On an Eigenvalue Problem for Some Nonlinear

Transformation of Multi-Dimensional Arrays

Sawinder Pal Kaur

Ph.D. Dissertation Defense

Advisors : Prof. Israel Koltracht

Prof. Yung S. Choi

Prof. Vadim Olshevsky

Department of Mathematics

University of Connecticut, Storrs, CT, USA

December 10,2007

Outline

Motivation

Previous Work

Key Contributions

Existence of a unique positive solution of some

nonlinear transformation in two and three variables and

properties of the solution

Sign preservation in the solution of discrete semi-linear

parabolic equation with general totally positive matrices

Positive Solution of Some Nonlinear

Transformation

Motivation: Bose-Einstein Condensation (BEC)

Fluid in the lowest energy state

Properties are not completely

understood

It provides the best control over the

motion and position of atoms

Motivation: Time Independent Gross-Pitaevskii Equation

Certain properties of BEC at zero temperature are described by time

independent Gross-Pitaevskii equation (GPE)

Discretization of one dimensional time independent GPE with zero

boundary conditions using finite difference method

A is irreducible Stieltjes matrix,

and

3Au ku u

1

, || || 1

n

u

u u

u

3

1

3

3

n

u

u

u

2 2 2

1

2

2 2 2

2 11

1 1

1 2

0

0 n

h a x

Ah

h a x

Existence of Unique Positive Solution in One Variable: 1

here A is a irreducible Stieltjes matrix, has a

positive solution

for λ > µ, where µ is the smallest positive eigen value of A

and

functions satisfying

If, in addition, for

whenever 0< s < t, then the positive solution is unique.

( ) ,Au F u u

1 1

2 2

( )

( )( ) ,

( )n n

f u

f uF U

f u

1for 1,..., , : (0, ) (0, ) are ii n f C

0

( ) ( )lim 0, limi i

t t

f t f t

t t

(

1,...,

)

,

,( )i i

i

t

n

f s f t

s

Properties of the Solution in One Variable

Properties of positive solution

if

is continuous on

For

For

Generalize some results of the Perron-Frobenius theorem for non-linear perturbation of the Stieltjes matrix.

1 2( ) ( )x x 1 2

( )x ( , )

1,..., , lim ( )ii n x

1,..., , lim ( ) 0ii n x

Finite Difference Method in Two Variables Case

GPE on a truncated square

here Γ is the boundary of the square –T ≤ x, y ≤ T and

Discrete GPE

here

similarly Ay is defined.

, ,T T T T

2 2 3

2

( )

( , ) 0, | 0, ( , ) 1

,

T T

T T

u u

u ax by u ku u

x y u u x y dxdy

xx yyu u u

3

x yA U UA kU U ijU u

3 3 2 2

11 1 1

3

2

3 3 2 21

2 11

, 1 1

1 2

0

0

n

x

n nn n

u u h ax

U Ah

u u h ax

Existence of the Unique Positive Solution in Two Variables: 1

Theorem :

Let be the smallest eigenvalue of

and be the corresponding eigen matrix,

here A and B are n x n irreducible Stieltjes matrices

µA and µB are smallest positive eigen values and pA and pB

are corresponding eigenvectors of A and B respectively.

A B ( )L U AU UB

T

A BP p p

Existence of the Unique Positive Solution in Two Variables: 2

Let λ > (µA + µB) and

where for are C1 functions

satisfying the condition

11 11 1 1

1 1

( ) ( )

( )

( ) ( )

n n

n n nn nn

f u f u

F U

f u f u

, 1,..., , : (0, ) (0, )iji j n f

0

( ) ( )lim 0, lim

ij ij

t t

f t f t

t t

Existence of the Unique Positive Solution in Two Variables: 3

then L(U)+F(U)=AU + UB + F(U) = λ U has a positive solution

If in addition for

then the positive solution is unique.

