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Wide Stencil for the Monge Amp` ere Equation by Jessey Lin A research paper presented to the University of Waterloo in partial fulfillment of the requirement for the degree of Master of Mathematics in Computational Mathematics Supervisor: Prof. Justin Wan W. L. Waterloo, Ontario, Canada, 2014 c Jessey Lin. Public 2014
Transcript

Wide Stencil for the Monge AmpereEquation

by

Jessey Lin

A research paperpresented to the University of Waterloo

in partial fulfillment of therequirement for the degree of

Master of Mathematicsin

Computational Mathematics

Supervisor: Prof. Justin Wan W. L.

Waterloo, Ontario, Canada, 2014

c© Jessey Lin. Public 2014

I hereby declare that I am the sole author of this report. This is a true copy of the report,including any required final revisions, as accepted by my examiners.

I understand that my report may be made electronically available to the public.

ii

Abstract

We propose a new numerical scheme to solve the elliptic Monge Ampere Equation (MAE)with Dirichlet boundary condition. The problem is motivated from applications of theMAE to image registration modelling. The MAE is challenging to solve, it is fully nonlinearand has non-unique solutions and a general and efficient numerical scheme is difficult toconstruct. Our numerical algorithm solves the MAE by transforming it to a Hamilton-Jacobi-Bellman (HJB) equation, which has the form of a linear PDE coupled nonlinearlywith two control parameters. The HJB equation is further discretized by a wide stencilmethod. We prove the Barles-Souganidis convergence of the numerical scheme to theviscosity solution by showing consistency, stability and monotonicity. The performance ofthe numerical method will be shown by examples of smooth and singular MAE problems.

iii

Acknowledgements

I would like to thank my supervisor, Professor Justin W. L. Wan for his guidance andencouragement. I would like to thank my CM classmates who are all going to be futureleaders of the 21st century! I have learnt a lot from them. In addition, I would like tothank all the teachers who taught me here at the university, I had a truly rewarding year.

iv

Dedication

This is dedicated to my family and to all my teachers whose influence, is a life time.

v

Table of Contents

List of Tables viii

List of Figures ix

1 Introduction 1

1.1 Image Registration and the MAE . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Numerical methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Overview of The Essay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 The Monge Ampere Equations 5

2.1 The Monge-Ampere Equation . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Viscosity Solutions of the Monge-Ampere Equation . . . . . . . . . . . . . 6

2.3 From the Monge-Ampere Equation to Hamilton-Jacobi-Bellman Equation . 7

3 Discretization of the Monge-Ampere Equation 10

3.1 Basic Set Up and Notations . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.2 The LQH Operator Discretization . . . . . . . . . . . . . . . . . . . . . . . 11

3.3 Bilinear Interpolation of points from the Wide Stencil Method . . . . . . . 13

3.4 Points near the Boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.5 The Matrix form of the Discretized Equations . . . . . . . . . . . . . . . . 17

vi

4 Convergence to the Viscosity Solution 20

4.1 Consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4.2 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4.3 Monotonicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

5 Numerical Results 30

5.1 Smooth Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

5.1.1 Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

5.1.2 Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

5.2 Non-smooth Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

5.2.1 Example 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

6 Conclusions 36

References 37

vii

List of Tables

viii

List of Figures

1.1 Registration of human knee. (a)-(c) : registration without transformation. (d)-

(f): registration with transformation y. (Image courtesy of IOP Publishing, [9]) 2

1.2 Reproduction of figures for the wide stencil scheme proposed by Obermanet al [11]. The number of stencil points (represented by circles) increaseswith decreasing mesh size. . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1 Illustrations for viscosity solutions. (Image courtesy of SIAM Review, [7]) . 7

3.1 Local grid rotation of angle φ about (xi, yj). Pmi,j, m = 1, . . . 4 are the new

stencil points. r1, r2 are the new axes, h′ is the new stencil length. . . . . . 12

3.2 Location of P 4i,j and its 4 nearest neighbours. . . . . . . . . . . . . . . . . . 14

3.3 Shrinking of stencil point P 1i,j . . . . . . . . . . . . . . . . . . . . . . . . . 15

5.1 Plots of example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

5.2 Plots of example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

5.3 Plots of example 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

ix

Chapter 1

Introduction

The Monge Ampere equation belongs to the category of fully nonlinear second order PDEs.It was first studied by Gaspard Monge in 1784 and later by Andre-Marie Ampere in 1820.It has wide applications in differential geometry problems such as the Minkowski problemand optimization problems such as the Monge-Kantorovich minimization problem. Areassuch as astrophysics, medical image analysis and reflector design apply models using theMAE [11]. In this chapter, we will motivate our study of the MAE and review some of theexisting methods.

1.1 Image Registration and the MAE

Our interest of the MAE stems from the image registration problem. Figure 1.1a andFigure 1.1b shows two images, the reference (R) and the template (T ), of an MR scan ofthe human knee. They are to be aligned or registered for clinical purposes. However R isbent to an angle while T is not bent. One cannot apply simple linear transformations toalign them so we need to find a good transformation y : R → T , in such a way that theirdifference T (y)−R in the resulting image is minimized.

To mathematically tackle the problem, we consider the model provided by the optimaltransport problem [13].

The optimal transport problem seeks to find an optimal mapping y between two densityfunctions R, T defined on Ω ⊆ R2 with the constraint that mass is preserved, i.e.

ˆΩ

R(x) dx =

ˆΩ

T (y(x))det(Dy(x)) dx (1.1)

1

Figure 1.1: Registration of human knee. (a)-(c) : registration without transformation. (d)-(f):registration with transformation y. (Image courtesy of IOP Publishing, [9])

so that their Monge-Kantorovich distance metric,

d(R, T ) = min

ˆΩ

‖x− y(x)‖pR(x) dx

is minimized.

In particular, when p = 2, we can write the optimal map, y, as:

y = ∇φ, (1.2)

φ convex on Ω,

where φ satisfies the Monge Ampere Equation (MAE):

det(D2φ(x)) =R(x)

T (∇φ)=: f.

This is the mass preserving requirement and can be easily observed when subsituting (1.2)into (1.1) above.

2

(a) 9 point stencil (b) 17 point stencil (c) 33 point stencil

Figure 1.2: Reproduction of figures for the wide stencil scheme proposed by Oberman et al[11]. The number of stencil points (represented by circles) increases with decreasing meshsize.

1.2 Numerical methods

Due to the high non-linearity and non-uniqueness of the MAE, it poses a number of nu-merical challenges to set up with the right numerical scheme.

Fortunately, the frame work provided by Barles and Souganidis [3] allows one to studynumerical schemes to overcome the aforementioned difficulties. It basically states that ifthe numerical scheme is consistent, stable and monotone in an appropriate sense, then itwould converge to the viscosity solution of the MAE.

