Course Details for A.Y. 2017/2018

Course Objectives

Goals of the course: Give the mathematical instruments for numerical optimization and numerical integration of differential equations. The course is of 6 credits and consists of 60 hours. Expected learning: Being able ti solve numerically and develop codes for general problems in optimization and in differential equations.

Course Content

• Iterative methods for the solution of nonlinear systems. Bisection method for nonlinear  equations. General theory of iterative methods. Convergence and order of convergence.  Newton method for a single equation. Extension to nonlinear systems. Newton Kantorovich  theorem. Quasi-Newton methods. Laboratory session: Newton method.
• Optimization methods. Minimization problems for general functions: conditions for relative minima; necessary first and second order conditions; sufficient second order conditions; convexity. Quadratic functions with symmetric positive definite structure; steepest descent gradient method (SD); convergence estimate; Kantorovich inequality; convergence speed of steepest gradient method. Laboratory session: steepest gradient method.
• Conjugate directions methods. Q-conjugacy and implications; expansion lemma; conjugate gradient method (CG); characterization theorem for the CG method; properties of the method; CG method optimality; comparison between CG and SD; general convergence estimates and spectral relationships; partial CG method; application to structured matrices; preconditioned gradient method. Nonlinear extension of CG method. Laboratory sessions (2); application of CG to boundary value elliptic problems.
• Penalty methods: properties and convergence; application of partial CG method to penalty methods; optimal scaling of penalty function. Laboratory session: penalty method applied to few test problems.
• Numerical methods for the numerical approximation of initial falue problems for ordinary differential equations. Euler explicit method; convergence; uniform convergence lemma for the numerical solution; convergence of the generated piecewise linear functions; a priori error estimates under C2-regularity; a posteriori error estimates; one step methods; examples; Taylor expansion based methods; Runge-Kutta methods. Laboratory session: implementation of an embedded Runge-Kutta method with stepsize control.
• Convergence and order conditions for one step schemes; discretization error; exact relative  increment; numerical relative increment; order of a one step method; convergence; consistence theorem; a priori error estimates; a posteriori error estimates; general form of Runge-Kutta methods; Butcher notation; order conditions; implicit methods; existence of numerical solution for an implicit Runge-Kutta scheme.  Laboratory sessions (2): implementation of an implicit integrator.
• Stability; dissipative problems and stability; stiff problems; A-stability; more general stability definitions; dense output; continuous methods; collocation methods; Gaussian collocation; implementative issues; stepsize control. Laboratory session: integration of a stiff problem arising in chemical kinetics.
• Numerical methods to approximate the siolution of boundary value problems for ordinary differential equations. Shooting method; finite difference method; variational methods; application to a second order elliptic problem. Laboratory sessions (2): implementation of shooting method; implementation of the variational method.

Learning Outcomes (Dublin Descriptors)

On successful completion of this course, the student should

Prerequisites and Learning Activities

Basic numerical analysis. Differential equations.

Assessment Methods and Criteria

Oral examination and a term paper with implementation issues.

Textbooks

• J. Stoer e R. Bulirsch, Introduction to numerical analysis. , Springer Verlag. 2002.    
• D.G. Luenberger, Linear and nonlinear programming , Kluwer Academic Publishers. 2003.    
• E. Hairer, S.P. Norsett e G. Wanner, Solving ordinary differential equations. I. Nonstiff problems. Second edition. , Springer Verlag.. 1993.    

Internet Resources

Course page updates

• A.Y. 2015/2016
• A.Y. 2016/2017
• A.Y. 2017/2018
• A.Y. 2018/2019
• A.Y. 2019/2020