INSTITUTE OF APPLIED MATHEMATICS

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
  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.

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
  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
  • A Posteriori Error Analysis: estimator, mesh refinement
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.

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
  1. N. Christianini and J. Shawe-Taylor, An Introduction to Support Vector Machines, Cambridge University Press, 2000.
  2. 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
  1. 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
  1. Stephen. Lynch, Dynamical Systems with Applications using MATLAB, Birkhäuser, Basel, 2014.
  2. James D. Meiss, Differential Dynamical Systems, SIAM, 2007.
  3. J. Jost, Dynamical Systems:Examples of Complex Behaviour, Springer, 2005.

Past PhD Qualifying Exams (Samples)