In reference to this framework, Oberman et al [11] has developed a monotone widestencil finite difference scheme to approximate the fully nonlinear PDE. Note that a mono-tone finite difference scheme even for a linear elliptic PDE, using a narrow stencil does notin general exist [7]. However, Oberman’s method may be computationally expensive sincethe number of stencil points quadruples as each dimension of the 2D domain increasestwo-folds; see Figure 1.2

Galerkin type methods based on approximating infinite spaces with finite ones such asthe augmented Lagragian and least squares methods were developed by Dean and Glowinski[6]. Their method involves formulating the MAE into a constrained minimization problemand solving the non-linear systems numerically. However, when the solution is not smooth,convergence may not always be guaranteed.

Finite element methods such as the vanishing moment method was studied by Fengand Neilan [8]. The method involves approximating a fully non-linear second order PDEby a sequence of higher order quasi-linear PDEs. However, highly non-linear systems needto be computed and the boundary conditions are hard to approximate well [7].

3

1.3 Overview of The Essay

In this essay, we will study the degenerate elliptic fully nonlinear second order MongeAmpere equation on the square domain with a positive source term f and Dirichlet bound-ary conditions. This essay first applies a transformation to the MAE in Chapter 2 to anonlinear HJB PDE with a linear objective function. Then it adopts a monotone widestencil scheme [16] in Chapter 3 to approximate the numerical solution. In Chapter 4, weprove that the numerical scheme converges to the unique viscosity solution. Finally, inChapter 5, we will present examples of numerical experiments and convergence results.

4

Chapter 2

The Monge Ampere Equations

2.1 The Monge-Ampere Equation

The Monge Ampere equation (MAE) belongs to the class of fully non-linear second orderpartial differential equations. Fully non-linear PDEs are the class of non-linear PDEs whichare non-linear in the highest order derivatives. Formally the operator is of the form:

F [u](x) = F(D2u(x),∇u(x), u(x), x

)= 0,

where F ∈ C(Rd×d × Rd ×R× Ω

)and Ω ⊆ Rd is a bounded domain. Here, D2u(x)

denotes the hessian matrix of u at x. Moreover, it is degenerate elliptic if

F (B, p.z, x) ≤ F (A, p.z, x)

for all x ∈ Ω, z ∈ R, p ∈ Rd andA,B are symmetric d × d matrices with A ≥ B whichmeans that A − B needs to be positive semi-definite. In this paper, we will consider thenon-homogeneous Dirichlet Monge Ampere equations, i.e.

Fu = uxxuyy − u2xy = f(x, y) in Ω, (2.1)

u = g on ∂Ω.

The MAE is a (degenerate) elliptic operator only if we impose the additional requirementthat

1. u is strictly convex,

5

2. f > 0.

Interested readers can refer to [5] for the reference. The domain Ω will be any convexbounded region in the two dimensional Euclidean space R2. In this paper, we will considerthe square domain given by

Ω = [ 0, 1 ]× [ 0, 1 ].

2.2 Viscosity Solutions of the Monge-Ampere Equa-

tion

In general, classical solutions of the MAE do not exist. We need weaker versions of theconcept of ‘solutions’.

Definition 2.1 (Viscosity solution). Let F [ϕ] = det(D2ϕ) − f .The function u ∈ C(Ω)is a viscosity subsolution (supersolution) of F if whenever ϕ ∈ C2(Ω) and x0 ∈ Ωmaximizes (minimizes) u− ϕ for all x in a neighborhood of x0, then we must have

F [ϕ](x0) ≥ (≤)0.

The function u is a viscosity solution if it is both a viscosity subsolution and supersolution.

Geometrically, u is a viscosity subsolution if for every test function φ ∈ C2 that touchesthe graph of u from above at x0 in Figure 2.1a, there holds F [φ](x0) ≤ 0 and if φ touchesthe graph from below at x0 in Figure 2.1b, there holds F [φ](x0) ≥ 0.

For the existence and uniqueness of the viscosity solution of problem (2.1), we need thefollowing theorem.

Theorem 2.1. [12] Let Ω ⊆ Rd be bounded and strictly convex, g ∈ C(∂Ω), f ∈ C(Ω) withf ≥ 0. Then there exists a unique convex viscosity solution u ∈ C(Ω) of problem (2.1).

In Chapter 4, we will see a general framework provided by Barles and Souganidis [3]allows one to show that their approximation schemes achieve convergence to the viscositysolutions.

6

(a) u subsolution (b) u supersolution

Figure 2.1: Illustrations for viscosity solutions. (Image courtesy of SIAM Review, [7])

2.3 From the Monge-Ampere Equation to Hamilton-

Jacobi-Bellman Equation

MAE has derivative terms which are quadratic and it is difficult to construct a numericalscheme without having to compute complex non-linear systems. If we can transform theMAE to HJB, which has a linear PDE objective function, then it would be much easier todiscretize. In fact, we shall see that the Monge-Ampere equation can be formulated in thefollowing form:

minα∈Z

[Lαu− f ] = 0, Lα is a linear operator,

where α ∈ Z is the set of admissible controls. The equivalent formulation above was firstproved in [15] but we will present the version from [18] here. First we need the followinglemma.

Lemma 2.1. Let H be symmetric, g > 0 and S+1 be defined as above. Then H satisfies

maxA∈S+

1

[Tr(AH) + g

√det(A)

]= 0

if and only ifH is negative definite,

2√det(−H) = g.

Interested readers may refer to [15], [18] for the proof. In fact it follows from a variant

7

of the AM-GM inequality:

A,B ≥ 0, 2√det(AB) ≤ Tr(AB),

and the properties of the set S+1 .

Theorem 2.2. Let u = u(x, y) and Ω be convex in R2. Then u solves the elliptic MAE:

det(D2u(x)) = f 2, (2.2)

D2u(x) positive definite on Ω, (2.3)

if and only if it solves the HJB:

minA∈S+

1

[Tr(AD2u(x))− 2f

√det(A)

]= 0, (2.4)

where S+1 = A ∈ Rd×d : A ≥ 0, T r(A) = 1.

