Siraj –ul – Islam
Laboratory for Multiphase Processes
University of Nova Gorica, Slovenia
Siraj –ul – Islam
Laboratory for Multiphase Processes
University of Nova Gorica, Slovenia
Some Applications of Wavelets
Siraj –ul – Islam
Laboratory for Multiphase Processes
University of Nova Gorica, Slovenia
Some Applications of Wavelets
Siraj –ul – Islam
Laboratory for Multiphase Processes
University of Nova Gorica, Slovenia
Khyber Pass
Some
Wavelets
"Khyber
is Applications
a Hebrew wordofmeaning
a fort"
Siraj –ul – Islam
Laboratory for Multiphase Processes
University of Nova Gorica, Slovenia
Khyber Pass
"Khyber is a Hebrew word meaning a fort"
• Alexander the Great and his army marched through the
Khyber to reach the plains of India ( around 326 BC)
• In the A.D. 900s, Persian, Mongol, and Tartar armies forced their
way through the Khyber
• Mahmud of Ghaznawi, marched through with his army as
many as seventeen times between 1001-1030 AD
• Shahabuddin Muhammad Ghaur, a renowned ruler of Ghauri
dynasty, crossed the Khyber Pass in 1175 AD to consolidate
the gains of the Muslims in India
• In 1398 AD Amir Timur, the firebrand from Central Asia, invaded India
through the Khyber Pass and his descendant Zahiruddin Babur made
use of this pass first in 1505 and then in 1526 to establish a mighty
Mughal empire
• January 1842, in which about 16,000 British and Indian troops
were killed
Some Applications of Wavelets
Siraj –ul – Islam
Laboratory for Multiphase Processes
University of Nova Gorica, Slovenia
Some Applications of Wavelets
Siraj –ul – Islam
Laboratory for Multiphase Processes
University of Nova Gorica, Slovenia
Some Applications of Wavelets
Siraj –ul – Islam
Laboratory for Multiphase Processes
University of Nova Gorica, Slovenia
What are Wavelets?
A wavelet is a function which
• maps from the real line to the real line
• has an average value of zero
• has values zero except over a bounded domain
10
What are Wavelets?
The word wavelet refers to the function h(t) that generates a
basis for the orthogonal complement of V0 in V1
• A small wave
• Extends to finite interval
Wavelets analysis is a procedure
through which we can decompose
a given function into a set of elementary
waveforms called wavelets
11
Types Of Wavelets
ICCES 2010
Las Vegas, March 28 - April 1, 2010
12
The Haar Scaling Functions and Haar Wavelets
a) Haar scaling function (Father function)
b) Haar Wavelet function (Mother wavelets)
13
The Haar Scaling Function and
14
The Haar Wavelets
15
The Haar Wavelets and its Integrals
with the collocation points
The repeated integral of Haar wavelet is given by
16
The Haar Wavelets and its Integrals
17
Some applications of wavelets
• Numerical Analysis
• Ordinary and Partial Differential
Equations
• Signal Analysis
• Image processing and Video Compression
(FBI adopting a wavelet-based algorithm as
a the national standard for digitized finger
prints)
• Control Systems
• Seismology
Highly Oscillating function
19
Multi-Resolution Analysis
20
Multi-Resolution Analysis
21
Multi-Resolution Analysis
The space L2 (R) can be decomposed as an infinite orthogonal direct sum
L2 (R)  V0  W0  W1  W2  .
In particular, each f  L2 (R) can be written uniquely as

