv to both sides we see that
(L+ µI) v = µv.
We invert the operator on the left hand side to obtain
v = Kv
or equivalently
(I −K)v = 0.
This is equivalent to the statement that v is in the nullspace of I −K.
Similarily, we can show that the adjoint operators L∗ and I − K∗ also have identical
kernels.
The range of the operators
Now we want to show that the operators L and I −K have identical ranges. We begin by
selecting any element f in the range of L. This means that there exists some element v ∈ Hsuch that
Lv = f.
As before, we add µv to both sides to obtain
(L+ µI) v = f + µv.
Inverting the operator on the left hand side we see that
v =1µKf +Kv.
Equivalently, we can write
(I −K)v =1µKf,
which means that 1µKf is in the range of I −K. By Theorem A.2, the range of I −K is
identical to the orthogonal complement of the nullspace of I −K∗. In other words,(1µKf,w
)= 0 ∀w ∈ N(I −K∗).
From the definition of the ajoint this holds if and only if(f,K∗w
)= 0 ∀w ∈ N(I −K∗).
APPENDIX A. THE FREDHOLM ALTERNATIVE 95
Since w ∈ N(I −K∗), this is equivalent to saying that
(f, w) = 0 ∀w ∈ N(I −K∗).
In other words, f is in the orthogonal complement of the nullspace of I −K∗ which, by the
Fredholm alternative, is the same as the range of I − K. Therefore we conclude that the
operators L and I −K have identical ranges.
The Fredholm alternative for L (Theorem A.4) now follows immediately from Theo-
rem A.2.
A.2.4 The Solvability Condition
Finally, we are ready to prove the solvability condition proposed by Theorem A.1. We begin
by assuming that the equation
Lu = f
has a solution. This is clearly true if and only if
f ∈ R(L).
By part (iii) of Theorem A.4 this holds if and only if
f ∈ N(L∗)⊥.
This is the same as saying that
f ⊥ v ∀v ∈ N(L∗),
which is equivalent to
(f, v) = 0 ∀v ∈ N(L∗).
This is precisely the claim of Theorem A.1.
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