# Course Details for A.Y. 2018/2019

#### Name:

**Advanced Numerical Analysis / Advanced Numerical Analysis **

### Basic information

##### Credits:

*:* Master Degree in Mathematics 6 CFU (b)

##### Degree(s):

Master Degree in Mathematics 2^{nd} anno curriculum Generale Compulsory

##### Language:

English

### 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

- mature a knowledge of the main algorithms for the numerical approximation of nonlinear optimization
problems as well as of main methods for the numerical solution of ordinary differential equations
(initial value problems and boundary value problems).

### 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

This course page is available (with possible updates) also for the following academic years:

** To read the *** current * information on this course, if it is still available, go to the university course catalogue .
*Course information last updated on: 28 settembre 2017, 10:40*