0.0 Introduction
 

PART I. MODELING TASKS / APPLICATIONS

1.0 Tomographic Image Reconstruction

1.1 Non-invasive medical imaging techniques
1.2 More on CT and its applications
1.3 Model problem
1.4 Least squares approach
1.5 Back projection (BP) method
1.6 Fourier transform (FT) method (alternate version)
1.7 Filtered BP method derived from the FT method

2.1 Freak wave episodes
2.2 Apparent mechanism
2.3 Derivation of governing equations
2.4 Test problem - circular current
2.5 Atlas Pride - incident revisited

3.1 Overview of eigenfaces

3.2 Representation in terms of eigenfaces

3.3 Facial recognition

3.4 Calculating the eigenfaces numerically

3.5 Updating the training set

4.1 A brief history of navigation
4.2 Principles of GPS
4.3 Test problem with numerical solution
4.4 Error analysis
4.5 Pseudorandom sequences

5.1 Origin and availability of hydrocarbons

5.2 Seismic modeling

PART II. SUPPORTING MATHEMATICAL TECHNIQUES

1.1 Projectile problem
1.2 Dimensional analysis
1.3 Scaling

2.1 Convergent vs. asymptotic expansions
2.2 Acceleration techniques
2.3 Expansions for polynomial roots
2.4 Expansions for integrals
2.5 Regular perturbations for ODEs
2.6 Singular perturbations for ODEs

3.1 Fourier series
3.2 Fourier transform
3.3 Discrete Fourier transform (DFT)
3.4 2-D Fourier transform

4.1 Wave function
4.2 Derivation of equations for waves along a string
4.3 Water waves
4.4 First order systems for some linear wave equations
4.5 Analytical solutions of the acoustic wave equation
4.6 Characteristics, Hamiltonian systems

PART III. SUPPORTING NUMERICAL TECHNIQUES

1.0 Linear systems, QR and SVD

1.1 Echelon form
1.2 Gaussian elimination
1.3 QR factorization
1.4 Singular value decompusition (SVD)
1.5 Overdetermined linear systems

2.1 Three iterative methods for scalar equations
2.2 Generalizations to systems of equations
2.3 Local optimization in 1-D
2.4 Local optimization in n-D
2.5 Global optimization in n-D

3.0 Polynomial interpolation / finite differences

3.1 Lagrange's interpolation formula
3.2 Error estimate, the Runge phenomenon
3.3 Newton's interpolation formula
3.4 Finite difference formulas

4.0 Splines

5.0 Radial basis functions (RBF)

6.0 The fast Fourier transform (FFT) algorithm

6.1 FFT implementations
6.2 A selection of FFT applications

8.0   Numerical methods for ODE initial value problems

8.1 Linear multistep methods

8.2 Runge Kutta methods

9.0   Finite differences for PDEs - the wave- and heat equations

10.0  Optimization / Simulated annealing
 

PART IV. MODELING PROJECTS

1.0    X-ray crystallography

        1.1  Model problem
        1.2  Analytical technique
        1.3  Computer implementation

2.0    Signature verification
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3.0   ...............................
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x.0    Brief descriptions of several additional projects

        x.1  .....
        x.2  .....
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PART V. EXERCISES

1.0  Exercises linked to sections in Part I: Modeling tasks / Applications

        1.1  Tomographic image reconstruction
        1.2  Freak ocean waves
        1.3  Facial inage recognition
        1.4  GPS satellite navigation
        1.5  Seismic forward modeling in oil/gas exploration

2.0  Exercises linked to sections in Part II: Supporting Mathematical Techniques

        2.1  Dimensional analysis and scaling
        2.2  Perturbation methods / Asymptotic expansions
        2.3  Fourier series / transforms
        2.4  Analysis of wave and heat equations

3.0  Exercises linked to sections in Part III: Supporting Numerical Techniques

        3.1  Linear systems, QR and SVD
        3.2  Polynomial interpolation / finite differences
        3.3  Zeros of functions
        3.4  Splines
        3.5  Radial basis functions
        3.6  The FFT algorithm
        3.7  Wavelets
        3.8  Numerical methods for ODE initial value problems
        3.9  Finite differences for PDEs - the wave and heat equation
        3.10 Optimization / Simulated annealing
 

PART VI. COMPUTER (MATLAB) CODES FOR MAIN MODELING TASKS IN PART I.

1.0  Codes for I.1: Tomographic image reconstruction

1.1  Logo
1.2  Scan
1.3 Least Square reconstruction
1.4 Back projection method
1.5 Fourier method

2.0  Codes for I.2: Modeling of freak ocean waves

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3.0  Codes for I.3: Facial recognition

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4.0  Codes for I.4: GPS satellite navigation

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5.0  Codes for I.5: Seismic forward modeling in oil/gas exploration

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REFERENCES