## PhD Qualifying Exam in Scientific Computing

##### Last Updated:

09/07/2017 - 15:29

## Rules and Regulations

- The exam has two parts: written and oral exams in English.
- The written examination includes a compulsory field and an elective field;
- Compulsory Field: Scientific Computing
- Elective Fields:
- Numerical Optimization
- Finite Element Methods
- Statistical Learning
- Inverse Problems
- Applied Nonlinear Dynamics

Each part is over 100 points and the overall written exam grade is equal to the average of the points earned from each part.

- To pass the written exam, the student must receive at least 65 points from the overall written exam.
- The student takes the oral examination, if he/she passes the written exam.
- The oral examination is over 100. The grade of the oral examination is determined by the average of the grades of each jury member who gives a grade between 0 and 100.
- Overall grade is calculated by taking 70% of the written exam and 30% of the oral exam grade.

(overall grade= 0.7*written exam grade + 0.3*oral exam grade.) - The passing grade for the Ph.D. qualification exam is 70 over 100. A student obtaining a passing grade is declared to have passed the PhD Qualification Exam.
- Although the student might have passed the PhD Qualification Exam and finished his/her courses for the programme, the committee members can suggest the student to take a course (or courses) in addition to his/her studies.

## Content of the Examination

Here is the list of topics covered in the PhD Qualifying Exam.

### Compulsory Field

All students in Scientific Computing PhD programme should take:

#### Scientific Computing

- Computer Arithmetic
- Linear Equations: Gauss elimination, LU decomposition
- Linear Least Squares: data fitting, normal equations, orthogonal transformations
- Eigenvalue Problems
- Singular Value Decomposition
- Nonlinear Equations: bisection, fixed-point iteration, Newton’s method, optimization
- Interpolation: polynomials, piecewise polynomials
- Numerical Differentiation and Integration
- Ordinary Differential Equations: Euler’s method, multistep methods, Runge-Kutta methods, stiff equations, adaptivity
- Boundary Value Problems: shooting, collocation, Galerkin methods
- Partial Differential Equations: parabolic, elliptic, and hyperbolic equations
- Iterative Methods for Sparse Linear Systems: splitting methods, descent methods, conjugate gradients, preconditioners, multigrid methods

##### Suggested References

- U. Ascher and C. Greif, A First Course in Numerical Methods, SIAM, 2011.
- A. Iserles, A First Course in the Numerical Analysis of Differential Equations, Cambridge University Press, 2009.
- R. J. LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations: Steady- State and Time-Dependent Problems, SIAM, 2007.
- M. T. Heat, Scientific Computing, McGraw Hill, 1997.

### Elective Fields

Students have to choose *one* of the following *five* elective topics:

#### Numerical Optimization

- Unconstrained optimization: Newton methods, line search methods, trust region methods, quasi-Newton methods, nonlinear least square problems
- Constrained optimization: linear programming, simplex method, equality and inequality constraints, Lagrange multiplier algorithms, active set strategy, interior point methods, sequential quadratic programming, penalty and barrier methods.

##### Suggested References

- I. Griva, S. G. Nash and A. Sofer, Linear and nonlinear programming, 2nd edition, SIAM, Philadelphia, 2009.
- J. Nocedal and S.J. Wright, Numerical Optimization, Springer.
- Lecture Notes are prepared by B. Karasözen and G.-W. Weber on Numerical Optimization: Constrained Optimization (available on Public Lecture Notes)

#### Finite Element Methods

- Abstract Finite Element Analysis: weak derivatives, Sobolev spaces, Lax-Milgram lemma
- Piecewise Polynomials Approximations 1D and 2D: interpolation, projection
- Finite Element Method 1D and 2D: weak formulation, derivation of linear system of equations, a priori estimates
- Time Dependent Problems: finite differences for systems of ODE, stability estimates
- Semi-elliptic equations
- A Posteriori Error Analysis: estimator, mesh refinement

##### Suggested References

- M. G. Larson and F. Bengzon, The Finite Element Method: Theory, Implementation, and Applications, Springer-Verlag Berlin Heidelberg, 2013.
- M. S. Gockenbach, Understanding and Implementing the Finite Element Method, SIAM, 2006.

#### Statistical Learning

- Foundations of Probablility Theory and Statistics
- Foundations of Statistical Learning and Simulation
- Linear Methods of Regression
- Linear Methods in Classification
- Model Assessment and Selection
- Model Inference and Averaging
- Additive Models, Trees and Related Methods
- Prototype Methods and Nearest Neighbours
- Cluster Algorithms
- Elements of Support Vector Machines
- Unsupervised Learning

##### Suggested References

- N. Christianini and J. Shawe-Taylor, An Introduction to Support Vector Machines, Cambridge University Press, 2000.
- T. Hastie, R. Tibshirani, and J. Friedman, The Elements of Statistical Learning, Springer Series in Statistics, 2nd edition, Springer, 2009.

#### Inverse Problems

- Foundations of Inverse Problems
- Linear Regression
- Least Squares Theory
- Discretizing Continuous Inverse Problems
- Rank Deficiency and Ill-Conditioning
- Tikhonov Regularization
- Iterative Method
- Fourier Techniques
- Nonlinear Inverse Problems
- Nonlinear Regression
- Nonlinear Least Squares
- Bayesian Methods

##### Suggested References

- A. Aster, B. Borchers, C. Thurber, Parameter Estimation and Inverse Problems, Academic Press, 2nd edition, 2012.

#### Applied Nonlinear Dynamics

- Linear Dynamical Systems: fundamental theorem of continuous systems, a qualitative investigation of planar systems, stability of linear systems
- Nonlinear Dynamical Systems; manifold approach, fixed points and phase flow, local nonlinear systems, linearization, stability of nonlinear systems, hyperbolic, nonhyperbolic cases, limit cycles, the Poincaré map, liapunov function, Hartman-Grobman theorems
- Bifurcations and Hamiltonian Systems: one dimensional bifurcations, saddle node, Pitchfork, transcritical bifurcations, Hopf bifurcations, Hamiltonian flows, classification of flows
- Chaos and Fractals: Lorenz equations, chaos on a strange attractor, Lorenz map, Cantor set, dimension of fractals

##### Suggested References

- Stephen. Lynch, Dynamical Systems with Applications using MATLAB, Birkhäuser, Basel, 2014.
- James D. Meiss, Differential Dynamical Systems, SIAM, 2007.
- J. Jost, Dynamical Systems:Examples of Complex Behaviour, Springer, 2005.