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. The student is required to submit his/her paper for the oral (area) exam as an attachment to his/her petition for taking the area 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 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
  • (Partially) Elective Part
    • Numerical Optimization                  % 30    (or  % 10)    
    • Finite Element Methods                  % 10   (or  % 30)    
• During the written examination, the student should choose one of options: Numerical Optimization or Finite Element Methods.  Each group consists of three questions. Student will be asked to attempt at least one of questions from each group.
• 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.
• The submitted document written by the student should be original. It should not be put together by coping and pasting from the literature.
• A study prepared for MSc Thesis cannot be directly used for this exam. A contribution is expected to be made during PhD studies even if the topic remains the same.
• An article (published or submitted) written by the student as the primary author in last 12 months before the exam can be used for this exam. However, the same article cannot be used by more than one student.
• In case of failure, a new topic can be chosen or the jury must indicate the expectations from the student for the second exam if the same topic will be used.
• In case of the student is taking the oral exam from the same topic, a supporting document explaining the changes from the previous one should be provided by the student to the jury along with the latest version of the study.

Expectation from Student before Exam

• To choose a topic within the student’s research field
• To conduct a literature survey on the chosen topic
• To make a contribution to the chosen topic as described below
• To prepare the study in the suggested format (LaTeX)
• To submit the study to the secretary of institute until at the end of October (or April) as an attachment to his/her petition for taking the qualification exam.

Expectations from Student during Exam

• To present the submitted study in the suggested format (at most 40 minutes)
• To answer questions about the study or the general questions related to the study (30 minutes)

Expected Contribution:  The student can make contributions by choosing one or more of the following types:

• Literature Survey: A literature survey is required in every study, but those students who select this category will be expected to conduct a more detailed literature survey by making comparison of the previous studies and to provide an analysis-synthesis of the literature in the selected topic.
• Implementation: The student make implementation of a paper in the selected topic which should be chosen together with the student’s advisor.  If applicable, student should produce results by changing various parameters and discuss the obtained results.
• Novel Approach: The student propose a novel approach to a selected problem and implement the approach.  The obtained results should be compared with the existing ones.
• Theoretical Contribution: The student propose a novel theoretical approach to the selected problems. The utility, correctness, and the reasoning behind the proposed approach should be displayed.
• Case Study: The student apply an existing method/algorithm to a realistic problem and discuss the obtained results.

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 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.