sawinder pal kaur phd thesis
DESCRIPTION
My PhD ThesisTRANSCRIPT
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