f  v0   w j where v0 belongs to V0 and w j belongs to Wj
j 0
22
Multi-Resolution Analysis
23
Multi-Resolution Analysis
Scaling function (Father wavelet) h0 basis in V
Wavelet function (Mother wavelets) hi basis in W
24
Gaussian Quadrature
 ( x)
25
Gaussian Quadrature
26
Gaussian Quadrature
27
Problems with Gaussian Quadrature
• Solution 2n by 2n system
• Search for better nodal values
• Finding optimized values for the unknown weights
28
Numerical Integration based on Haar wavelets
Inter. J. Computer Math.
2010
29
Numerical Integration based on Haar wavelets
30
Numerical Integration based on Haar wavelets
31
Numerical Integration based on Haar wavelets
32
Numerical integration for double and triple integrals
33
Numerical integration for double and triple integrals
34
Numerical double integration with variable limits
To extend the present idea to numerical integration with
variable limits and make it more efficient, we use an iterative
approach instead of using two and three dimensional wavelets
35
Numerical triple integration with variable limits
36
Numerical results
37
Numerical results
38
Numerical results
39
Numerical results
40
Numerical results
41
Numerical results
Symmetric Gauss
Legendre
107
Symmetric Gauss
Legendre
1011
42
Convergence of the method
43
Numerical Solution of Ordinary Diff. Eqs.
Existing Methods
• Runge-Kutta family of Methods (Need shooting like to convert
BVP into IVP, Stability limits)
• Finite difference Methods (Low accuracy and large matrix
inversion)
• Asymptotic Methods (Series solution convergence problem)
44
Shooting method
• Idea: transform the BVP in an initial value problem (IVP), by
guessing some of the initial conditions and using the B.C. to
refine the guess, until convergence is reached
Target
Too high: reduce the initial velocity!
Too low: increase the initial velocity!
Convergence can be problematic
Use the same algorithms used for IVP
Shooting Method for Boundary Value Problem ODEs
Definition: a time stepping algorithm along with a root finding method for
choosing the appropriate initial conditions which solve the boundary value
problem.
Second-order Boundary-Value Problem
y1 ' '  f ( x, y, y' ), y(a)=A and y(b)=B
Computational Algorithm Based on Haar Wavelets
Computer Math. Model.
2010
1. Contrary to the existing methods, the new method based on
wavelets can be used directly for the numerical solution of
both boundary and initial value problems
2.
Stability in time integration is overcome.
3. Variety of boundary condition can be implemented with
equal ease
4. Simple applicability along with guaranteed convergence.
Haar Wavelets for Boundary Value Problem in ODEs
Consider the following
coupled nonlinear ODEs
Along with boundary conditions
Haar Wavelets for Boundary Value Problem in ODEs
Wavelets approximation for f and  can be given
by,
Haar Wavelets for Boundary Value Problem in ODEs
Haar Wavelets for Boundary Value Problem in ODEs
Haar Wavelets for Boundary Value Problem in ODEs
Haar Wavelets for Boundary Value Problem in ODEs
Haar Wavelets for Boundary Value Problem in ODEs
Haar Wavelets for Boundary Value Problem in ODEs
Haar Wavelets for Boundary Value Problem in ODEs
Haar Wavelets for Boundary Value Problem in ODEs
Haar Wavelets for Boundary Value Problem in ODEs
Haar Wavelets for Boundary Value Problem in ODEs
Haar Wavelets for Boundary Value Problem in ODEs
Haar Wavelets for Boundary Value Problem in ODEs
Haar Wavelets for Boundary Value Problem in ODEs
Haar Wavelets for Boundary Value Problem in ODEs
Haar Wavelets for Boundary Value Problem in ODEs
Haar Wavelets for Boundary Value Problem in ODEs
Adaptivity Through Non-uniform Haar Wavelets
Inter. J. Comput. Method Eng
Science & Mechanics (2010)
Adaptivity Through Non-uniform Haar Wavelets
Where
 0
Adaptivity Through Non-uniform Haar Wavelets
Where
 0
Adaptivity Through Non-uniform Haar Wavelets
Where
 0
Adaptivity Through Non-uniform Haar Wavelets
Where
 0
Nodes Generations Through Cubic Spline
Nodes Generations Through Cubic Spline
Nodes Generations Through Cubic Spline
1
1
0.5
0.5
data 1
spline
Cubic spline interpolant
0
0
-0.5
-0.5
-1
1
1.5
2
2.5
residuals
3
3.5
2
1
data 1
linear
quadratic
cubic
4
4th degree
5th degree
6th degree
7th degree
8th degree
-1
-1.5
1
1.5
2
2.5
residuals
3
3.5
4
1.5
2
2.5
3
3.5
4
-15
4
x 10
2
0
-2
0
-4
-1
-2
1
1.5
2
2.5
3
3.5
4
1
Nodes Generations Through Cubic Spline
Cubic spline interpolant
5
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
-5
-4
-3
-2
-1
0
1
2
3
4
5
Nodes Generations Through Cubic Spline
3
data
spline
2.5
2
1.5
1
0.5
0
-0.5
-1
-1.5
-2
0
2
4
6
8
10
12
14
16
Nodes Generations
initial velocity
In rolling direction
initial
temperature
initial
shape
solve temperature
of the slice
at the new position
calculate deformation
of the slice
at the new position
initial shape
v0
final velocity
in rolling direction
initial nodes
final shape
v
final nodes
renoding
Nodes Generations
Nodal points are generated through the following procedures:
Transfinite Interpolation
Elliptic Grid Generation
Nodes Generations
TRANSFINITE INTERPOLATION
Through this technique we can generate initial grid which is confirming to the geometry
we encounter in different stages of plate and shape rolling.
We suppose that there exists a transformation
r( , )  [ x( , ), y( , )]t
which maps the unit square, 0    1, 0    1 in the computational domain onto the
interior of the region ABCD in the physical domain such that the edges
  0, 1 map to the boundaries AB, CD and the edges   0,1 are mapped to the
boundaries AC, BD.
The transformation is defined as
r ( , ) = (1-  )rl ( ) +  rr () +(1- )rb ( ) +rt ( ) - (1-  )(1-  )rb (0) - (1-  )rt (0) - (1- ) rb (1) - rt
Where rb , rt , rl , rr
respectively
represents the values at the bottom, top, left and right edges
Nodes Generations
rt ( )