Proof. : Let g = 2f , w = −u and H = D2w. Applying Lemma 2.1, we have,det(D2u) = f 2

D2u(x) positive definite on Ω,⇐⇒

2√det(−D2w(x)) = g

D2w(x) negative definite on Ω,

⇐⇒ maxA∈S+

1

[Tr(AD2w(x)) + g

√det(A)

]= 0,

⇐⇒ minA∈S+

1

[Tr(AD2u(x))− 2f

√det(A)

]= 0

Hence the MAE with a convex solution u is equivalent to the HJB equation (2.4). Inorder to compute (2.4) numerically in R2, we need an explicit form of S+

1 . For example,S+

1 in R2 can be parametrized by the set below:[cos θ sin θ− sin θ cos θ

] [a 00 1− a

] [cos θ − sin θsin θ cos θ

]: 0 ≤ a ≤ 1, 0 ≤ θ ≤ 2π

. (2.5)

Finally, if we let Z = [0, 1]× [0, 2π] be the set of all admissible controls and Q = (a, θ) ∈ Z,then (2.4) can be written as:

8

minQ∈Z

LQHu− 2

√a(1− a)f

= 0, (2.6)

whereLQHu = d11uxx + 2d12uxy + d22uyy. (2.7)

Here f is given in (2.1) and

d11 = a cos θ + (1− a) sin2 θ, (2.8)

d22 = a sin θ + (1− a) cos2 θ,

d12 = (1− 2a) cos θ sin θ.

Remark 2.1. It is not hard to verify that

D =

(d11 d12

d12 d22

)is a semi-positive definite matrix.

9

Chapter 3

Discretization of the Monge-AmpereEquation

In this chapter, we will describe our discretization of the transformed Monge-AmpereEquation (2.6). As mentioned, we need the discretization to ensure a monotone schemewhich guarantees the convergence to the desired viscosity solution. The discretization thatwe will adopt is a wide stencil method based on the rotation of the local grid. Additionalproblems are further addressed due to this method, such as the shrinking of stencil pointsthat fall outside Ω.

3.1 Basic Set Up and Notations

The solution of the HJB equation (2.6) is a function defined in Ω ⊆ R2, it will be solvedon a set of n × n grid points. Let Ui,j be the approximate solution of u(xi, yj) wherei, j = 0, 1, . . . , n + 1. Note that since the Dirichlet boundary condition is imposed (2.1),Ui,j are given by the corresponding values of g(xi, yj) when i, j = 0 or n+ 1. We will alsobe using a uniform grid, so the size of our grid at each dimension is

h =1

n+ 1.

For the purpose of implementing the approximation on a computer, we will compute (2.6)via constructing the linear system:

LQ∗U = FQ∗ , (3.1)

10

whereU = (U1,1,U1,2, . . . ,Un,1, . . . ,U1,n, . . . ,Un,n )

is the solution vector. LQ is the n2×n2 matrix consisting of the coefficients of the discretizedHJB and Q∗ = (a∗, θ∗) is the optimal control in the set of discrete admissible controls Zh.For computational purposes, we have to use a single index to reference an entry of U above:

Ul = Ui,j, l = i+ (j − 1)n i, j = 1, . . . , n.

We also let LQl,k be the (l, k)− th entry of the matrix where k = 1, . . . , n2. We give details

on how LQ and FQ are constructed below.

3.2 The LQH Operator Discretization

Observe that LQH , we see that it consists of second derivatives of u: uxx, uxy, uyy. Thestandard approach is to approximate them by central differencing:

uxx(xi, yj) ≈Ui−1,j − 2Ui,j + Ui+1,j

h2,

uyy(xi, yj) ≈Ui,j−1 − 2Ui,j + Ui,j+1

h2,

uxy(xi, yj) ≈2Ui,j + Ui+1,j+1 + Ui−1,j−1

2h2− Ui+1,j + Ui−1,j + Ui,j+1 + Ui,j−1

2h2,

or

uxy(xi, yj) ≈ −2Ui,j + Ui+1,j−1 + Ui−1,j+1

2h2+Ui+1,j + Ui−1,j + Ui,j+1 + Ui,j−1

2h2. (3.2)

Substituting (3.2) into (2.6) and collecting terms, we will see that the coefficients ofUp,q, p 6=i or q 6=j are positive but the coefficient of Ui,j is given by the term

2d11 + 2d22 − 2d12

h2,

which may be positive or negative. As a result, the scheme may not be monotone. Thisis due to the presence of the crossed derivative, creating a non-zero d12 term in the abovediscretization. If we can eliminate the uxy term from LQH , we can guarantee a positivecoefficient discretization and hence a monotone scheme.

11

Figure 3.1: Local grid rotation of angle φ about (xi, yj). Pmi,j, m = 1, . . . 4 are the new

stencil points. r1, r2 are the new axes, h′ is the new stencil length.

We can eliminate the uxy term by a change of variables at each grid point. This isequivalent to an appropriate rotation φ about each (xi, yj)( Figure 3.1) such that when the

corresponding transformation X(φ) is applied to LQH , only the terms uxx, uyy remain, i.e.find φ such that

X(φ)−1LQHX(φ) = d′11∂2

∂w2 + d′22∂2

∂z2, (3.3)

where d′11, d′22 are the corresponding coefficients of the transformed equation. Solving the

above, we find that

φ =1

2arctan

(2d12

d11 − d22

),

X(φ) =

(cosφ − sinφsinφ cosφ

),

and

d′11 = d11 cos2(φ)d22 sin2(φ) + 2d12 cos(φ) sin(φ),

d′22 = d11 sin2(φ)d22 cos2(φ)− 2d12 cos(φ) sin(φ). (3.4)

Consider Figure 3.1. Denote the new axes by r1 and r2, where

r1 =

(cosφsinφ

), r2 =

(− sinφ− cosφ

).

12

The new stencil points are Pmi,j, m = 1, . . . 4, where

P 1i,j = (xi, yj) + h′r1,

P 2i,j = (xi, yj) + h′r2,

P 3i,j = (xi, yj)− h′r1,

P 4i,j = (xi, yj)− h′r2.

Here h′ is the new stencil length. A suitable value will be assigned to it in Chapter 4. As itwill be made more clear in Chapter 4, the stencil length h′ needs to be greater than h andhence the name wide stencil method. In contrast, the wide stencil method by Oberman etal [4], [11], [17] refers to the use of many neighbours, not all of which are the nearest ones.

Solving (2.6) is then equivalent to solving

minQ∈Z

LQT v − 2

√a(1− a)f

= 0, (3.5)

where v = v(w, z) is the representation of u in its new coordinates (w, z).

