Siraj –ul – Islam Laboratory for Multiphase Processes University of Nova Gorica, Slovenia

Siraj –ul – Islam

Laboratory for Multiphase ProcessesUniversity of Nova Gorica, Slovenia



Some Applications of Wavelets









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

• January 1842, in which about 16,000 British and Indian troops were killed

• 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

Khyber Pass"Khyber is a Hebrew word meaning a fort"










What are Wavelets?

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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

What are Wavelets?

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• A small wave• Extends to finite interval

The word wavelet refers to the function h(t) that generates a basis for the orthogonal complement of V0 in V1

Wavelets analysis is a procedurethrough which we can decompose a given function into a set of elementarywaveforms called wavelets

Types Of Wavelets

ICCES 2010Las Vegas, March 28 - April 1, 2010

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The Haar Scaling Functions and Haar Wavelets

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a) Haar scaling function (Father function) b) Haar Wavelet function (Mother wavelets)

The Haar Scaling Function and

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The Haar Wavelets

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The Haar Wavelets and its Integrals

16

with the collocation points

The repeated integral of Haar wavelet is given by

The Haar Wavelets and its Integrals

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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

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Multi-Resolution Analysis

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21


22

2

20 0 1 2

2

0 0 00

The space L (R) can be decomposed as an infinite orthogonal direct sum

L (R) V W W W .

In particular, each L (R) can be written uniquely as

where v belongs to V and belongs j jj

f

f v w w

jto W


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Scaling function (Father wavelet) basis in V 0h

Wavelet function (Mother wavelets) basis in Wih

Gaussian Quadrature

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( )x

Gaussian Quadrature

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Gaussian Quadrature

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Problems with Gaussian Quadrature

28

• Solution 2n by 2n system• Search for better nodal values• Finding optimized values for the unknown weights

Numerical Integration based on Haar wavelets

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Inter. J. Computer Math. 2010


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31


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Numerical integration for double and triple integrals

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Numerical integration for double and triple integrals

34

Numerical double integration with variable limits

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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

Numerical triple integration with variable limits

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Numerical results

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Numerical results

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Numerical results

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Numerical results

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Numerical results

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Numerical results

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1110Symmetric Gauss Legendre

710 Symmetric Gauss Legendre

Convergence of the method

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Numerical Solution of Ordinary Diff. Eqs.

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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)

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!

Use the same algorithms used for IVPConvergence can be problematic

Shooting Method for Boundary Value Problem ODEsShooting 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

),',,(''1 yyxfy y(a)=A and y(b)=B

Computational Algorithm Computational Algorithm Based on Haar Wavelets

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.

Computer Math. Model. 2010

Haar Wavelets for Boundary Value Problem in ODEsHaar Wavelets for Boundary Value Problem in ODEs

Consider the following coupled nonlinear ODEs

Along with boundary conditions


Wavelets approximation for and can be given by,

f

















Adaptivity Through Non-uniform Haar Wavelets Adaptivity Through Non-uniform Haar Wavelets

Inter. J. Comput. Method Eng Science & Mechanics (2010)


Where 0


Where 0


Where 0


Where 0

Nodes Generations Through Cubic SplineNodes Generations Through Cubic Spline



1 1.5 2 2.5 3 3.5 4-4

-2

0

2

4x 10

-15 residuals1 1.5 2 2.5 3 3.5 4

-1.5

-1

-0.5

0

0.5

1

Cubic spline interpolant

data 1

spline

1 1.5 2 2.5 3 3.5 4-2

-1

0

1

2residuals

1 1.5 2 2.5 3 3.5 4-1

-0.5

0

0.5

1

data 1 linear

quadratic

cubic

4th degree 5th degree

6th degree

7th degree 8th degree


-5 -4 -3 -2 -1 0 1 2 3 4 50

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5Cubic spline interpolant


0 2 4 6 8 10 12 14 16-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

3

data

spline

Nodes GenerationsNodes Generations

initial temperature

calculate deformation of the slice

at the new position

solve temperature of the slice

at the new position

initial shape

initial velocityIn rolling direction

final velocityin rolling direction

0v

v

initial shape

final shape

initial nodes

final nodes renoding


Nodal points are generated through the following procedures:

Transfinite Interpolation

Elliptic Grid Generation


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

which maps the unit square, in the computational domain onto the interior of the region ABCD in the physical domain such that the edges

map to the boundaries AB, CD and the edges are mapped to the boundaries AC, BD.

The transformation is defined as

Where represents the values at the bottom, top, left and right edges respectively

( , ) [ ( , ), ( , )]tx y r

0 1, 0 1

0, 1 0,1

0 0 1l r b t b t b t, r ( ) = (1- )r ( ) + r ( ) + (1- )r ( ) + r ( ) - (1 - )(1 - )r ( ) - (1 - ) r ( ) - (1 - ) r ( ) - r

, , ,b t l rr r r r


An example of transformation from computational domain to physical domain.

( )tr

( )br

( )lr ( )rr


ELLIPTIC GRID GENERATION

The mapping procedure defined above form the physical domain to the computational domain is described by are continuously differentiable maps of all order.

The grid generated through transfinite interpolation can be made more conformal to the geometry by using the following elliptic grid generators

where is the Jacobean of the transformation.

2 2 2

22 12 112 2

2 2 2

22 12 112 2

2 0

2 0

x x xg g g

y y yg g g

2 2

22 122 2

2 2

11 2

1 1, ,

1

x x x x y yg g

J J

x xg

J

J

( , ), ( , )x y x y


Transfinite Interpolation Eliptic Grid Generation


0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8title here

x

y

0 0.5 1 1.5 2 2.50

0.5

1

1.5title here

x

y

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-0.5

0

0.5

1

1.5

2title here

x

y

0 0.5 1 1.5 2 2.5 3 3.5 4-4

-3

-2

-1

0

1

2title here

x

y

Application of Meshless Method to Hyperbolic PDEsApplication of Meshless Method to Hyperbolic PDEs

Submitted to journal






Comparison of Local and Global Meshless MethodsComparison of Local and Global Meshless Methods

CMES. 2010



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

Documents

Siraj –ul – Islam Laboratory for Multiphase Processes University of Nova Gorica, Slovenia