, 1,...,

(0

) )

,

(,

ij ijf s f t

s t

i j n

s t

Existence of the Unique Positive Solution in Two Variables: 4

Proof

proved is monotone using

Kronecker product of matrices and using Vec-function

Kronecker product of two matrices A and B

( )L U AU UB

11

11 1 1

11 1 1

1

1

1 1

1

1

,

, ,

( )

n m

n q

m mn p

m mn

n

pq

n

m

a

a B a B a

A B mp nq Vec A

a a b b

A m n B p q

a a b b

a B a B a

a

Existence of the Unique Positive Solution in Two Variable: 5

cU+ L(U) + F(U) = (cI +λ) U, c>0

write as an equivalent fixed point problem

here

using Kantorovich theorem It is shown that the iteration

converges monotonically to a positive solution for an appropriate

choice of initial iterate and sufficiently large c.

1( ) (( ) ( )),

( ) ,

U S U L c U F U

L U cU AU UB

Kantorovich theorem

Result of Kantorovich

Theorem : Let be defined and continuous on an interval [y, z] and let

(i): y < S(y) < z

(ii): y < S(z) < z

(iii): implies

(iv) If y ≤ x1≤ … ≤ xn ≤ …≤ z and xn ↑ x then S(xn)↑S(x)

Then

(a): the fixed point iteration with converges:

(b): the fixed point iteration with converges:

(c): if x is a fixed point of S in [y, z] then

(d): S has a unique fixed point in [y, z] if and only if

1 2y x x z 1 2( ) ( )y S x S x z

1( )k kx S x

* * * *, ( ) , ;kx x S x x y x z

1( )k kx S x 0x z

* * * *, ( ) , ;kx x S x x y x z *

*x x x

*

*x x

: n nS R R

0x y

Existence of the Unique Positive Solution in Two Variables: 6

y = β1pA (pB)T= β1P, and z = β2 pA (pB)T= β2P, here pA (pB)T is the

eigen matrix corresponding to smallest eigenvalue of L(U), and

0<β1 < β2

c = max[sup |fij|]-λ

proved that S(U) satisfies all the conditions of Kantorovich theorem

in the interval [β1P, β2P]

Positivity of the solution follows from the fact that all entries of P are

positive and the the fixed point of the transformation lies in the

interval [β1P, β2P]

Existence of the Unique Positive Solution in Three Variables: 1

Discrete GPE on a truncated cube

U is a 3-D array

Given

and are defined similar to

1, , 1, ,

, , , ,

( )

( )

( )

x j k j k

x

x N j k N j k

A U U

A U A

A U U

3( ) ( ) ( )x y zA U B U C U kU U

, 1,...j k N

( )yB U ( )zC U ( )xA U

x

y

z

A

Theorem :

