General Information

• Ph.D. Qualifying exam consists of a written part and an oral part in English. The candidate is considered successful when he/she passes both parts.
• Ph.D. Qualifying exam is given twice a year each May and November.
• The candidate should get the approval of his/her advisor and petition the department at least one month before the exam.
• For every student who will take the qualification exam, a jury comprised of 5 people with Ph.D. degree in the selected field, including the student's advisor, will be formed by the Qualifying Exam Committee. At least two members of jury should be outside of the student's higher education institution.
• The candidate failing to pass the Ph.D. Qualifying exam (5th term at the latest) is given a second chance in the subsequent offering of the exam. Failure in the second attempt leads to the dismissal of the student from the Ph.D. program.

Written Exam

• The written examination consists of   a core part and a (partially)  elective part:
  • Core  Part:
    • Scientific Computing I-II                 60%
  • Elective Part - Options:
    
    • Numerical Optimization                  40%      
    • Finite Element Methods                  40%     
• Before the written examination, the students must choose and declare one of the options, Numerical Optimization or Finite Element Methods, in their petition.
• To pass the written exam, the student must receive at least 65 points over 100 points from the overall written exam.

Oral (Area) Exam:

• The purpose of the oral examination is to evaluate student’s ability and potential conduct research at the doctoral level and to encourage student to have an earlier involvement in research.
• 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.
• To pass the oral exam, the student must receive at least 50 points over 100 points.

Overall Grading:

• 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 minimum 65 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 program, the committee members can suggest the student to take a course (or courses) in addition to his/her studies.

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

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
• Interpolation: polynomials, piecewise polynomials
• Numerical Differentiation and Integration
• Ordinary Differential Equations: Euler’s method, multistep methods, Runge-Kutta methods, stiff equations
• Partial Differential Equations: parabolic, elliptic, and hyperbolic equations
• Iterative Methods for Sparse Linear Systems: splitting methods

Suggested References

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

Numerical Optimization

• Unconstrained optimization:  Newton methods, line search methods, trust region methods, quasi-Newton methods
• 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

1. I. Griva, S. G. Nash and A. Sofer, Linear and nonlinear programming, 2nd edition, SIAM, Philadelphia, 2009.
2. J. Nocedal and S.J. Wright, Numerical Optimization, Springer.
3. 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

Suggested References

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