Let (wi, zj) be the grid points of the new coordinate plane. Note that since (xi, yj) is

the rotation center, we have (xi, yj) = (wi, zj). We discretize LQT by central differencing:

LQT v(wi, zj) ≈ d′11

[u(P 2

i,j) + u(P 4i,j)− 2Ui,j

(h′)2

]+ d′22

[u(P 1

i,j) + u(P 3i,j)− 2Ui,j

(h′)2

]. (3.6)

3.3 Bilinear Interpolation of points from the Wide

Stencil Method

As shown in Figure 3.1, the stencil points Pmi,j, m = 1, . . . 4 do not in general lie on the

original grid points. Let us zoom into the point P 4i,j in Figure 3.2. P 4

i,j lies in the grid squareof its neighbours (p4 + s, q4 + t), s, t = 0, 1. The value of u at P 4

i,j can be interpolated fromthe values at the four points. Let Jh be a bilinear interpolation operator on a domain withgrid size h. To approximate u(P 4

i,j), we have the following form

Jhu(P 4i,j) =

∑s=0,1t=0,1

ωp4+s,q4+tij u(p4 + s, q4 + t). (3.7)

13

(p4, q4) (p4 + 1, q4)

(p4 + 1, q4 + 1)(p4, q4 + 1)

P 4i,j

Figure 3.2: Location of P 4i,j and its 4 nearest neighbours.

Combining the above, we have:

u(P 4i,j) ≈ Jhu(P 4

i,j)

=∑s=0,1t=0,1

ωp4+s,q4+tij u(p4 + s, q4 + t)

=∑s=0,1t=0,1

ωp4+s,q4+tij Up4+s,q4+t.

(3.8)

From (3.6), we have

LQT v(wi, zj) ≈ d′11

[ Jhu(P 2i,j) + Jhu(P 4

i,j)− 2Ui,j(h′)2

]+ d′22

[ Jhu(P 1i,j) + Jhu(P 3

i,j)− 2Ui,j(h′)2

]=

d′11

(h′)2Jhu(P 1

i,j) +d′11

(h′)2Jhu(P 3

i,j) +d′22

(h′)2Jhu(P 2

i,j) +d′22

(h′)2Jhu(P 4

i,j)

− 2d′11 + d′22

(h′)2Ui,j.

(3.9)

3.4 Points near the Boundary

The previous section has covered the case where Pmi,j is inside the computational domain.

When it falls outside, we would need to shrink the point back to the boundary by anappropriate distance h∗, i.e. find h∗ such that (xi, yj)± h∗rk ∈ ∂Ω.

14

(xi, yj)

P 1i,j

P 3i,j

(xi, yj)+2

h′

h+2

Figure 3.3: Shrinking of stencil point P 1i,j

For k = 1, 2, define (xi, yj)+k = (xi, yj) + h+

k rk to be the updated stencil point positionsin the direction of rk and (xi, yj)

−k = (xi, yj) − h−k rk, in the negative direction. Here

h±k , k = 1, 2 are the values assigned to h∗.

Consider Figure 3.3. Let k = 1, P 3i,j lies in the computational grid, so (xi, yj)

−1 = P 3

i,j.However P 1

i,j falls outside. We will shrink the stencil distance from the default h′ toh∗ = h+

2 such that the new stencil point (xi, yj)+1 lies on the boundary. Other cases are

treated similarly, see Algorithm 1.

After obtaining h+k , h

−k from Algorithm 1, we use central differencing to approximate

the second derivatives ∂2v∂w2 or ∂2v

∂z2of LQT to get (3.10). It is easy to see that (3.10) reduces

to (3.9) whenh+k = h−k = h′

is satisfied. If we let LQW be the discretized form of the operator LQT , it can be writtencompactly as

15

Algorithm 1 Shrink points to boundary for grid point (xi, yj), k = 1 or 2

Let (xi, yj)+k = (xi, yj) + h′rk and h+

k = h′

if (xi, yj)+k /∈ Ω then

solve (xi, yj) + h∗rk = (xb, yb) for h∗, such that (xb, yb) ∈ ∂Ωh+k = h∗

(xi, yj)+k = (xb, yb)

end ifLet (xi, yj)

−k = (xi, yj)− h′rk and h−k = h′

if (xi, yj)−k /∈ Ω then

solve (xi, yj)− h∗rk = (x′b, y′b) for h∗, such that (x′b, y

′b) ∈ ∂Ω

h−k = h∗

(xi, yj)−k = (x′b, y

′b)

end ifThe second derivative terms ∂2v

∂w2 or ∂2v∂z2

are approximated as

Jhu((xi,yj)−k )−Ui,jh−k

+Jhu((xi,yj)+k )−Ui,j

h+k

h+k +h−k2

(3.10)

16

LQWv(wi, zj) =2d′11

(h+1 + h−1 )h−1

Jhu((xi, yj)−1 ) +

2d′11

(h+1 + h−1 )h+

1

Jhu((xi, yj)+1 )

+2d′22

(h+2 + h−2 )h−2

Jhu((xi, yj)−2 ) +

2d′22

(h+2 + h−2 )h+

2

Jhu((xi, yj)+2 )

− 2

(d′11

(h+1 h−1 )

+d′22

(h+2 h−2 )

)Ui,j,

(3.11)

for all i, j = 1, . . . n.

In addition, it is worth noting that if (xi, yj) is a point near the corners of the squaredomain, then more than one stencil point will fall outside the domain. This creates prob-lems in the consistency of LQW approximation. However, in Chapter 4, we will see that bychoosing an appropriate h′ and together with Algorithm 1, we can still retain consistency.

3.5 The Matrix form of the Discretized Equations

From(3.5) and (3.11), (3.5) has the following discretized form for all grid points (xi, yj)∈ Ω:

minQ∈Zh

LQWUi,j − 2

√a(1− a)f(xi, yj)

= 0, (3.12)

where Zh is the set of controls discretized to order h to retain consistency. From thisequation, we will compute numerically the values of Ui,j through assembling a matrixwhere each row represents one grid point. This is shown in detail below.

Let us fix the grid point (xi, yj), observing from (3.11), we see that its corresponding

entry, LQl,l, l = i+ (j − 1)n is the coefficient of the term Ui,j above, i.e.,

LQl,l = − 2

(d′11

(h+1 h−1 )

+d′22

(h+2 h−2 )

).

From the previous two sections, stencil points of point (xi, yj) is either inside Ω or outsideit. It falls into the two cases below and for simplicity, let us consider the stencil point P 4

i,j.Recall that its updated position is (xi, yj)

−2 . The other cases are treated similarly.

Case 1: (xi, yj)−2 ∈ Ω.

17

From (3.8) and (3.9), we have

LQl,k =

d′11

(h′)2ωp4+s,q4+tij , l = i+ (j − 1)n, k = p4 + s+ (q4 + t− 1)n, s, t = 0, 1.

In general, we have

LQl,k =

d′11

(h′)2ωp4+s,q4+tij if k = pm + s+ (qm + t− 1)n, s, t = 0, 1, m = 1, 3,

d′22(h′)2

ωp4+s,q4+tij if k = pm + s+ (qm + t− 1)n, s, t = 0, 1, m = 2, 4,

0 otherwise.

