Course Details for A.Y. 2015/2016

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.

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.    

Course page updates

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