Let and Let µA + µB+ µC be the smallest eigenvalue and

be the corresponding eigen array of

A, B, and C are n x n irreducible Stieltjes matrices

µA, µB ,and µC are smallest positive eigenvalues

are corresponding eigen vectors of A, B,

and C respectively

let λ > (µA + µB+ µC),

( ) ( ) ( ) ( )x y zL U A U B U C U

( ) ( ) ( )ijk A i B j C kp p p p p

, and A B Cp p p

Existence of the Unique Positive Solution in Three Variables: 2

Let here for

are C1 functions satisfying the condition

then has a positive solution

If in addition for

then the positive solution is unique

0

( ) ( )lim 0, lim

ijk ijk

t t

f t f t

t t

( ) ( )ijk ijkF U f u , , 1,..., , : (0, ) (0, )ijki j k n f

( ) ( ) ( ) ( )x y zA U B U C U F U U

(

, , 1,...,

) ( ),

,

0ijk ijkf s f t

s t

i j k n

s t

Existence of the Unique Positive Solution in Three Variables: 3

Proof

prove is monotone

write L(U) as

using Kronecker product of 3 matrices and expanding the

three dimensional arrays into n3 –vectors

show the matrix

is monotone

Rest of the proof is similar to two variable case

( ) ( ) ( ) ( )x y zL U A U B U C U

( ( ( )( ))) n n n n n nA I I I B IVe I IU ec UCc L V

n n n n n nA I I I B I I I C

Existence of the Unique Positive Solution in Three Variables: 4

Kronecker Product of Three Matrices

Example: let be an n x n matrix

and In be an identity matrix

then

11 1

1

n

n nn

g g

G

g g

3 3

11 1

2

1

2

is matrix

here is

0

0

matrix

n

n nn

ij

ij

j

n

i

n

G G

n n

G G

g

I

G

g

I

n n

G

111

11

1 1

1

1

( )

n

n

n

ijk

n

nn

nnn

u

u

u

Vec U

u

u

u

U u

Generalization of Perron-Frobenius Theorem

Properties of Positive solution in Two and Three variable case are similar to

the properties in one variable case

Let B be an irreducible and non-negative n×n matrix and ρ(B) be its

spectral radius, then

ρ(B) > 0

ρ(B) is an eigenvalue of B

There is a unique positive eigenvector x such that Bx =ρ(B)x

Generalization in higher dimensions

If F(X) satisfies the conditions of the theorem, then

Any number greater than the smallest positive eigenvalue of the

operator L(X) is an eigen value of

There is a unique positive eigen array corresponding to the

every eigenvalue

( ) ( )L X F X X

Sign Preservation Properties of Some Nonlinear

Transformations

Motivation: Non-increasing Oscillations in the

Solution of One Variable Heat Equation

One variable prototype of heat equation

here and α is the constant of diffusivity.

The number of oscillations in the solution u(x,t) does not increase

with time.

2

0

,0 1, 0

0, 1, 0,

,0

t xxu u x t

u t u t

u x u x

( , )u u x t

Semi-linear Parabolic Equation

One dimensional semi-linear parabolic equation

here and are

continuous functions on

0

0 1, 0

0, 1, 0,

,0

( , ) ( ),t xx x

x t

u u b

u t u t

u x u x

x t u f u

[0, ] .T R

( , ),b

b x tx

2

2, ,

f ff

u u

Discrete Semi-linear Parabolic Equation

Discretizing using

k, h are time and space mesh respectively

1 1 1

0

1

1

2

( ) ,

, ;

(0, ) (1, ) 0,

2

( ,0)

2j j j j jjj i i

j jj j

i

j j ji ii i i i i

j j i

iii

i

i

D

u

u uu b u f u u

k

u

u u

u x u

u

t u t

hu

h

uDu

( , ) j

i j iu x t u

,i jx ih t jk

Sign Preservation in Discrete Semi-linear Parabolic Equation

Theorem : Suppose f ,f‘, f” are continuous and

in for some real number then for every

ε>0 there exists a number such that under the

condition and

then at any time step j the number of peaks in are

less than or equal to he number of peaks in

0 ,f

Mu

[0, ]T R 0 ,M

0 0h

0h h

2

4 2

hk

h

j

iu

0.iu

0(( ), ,0) j

i j i i iu x t u u x u

Semi-linear Parabolic Equation

One dimensional semi-linear parabolic equation

here and is a sign preserving

continuous functions on

0

0 1,

( ) ( )

0

0, 1, 0,

,0

,t xx

x t

u t u t

u x u

u u v x u f u

x

[0, ] .T R

( ) 0v x f u

Forward Difference Method: 1

Discretizing using forward difference scheme

where

( 1) ( ) ( ) ( )( )k k k k

vU AU D U F U

1 1

( )

2

2

1

( )

1 2 0 0

1 2

0

0 0 1 2

( ) ( )

( )

( ) ( )

,, ,

,

0

0

k

k

k

n n

k

k

k

n

v

r r

r r

r

r r

v x f u

F U

v x f u

kr

h

u

U A

u

D and

Forward Difference Method : 2

Theorem : Suppose

k denotes the iteration index and

matrix with

and with

then the number of sign changes in are less than

or equal to the number of sign changes in for any iteration step k.