(3.13)

To handle cases where we have to use a boundary value (instead of the computationaldomain) from the Dirichlet condition, we need to define the vector BQ. First, let

χmij =

1 if Pm

i,j 6∈ Ω,

0 otherwise.

be the indicator function of whether the stencil points Pmi,j fall outside Ω. Define

BQI = χ1

ijd′11

(h+1 +h−1 )h−1u(P 1

i,j) +χ2ij

d′11(h+1 +h−1 )h+1

u(P 2i,j) +χ3

ijd′22

(h+2 +h−2 )h−2u(P 3

i,j)

+ χ4ij

d′22(h+2 +h−2 )h+2

u(P 4i,j).

(3.14)

Only when Pmi,j falls outside Ω will its corresponding term be added to BQ

I .

Case 2: (xi, yj)−2 ∈ ∂Ω

In this case, we have χ4ij = 1 in (3.14) above. Other instances of m are treated similarly

and BQI is updated.

Finally we are now in the position to describe our matrix system explicitly. Let

Q∗ = ( a∗, θ∗ )

∈ argminQ∈Z

[ LQU + BQ ]I

.

Then, the matrix entries on the l-th row, [LQ∗U]l where l = i+ (j − 1)n, i, j = 1, . . . , n is

18

given by the terms of LQ∗

W Ui,j, i.e. (3.13) above. The right hand side of (3.1) is given by

FQ∗

I = BQ∗

I + 2√a∗(1− a∗)f(xi, yj), (3.15)

which is the sum of vector BQI and the constant term in (3.5).

Hence we have constructed the linear system (3.1) as desired at the beginning of thischapter.

19

Chapter 4

Convergence to the ViscositySolution

As mentioned in previous chapters, we are interested in computing the viscosity solution to(2.1). In [3], a sufficient condition which guarantees convergence to the viscosity solutionis provided. We give a proof that our numerical scheme given in chapter 3 satisfies all ofthe requirements.

For clarity, let us first define the following notations.

x = (x, y),

D2u(x) =

(∂2u∂x2

∂2u∂x∂y

∂2u∂x∂y

∂2u∂y2

).

Then the value function (3.5) is denoted by

Fu = F(x, u(x), D2u(x)) = 0, (4.1)

whereFu = min

Q∈Z

LQT u− 2

√a(1− a)f

. (4.2)

The discrete value function (3.12) will be denoted by

K

(h, (xi, yj),Ui,j, Up,qp 6=i or q 6=j

)= 0, (4.3)

20

for all i, j = 1, . . . n, where

K

(h, (xi, yj),Ui,j, Up,qp6=i or q 6=j

)= min

Q∈Zh

LQWUi,j − 2

√a(1− a)fi,j

. (4.4)

Here fi,j = f(xi, yj) and h is the mesh size.

The convergence theorem in [3] that allows us to guarantee convergence to the viscositysolutions is given below. Interested readers can refer to [3].

Theorem 4.1. Consider a degenerate elliptic equation for which there exist unique viscositysolutions. A consistent (in the viscosity sense), l∞ stable and monotone approximationscheme converges on compact subsets to the viscosity solution.

4.1 Consistency

To be able to state our analysis in a rigorous way, we need the following definition of thesupremum and infimum of a function f which may not always be a continuous function.

Definition 4.1. Let f be a real-valued function on Ω ⊆ R2. The upper semi-continuousenvelope of f , f ∗, is

f ∗ = limr→0

sup f(y) | y ∈ B(x, r) ∩ Ω ,

where B(x, r) denotes the open ball centered at x with radius r.

The lower semi-continuous envelope of f , denoted by f∗, can be defined simi-larly. Based on the meaning of a viscosity solution in Definition 2.1, we give below thecorresponding definition of consistency :

Definition 4.2. [2] A numerical scheme is consistent in the viscosity sense if forany function φ ∈ C∞ with φi,j = φ((xi, yj)) and for all x ∈ Ω, (xi, yj) ∈ Ω, we have

lim suph→0ξ→0

(xi,yj)→x

K

(h, (xi, yj), φi,j + ξ, φp,q + ξp 6=i or q 6=j

)≤ F∗(x, φ(x), D2φ(x)) (4.5)

21

and

lim infh→0ξ→0

(xi,yj)→x

K

(h, (xi, yj), φi,j + ξ, φp,q + ξp 6=i or q 6=j

)≥ F∗(x, φ(x), D2φ(x)), (4.6)

where h and ξ are arbitrary small constants independent of x.

It has similar meanings to the standard consistency definition, that is, the discretizationerror will be negligible as mesh size h decreases.

To prove that our numerical scheme is consistent, we first prove that it is so locally.

Lemma 4.1. (local consistency conditions) Suppose the mesh size is h, and the controldiscretization is of order O(h), and if we take the stencil length h′ (defined in section 3.2),to be

√h, then for any function φ ∈ C∞ and using the notations above, we have that

K

(h, (xi, yj), φi,j+ξ, φp,q+ξp 6=i or q 6=j

)=

Fφi,j + O(h) + O(ξ), Pm

i,j ∈ Ω ∀m = 1, . . . , 4.

Fφi,j + O(√h) + O(ξ), otherwise.

(4.7)

Proof. Case 1: Pmi,j ∈ Ω for all m.

In this case, we have h+1 = h−1 = h+

2 = h−2 =√h. For all (xi, yj) belonging to Ω, we

will show:LQWφi,j = LQT φi,j + O(h).

22

LQWφi,j − LQT φi,j = d′11

[Jhφ((wi, zj) +

√hr1) + Jhφ((wi, zj)−

√hr1)− 2φi,j

h

]

+ d′22

[Jhφi,j( (wi, zj) +

√hr2 ) + Jhφi,j( (wi, zj)−

√hr2 )− 2φi,j

h

]

− d′11

∂2φi,j∂w2

− d′22

∂2φi,j∂z2

= d′11

[φ((xi, yj) +

√he1) + O(h2) + φ((xi, yj)−

√he1) + O(h2)− 2φi,j

h

]

+ d′22

[φ((xi, yj) +

√he2) + O(h2) + φ((xi, yj)−

√he2) + O(h2)− 2φi,j

h

]

− d′11

∂2φi,j∂x2

− d′22

∂2φi,j∂y2

= O(h) + O(h)

= O(h),

where e1, e2 are the canonical axes in the new coordinate grid. Note that the secondequality follows from the O(h2) accuracy of the bilinear interpolation and the second lastequality follows from the error when central differencing with stencil length

√h was used

to approximate the second derivatives. It remains to show that (4.7) is true. For all (xi, yj)such that Pm

i,j ∈ Ω,

K

(h, (xi, yj), φi,j + ξ, φp,q + ξp 6=i or q 6=j

)= min

Q∈Zh

LQT φi,j − 2

√a(1− a)fi,j

+ O(h) + O(ξ)

= minQ∈Z

LQT φi,j − 2

√a(1− a)fi,j

+ O(h) + O(h) + O(ξ)

= Fφi,j + O(h) + O(ξ). (4.8)

The first equality follows from the above analysis and the second, from the discretizationof Z.