rr ( )
rl ( )
rb ( )

An example of transformation from computational domain to physical
domain.
Nodes Generations
ELLIPTIC GRID GENERATION
The mapping procedure defined above form the physical domain to the computational
domain is described by    ( x, y),    ( x, y) are continuously differentiable maps of
all order.
2 x
2 x
2 x
g 22
 2 g12
 g11 2  0
 2
 

2 y
2 y
2 y
g 22
 2 g12
 g11 2  0
 2
 

The grid generated through transfinite interpolation can be made more conformal to the
geometry by using the following elliptic grid generators
2
2
1   x   x  
g 22  2  
 
 ,
J        
g12 
2
2
1   x   x  
g11  2  
 
 
J        
where J is the Jacobean of the transformation.
1  x x y y 


,
J 2      
Nodes Generations
Transfinite Interpolation
Eliptic Grid Generation
Nodes Generations
title here
title here
1.5
1.8
1.6
1.4
1
1.2
y
y
1
0.8
0.5
0.6
0.4
0.2
0
0.4
0.6
0.8
1
1.2
x
1.4
1.6
1.8
0
0.5
1
1.5
2
2.5
x
2
title here
title here
2
2
1
1.5
0
1
y
y
0
0.2
-1
0.5
-2
0
-3
-0.5
-4
0
0.5
1
1.5
2
x
2.5
3
3.5
4
0
0.2
0.4
0.6
0.8
1
x
1.2
1.4
1.6
1.8
2
Application of Meshless Method to Hyperbolic PDEs
Submitted to journal
Application of Meshless Method to Hyperbolic PDEs
Application of Meshless Method to Hyperbolic PDEs
Application of Meshless Method to Hyperbolic PDEs
Application of Meshless Method to Hyperbolic PDEs
Application of Meshless Method to Hyperbolic PDEs
Comparison of Local and Global Meshless Methods
CMES. 2010
Comparison of Local and Global Meshless Methods
Comparison of Local and Global Meshless Methods
Thank you
Progress is a tide. If we stand still we will surely be drowned.
To stay on the crest, we have to keep moving. ~ Harold Mayfield