( 1) ( ) ( )( )k k kU U F U

1 1

2 2

1

0 0

,0

0 0

n

n n

a b

c a

b

c a

n n , , 0 det( ) 0i i i ia b c and

1 1( )

( )

( )n n

f u

F U

f u

( ) 0 0,

( ) 0 0,

( ) 0 0

i i i

i i i

i i i

f x if x

f x if x and

f x if x

( 1)kU

( )kU

Forward Difference Method : 3

There is a sign change whenever two consecutive non zero entries

in a vector have opposite signs and zero has no sign

e.g changes signs only once

Proof : Using Crout factorization write

shown that all entries in lower and upper triangular matrices are

positive by using

, 0

1 1

2

10 0

0 0 1n

n

n

f

b f

b

d

d

, , 0and ( ) 0i i i ia b c det

Forward Difference Method : 4

factorize

eij ,gij >0 and D is a diagonal matrix with all positive entries

it is shown that multiplication with each and and addition of

will not increase number of sign changes in

21 1 12 1 ,

where

1 0 1 0

1,

1

0 1 0 1

nn n n

ij

ij ij

ij

E E DG G

gE G

e

ijEijG

( )kU( )( )kF U

Backward Difference Method : 1

Discretizing using backward difference scheme

where

( 1) ( ) ( ) ( )( )k k k k

vAU U D U F U

2

2

1 2 0 0

1 2, ,

0

0 0 1 2

r r

r r kA r

r h

r r

Backward Difference Method : 2

Theorem : Suppose

k denotes the iteration index and

matrix with

and with

then the number of sign changes in are less than

or equal to the number of sign changes in for any iteration step k.

( 1) ( ) ( )( )k k kU U F U

1 1

2 2

1

0 0

,0

0 0

n

n n

a b

c aP

b

c a

n n , , 0 and det( ) 0i i i iPa b c

1 1( )

( )

( )n n

f u

F U

f u

( ) 0 0,

( ) 0 0,

( ) 0 0

i i i

i i i

i i i

f x if x

f x if x and

f x if x

( 1)kU

( )kU

Backward Difference Method : 3

Proof : factorize

Show eij ,gij >0 and D is a diagonal matrix with all positive entries

Matrices will have all positive entries

it is shown that multiplication with each , and addition of

will not increase number of sign changes in

21 1 12 1 ,

where

1 0 1 0

1,

1

0 1 0 1

nn n n

ij

ij ij

ij

E E DG G

gE G

e

1

ijE 1

ijG

( )kU( )( )kF U

1 -1 and ij ijE G

Crank-Nicolson Method: 1

Theorem : Suppose

be an equation where k denotes the iteration index and

be matrices with

( 1) ( ) ( )( )k k kAU BU F U

1 1 1 1

2 2 2 2

1 1

0 0 0 0

,0 0

0 0 0 0

n n

n n n n

a b a b

c a c aA B

b b

c a c a

n n

, , , , , 0 and det( ) 0, det( ) 0i i i i i i i ia b c a b c A B

Crank-Nicolson Method: 2

and with

then the number of sign changes in are less than

or equal to the number of sign changes in for

each iteration step k.

1 1( )

( )

( )n n

f u

F U

f u

( ) 0 0,

( ) 0 0,

( ) 0 0

i i i

i i i

i i i

f x if x

f x if x

f x if x

( 1)kU

( )kU

Conclusions

Discrete time independent GPE and certain

transformations which are similar to discrete GPE have

unique positive solution

The solution is monotonic, continuous and bounded from

below in a semi infinite interval

Generalize some results of Perron-Frobenius Theorem

to higher dimensions

Shown that the number of oscillations in the solution of

discrete one- dimensional semi linear parabolic equation

with homogeneous dirichlet boundary conditions does

not increase as time propagates