Case 2: ∃Pmi,j /∈ Ω.

23

For all (xi, yj) belonging to Ω, we will show:

LQWφi,j = LQT φi,j + O(√h).

Consider the approximation for∂2φi,j∂w2 and suppose m = 1 (or 3), (

∂2φi,j∂z2

is just the samebut m = 2 or 4 replaced). With an abuse of notation here, write φ(wi) as φ(wi, zj) andsimilarly for φ(xi). Let us first consider the following analysis based on the Taylor seriesexpansion:

φ(xi − h−1 e1)− φ(xi)

h−1= −φ′(xi) +

h−12!φ′′(xi)−

(h−1 )2

3!φ(3)(xi) +

(h−1 )3

4!φ(4)(xi),

φ(xi + h+1 e1)− φ(xi)

h+1

= φ′(xi) +h+

1

2!φ′′(xi) +

(h+1 )2

3!φ(3)(xi) +

(h+1 )3

4!φ(4)(xi).

If we sum the above, the term on the left hand side is in fact the discretization of thesecond order derivatives from Algorithm 1, i.e. (3.10) which is the local truncation error:

φ(xi−h−1 e1)−φ(xi)

h−1+

φ(xi+h+1 e1)−φ(xi)

h+1

h−1 +h+12

−φ′′(xi) =h+

1 − h−13

φ(3)(xi)+(h+

1 )2 − h−1 h+1 + (h−1 )2

12φ(4)(xi).

There are several cases to consider. Firstly, suppose that h+1 = h−1 , then from the above

equation, the local truncation error is O((h−1 )2), but since h−1 is order h, we have the errorto be O(h2). The second case is when h+

1 6= h−1 , h+1 (orh−1 ) =

√h. In this case, h−1 = O(h),

so the local error is O(h −√h), which is just O(

√h). Finally, suppose h+

1 6= h−1 , h−1 6=√

h, h+1 6=

√h, then h−1 = O(h) and h+

1 = O(h) which makes the local error to be orderO(h). Combining the three cases, we have:

φ(xi−h−1 e1)−φ(xi)

h−1+

φ(xi+h+1 e1)−φ(xi)

h+1

h−1 +h+12

− φ′′(xi) = O(√h).

The intermediate steps above can all be verified if one works out the details of Algorithm1. From the above analysis, we have the following:

24

LQWφ(wi)− LQT φ(wi) = d′11

[Jhφ(wi + h+

1 r1) + Jhφ(wi − h−1 r2)− 2φ(wi)h−1 +h+1

2

]− d′11

∂2φ(wi)

∂w2

= d′11

[φ(xi + h+

1 e1) + O(h2) + φ(xi − h−1 e1) + +O(h2)− 2φ(xi)h−1 +h+1

2

]− d′11

∂2φ(xi)

∂x2

= O(h) + O(√h)

= O(√h).

To show that (4.7) is true, we follow similar steps as in (4.8) above. Hence, our lemma isproved.

Proposition 4.1. Suppose the numerical scheme 4.3 satisfies the conditions in Lemma4.1, then it is consistent in the viscosity sense.

Proof. One may follow similar steps in [14] .

4.2 Stability

Stability of a numerical scheme is when it produces an approximiate solution that isbounded independent of the mesh size h. It turns out here in this case that stability haslarge relations with the M-matrix property of the matrix LQ constructed in the previouschapter.

Definition 4.3. Let A be an n× n matrix. It is an M-matrix if

i. aii > 0 for all i,

ii. aij ≤ 0 for all i 6= j,

iii. A is nonsingular,

iv. A−1 ≥ 0.

A sufficient condition will be given below, let us first define the following:

25

Definition 4.4. An n × n matrix A is irreducible if there exists an n × n permutationmatrix P such that

PAP T =

(A11 A12

O A22

),

where A11 is an r×r submatrix and A22 is an (n−r)×(n−r) submatrix, where 1 ≤ r < n.If no such permutation matrix exists, then A is irreducible.

Proposition 4.2. (Axelsson, 1996) [1] If A is a real n× n matrix and satisfies

1. aii > 0 for all i,

2. aij ≤ 0 for all i 6= j,

3. A is irreducible,

4. diagonally dominant with at least one i strictly diagonally dominant.

then A is an M-matrix.

Lemma 4.2. If LQW is defined as in (3.11) and if a linear interpolation operator Jh is usedin (3.7), such that

ωpm+s,qm+tij ≥ 0 ∀m = 1, . . . , 4 , s, t = 0, 1, (4.9)∑

s=0,1t=0,1

ωpm+s,qm+tij = 1,

then LQ in (3.15) is an M-matrix for all Q ∈ Z.

Proof. We prove that LQW is an M-matrix by verifying the four conditions in the propositionabove. From (3.13) and (3.5), properties 1, 2 can be verified if we can show that LQ

l,l nevervanishes for all l. This is a bit technical and we shall outline the idea here instead. Noticethat if the optimal control parameter a∗ (c.f. 2.5) is neither 0 nor 1, then the matrix Din 2.1 is strictly positive definite, hence d′11 6= 0 and d′22 6= 0 by definition of positive defi-

niteness (for example, to verify the case for d′11, note that d′11 =

(sin(φ)cos(φ)

)′D

(sin(φ)cos(φ)

)).

However, when a = 0, we claim that if d′11 = 0 then d′22 6= 0 and vice versa. For a = 1, itis similar. If what we have just claimed is true, then by (3.5), LQ

l,l never vanishes. So when

26

a = 0, it is not hard to verify that d11 = sin(θ)2, d22 = cos(θ)2 and d12 = cos(θ) sin(θ). Ifit occurs that d′11 = 0, then by (3.4), we have

−2 cos(θ) sin(θ) cos(φ) sin(φ) = sin(θ)2 cos(φ)2 + cos(θ)2 sin(φ)2,

then also by (3.4),

d′22 = sin(θ)2 sin(φ)2 + cos(θ)2 cos(φ)2 + sin(θ)2 cos(φ)2 + cos(θ)2 sin(φ)2 = 1 6= 0.

We will concentrate on problems such that the resulting matrix LQW satisfies property 3.Let l = i+ (j − 1)n, now we prove property 4. We have the following two cases:

Case 1: Pmi,j ∈ Ω for all m.

From (3.13),

|[LQ]l,l| −∑k 6=l

|[LQ]l,k| = 2(

d′11

h+d′22

h)− d′11

h

∑s=0,1t=0,1

ωp1+s,q1+tij − d′11

h

∑s=0,1t=0,1

ωp2+s,q2+tij

− d′22

h

∑s=0,1t=0,1

ωp3+s,q3+tij − d′22

h

∑s=0,1t=0,1

ωp4+s,q4+tij

= 0.

Case 2: ∃Pmi,j /∈ Ω.

In general, we have

|[LQ]l,l| −∑k 6=l

|[LQ]l,k| = 2(

d′11

h+1 h−1

+d′22

h+2 h−2

)− χ1ij

2d′11(h+1 +h−1 )h−1

−χ2ij

2d′11(h+1 +h−1 )h+1

− χ3ij

2d′22(h+2 +h−2 )h−2

−χ4ij

2d′22(h+2 +h−2 )h+2

≥ 2(d′11

h+1 h−1

+d′22

h+2 h−2

)− 2(d′11

h+1 h−1

+d′22

h+2 h−2

)

= 0.

However, note that in the boundary points (e.g. when l = 1, n2), there must be a stencilpoint falling outside the computational domain and hence χmij = 0 for some m = 1, . . . , 4,

27

e.g. let m = 1:

|[LQ]l,l| −∑k 6=l

|[LQ]l,k| = 2(

d′11

h+1 h−1

+d′22

h+2 h−2

)−χ2ij

2d′11(h+1 +h−1 )h+1

− χ3ij

2d′22(h+2 +h−2 )h−2

−χ4ij

2d′22(h+2 +h−2 )h+2

> 2(d′11

h+1 h−1

+d′22

h+2 h−2

)− 2(d′11

h+1 h−1

+d′22

h+2 h−2

)

= 0.

So in cases like that , we have strict diagonal dominance on that row. Combining the twocases, property 4 is verified.

From the proposition below we see that our scheme is a stable scheme. For the proof,interested readers may refer to [16].

Proposition 4.3. If the conditions for Lemma 4.2 are satisfied then the discretizationscheme (4.3) is l∞ stable. And as mesh size h→ 0, we have

‖U‖∞ ≤ max(∥∥U0

∥∥∞ , ‖g‖∞

),

where g is the given Dirichlet boundary condition of (2.1).

4.3 Monotonicity

Monotonicity is an essential requirement for our scheme to converge to the viscosity solutionas mentioned earlier. We give the definition based on [16].

Definition 4.5. The discrete scheme is monotone if Vi,j ≥ Ui,j for all i, j, we have

K

(h, (xi, yj),Vi,j, Vp,qp 6=i or q 6=j

)≥ K

(h, (xi, yj),Ui,j, Up,qp6=i or q 6=j

).

Proposition 4.4. If the scheme (4.3) satisfies the condition of Lemma 4.2, then ourdiscretization consists of positive coefficients only and thus results in a monotone scheme.

Proof. From (3.13), all the coefficients involved in the discretization are positive for allQ ∈ Z, since linear interpolation satisfying (4.9) was used and the coefficients d′11 and

28

d′22 are positive for all Q ∈ Z, φ ∈ [−π2, π

2] (remark 2.1). Hence monotonicity follows from

similar derivations in [10].

29

Chapter 5

Numerical Results

In this chapter, we apply the proposed discretization method for the MAE on three ex-amples with smooth to mildly singular solutions. The computations were performed on aMac desktop with 2.8GHz Intel Core Duo processor and 4GB memory, using MATLABrunning in Mac OS X.6.

All the examples were used by [4] and [6]. The method used in [6] converged for thefirst two but not for the last one. We will follow [4] and use the approximate solution to

uxx + uyy =√

2f,

as the initial estimation for all the examples. This is to minimize the time needed to obtainthe final convergence result. In each iteration, we use policy iteration to numerically solvethe HJB equation (3.5). It is an iterative process and at each iteration, the value functionfrom the previous policy is updated and then an improved policy is found via the new valuefunction. In theory, if enough iterations are evaluated, the optimal policy should convergeto the optimal control of the HJB and the value function should converge towards itssolution. In our numerical experiments, we will just fix the number of iterations due tothe computational time using MATLAB.

Error is measured as the L2 norm of the difference between the computed and the exactsolutions. Formally, let Uh be the approximate solution and eh be the error at the gridlevel h, then

eh = h∥∥Uh − u

∥∥2.

The loglog plot of the graph of error versus mesh size h is plotted to analysis the convergencerate and the slope of the best fitted line is computed which represents this rate. In nearly

30

all the examples, we observed convergence rates of linear convergence O(h) or near linearconvergence.

5.1 Smooth Examples

5.1.1 Example 1

Consider the problem

Lu = (1 + x2 + y2) exp(x2 + y2) in Ω, (5.1)

u = exp(

12(x2 + y2)

)on ∂Ω,

(5.2)

where Ω is the square domain [−1/2, 1/2]× [−1/2, 1/2].

An exact solution isu(x, y) = exp

(12(x2 + y2)

). (5.3)

Our method converges to the same numerical solution (5.3). A plot of the computed andexact solutions is given in Figures 5.1a and 5.1b respectively.

Figure 5.1c is a loglog plot of mesh size h versus the L2 error and the slope of the bestfitted line was found to be 1.02 which implies that the convergence rate is linear.

5.1.2 Example 2

Consider the problem

Lu = 2 in Ω, (5.4)

u = x2 + y2 on ∂Ω,

where Ω is the square domain [0, 1]× [0, 1].

An exact solution isu(x, y) = x2 + y2. (5.5)

A plot of the computed and exact solutions is given in Figures 5.2a and 5.2b respectively.The computed solution closely approximates the exact solution (5.5). Figure 5.2c is a loglog

31

−0.5

0

0.5

−0.5

0

0.5

1

1.05

1.1

1.15

1.2

1.25

(a) Surface plot of computed solution on32× 32 grid.

−0.5

0

0.5

−0.5

0

0.5

1

1.05

1.1

1.15

1.2

1.25

(b) Surface plot of exact solution on 32×32grid.

−4.5 −4 −3.5 −3 −2.5 −2 −1.5 −1−8.5

−8

−7.5

−7

−6.5

−6

−5.5

−5Logarithmic Plot of L2 Error vs Mesh Size h

log(h)

log(e

rror)

(c) Best fit line in the loglog plot of meshsize h vs error, slope of line is 1.02.

Figure 5.1: Plots of example 1

plot of the mesh size h versus the L2 error and the slope of the line was found to be 1.51.Although the convergence was oscillatory, the overall trend implies that the convergenceis approximately linear.

32

0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

0

0.5

1

1.5

2

(a) Surface plot of computed solution on32× 32 grid.

0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

0

0.5

1

1.5

2

(b) Surface plot of exact solution on 32×32grid.

−4.5 −4 −3.5 −3 −2.5 −2 −1.5 −1−9.5

−9

−8.5

−8

−7.5

−7

−6.5

−6

−5.5

−5

−4.5Logarithmic Plot of L2 Error vs Mesh Size h

log(h)

log(e

rror)

(c) Best fit line in the loglog plot of meshsize h vs error, slope of line is 1.51.

Figure 5.2: Plots of example 2

33

5.2 Non-smooth Solutions

5.2.1 Example 3

Consider the problem

Lu =2

(2− x2 − y2)2in Ω,

u = −√

2− x2 − y2 on ∂Ω,

where Ω is the square domain [0, 1]× [0, 1].

An exact solution isu(x, y) = −

√2− x2 − y2. (5.6)

From Figure 5.3a, the gradient of f is unbounded at (1, 1), making the equation moderatelysingular. This example was also used by [6] and their method was known to diverge. Asolution plot on grid 32 × 32 is plotted in Figure 5.3b. Figure 5.3d is a loglog plot ofthe mesh size h versus the L2 error and the slope of the best fitted line was found to beapproximately 0.82 which implies that the convergence is near linear. The error is mainlylocated at the region of blow up, see Figure 5.3c and does not affect the overall convergenceof our solution as much as the method used in [6].

34

0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

0

20

40

60

80

100

120

140

(a) Surface plot of f on 32× 32 grid.

0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

−1.6

−1.4

−1.2

−1

−0.8

−0.6

−0.4

−0.2

(b) Surface plot of computed solution on32× 32 grid.

0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

0

1

2

3

4

5

6

7

x 10−3

(c) Surface plot of pointwise error on 32×32grid.

−4.5 −4 −3.5 −3 −2.5 −2 −1.5 −1−7

−6.5

−6

−5.5

−5

−4.5Logarithmic Plot of L2 Error vs Mesh Size h

log(h)

log(e

rror)

(d) Best fit line in the loglog plot of meshsize h vs error, slope of line is 0.82.

Figure 5.3: Plots of example 335

Chapter 6

Conclusions

In this paper, we proposed a numerical scheme to compute the unique viscosity solutionof the elliptic MAE with Dirichlet boundary condition. We first transformed the MAE toa Hamilton-Jacobian-Bellman equation whose objective function is a linear second orderPDE coupled with non-linear control parameters. This allows ease of discretization. Toobtain a monotone scheme, further work was done to transform it by a local grid rotation toeliminate the crossed derivatives terms, thus resulting in a wide stencil discretization. Thiscaused new challenges such as the stencil points lying outside the computational domain,however we modified the algorithm in [16] to shrink the points back to its boundary andat the same time retained consistency of the discretization.

Our numerical method is also stable and monotone in addition to being consistent withthe original MAE, hence by the Barles Souganidis convergence theorem, we proved that itconverged to the unique viscosity solutions. In the analysis of consistency, we showed thatthe convergence rate was either linear or order O(

√h).

The numerical experiments performed had solutions ranging from smooth to moderatelysingular. They all showed convergence and at a linear or near linear rate (O(h)) and theworser case of convergence at the boundary had minimal effect.

Directions for further work will be to design more efficient algorithms to solve thenonlinear HJB as policy iteration alone took too long for practical applications. To applyour numerical scheme to real life image registration problems especially medical imagingis also one direction of pursuit.

36

References

[1] Owe Axelsson. Iterative solution methods. Cambridge University Press, 1996. 26

[2] G. Barles. Convergence of numerical schemes for degenerate parabolic equations aris-ing in finance. Cambridge University Press, Cambridge, 1997. 21

[3] G. Barles and P.E. Souganidis. Convergence of approximation schemes for fully non-linear second order equations. Asymptotic Anal., 4:271–283, 1991. 3, 6, 20, 21

[4] Jean-David Benamou, Brittany D Froese, and Adam M Oberman. Two numericalmethods for the elliptic monge-ampere equation. ESAIM: Mathematical Modellingand Numerical Analysis, 44(04):737–758, 2010. 13, 30

[5] Shiu-Yuen Cheng and Shing-Tung Yau. On the regularity of the monge-ampere equa-tion det ( 2 u/ xi xj)= f (x, u). Communications on Pure and Applied Mathematics,30(1):41–68, 1977. 6

[6] E.J. Dean and R. Glowinski. Numerical methods for fully nonlinear elliptic equationsof the monge-ampere type. Computer Methods in Applied Mechanics and Engineering,195:1344–1386, 2006. 3, 30, 34

[7] Glowinski R. Feng, X. and M. Neilan. Recent developments in numerical methods forfully nonlinear second order partial differential equations. SIAM Review, 55:205–267,2013. ix, 3, 7

[8] X. Feng and M. Neilan. Mixed finite element methods for the fully nonlinear mongeam-pre equation based on the vanishing moment method. SIAM Journal on NumericalAnalysis, 47(2):1226–1250, 2009. 3

[9] Bernd Fischer and Jan Modersitzki. Ill-posed medicinean introduction to image reg-istration. Inverse Problems, 24(3):034008, 2008. ix, 2

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[10] P. A. Forsyth and G. Labahn. Numerical methods for controlled Hamilton-Jacobi-Bellman partial differential equations in finance. Journal of Computational Finance,11(2):1, 2007. 29

[11] Brittany D Froese and Adam M Oberman. Convergent finite difference solvers forviscosity solutions of the elliptic monge-ampere equation in dimensions two and higher.SIAM Journal on Numerical Analysis, 49(4):1692–1714, 2011. ix, 1, 3, 13

[12] C. E. Gutierrez. The Monge Ampere Equation. Birkhuser Mathematics, Basel, 2001.6

[13] Steven Haker, Lei Zhu, Allen Tannenbaum, and Sigurd Angenent. Optimal masstransport for registration and warping. International Journal of Computer Vision,60(3):225–240, 2004. 1

[14] Y. Huang and P. A. Forsyth. Analysis of a penalty method for pricing a guaranteedminimum withdrawal benefit (GMWB). Journal of Numerical Analysis, 32(1):320–351, 2012. 25

[15] N. V. Krylov. On control of the solution of a stochastic integral equation with degen-eration. Math. USSR Izv., 6(1):249, 1972. 7

[16] K. Ma and P.A. Forsyth. An unconditionally monotone numerical scheme for the twofactor uncertain volatility model. 2014. 4, 28, 36

[17] A. M. Oberman. Wide stencil finite difference schemes for the elliptic monge-ampereequation and functions of the eigenvalues of the hessian. Discrete Contin. Dyn. Syst.Ser. B, 10(1):221–238, 2008. 13

[18] I. Smears. Hamilton-Jacobi-Bellman Equations. Analysis and Numerical Analysis. 7

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