Math Courses

Numerical Methods

by Nicola Guglielmi (GSSI)

→ Europe/Rome
Ex-ISEF/Building-Main Lecture Hall (GSSI)

Ex-ISEF/Building-Main Lecture Hall

GSSI

Time-table Tue, Wed 8.30 -- 10.30 Wed 14.30 -- 16.30
20
Description

Lecturers
Nicola Guglielmi, nicola.guglielmi@gssi.it (Lectures),
Francesco Paolo Maiale, francescopaolo.maiale@gssi.it and Angelo Alberto Casulli angelo.casulli@gssi.it (Lab)

with possible further specific contributions


Timetable and workload
Lectures:  60 hours
Labs:  25 hours
Final project assignment: 24 hours


Course description and outcomes
This course is an introduction to modern numerical analysis. The primary objective of the course is to develop graduate-level  understanding  of methods of computational mathematics and skills to solve a range real-world mathematical problems on a computer by implementing advanced numerical algorithms  using a scientific computing language. The main focus is on numerical integration of differential equations and numerical optimization.


Course requirements
Calculus and basic linear algebra and numerical analysis. Previous programming experience in any language may help.


Course content
The course will cover the following topics

 

Numerics for ODEs and DDEs

Quadrature

  • I.1. Quadrature formulas, order conditions.
  • I.2. Error analysis
  • I.3. Superconvergence and orthogonal polynomials 
  • I.4. Gauss quadrature.

Initial Value Problems for ODEs

  • II.1. One-step (Runge-Kutta) methods 
  • II.2. Stiff problems
  • II.3. Collocation methods.

Delay differential equations

  • III.1. Models with delays (discrete and distributed).
  • III.2. Main difficulties with respect to ODEs
  • III.3 Implicit integration
  • III.4 Extension to distributed delays 
  • III.5 Volterra integrodifferential equations.
  • III.6 Applications

 

Numerical optimization

Gradient methods

  • IV.1. Gradient systems.
  • IV.2. Classical gradient methods. 
  • IV.3. Conjugate direction methods.

Constrained optimization

  • V.1. Penalty methods. 
  • V.2. Projection methods.

Applications

  • VI.1. Eigenvalue optimization

 
Books of reference
E. Hairer, G. Wanner, S. P. Nørsett; Solving Ordinary Differential Equations I, Springer
E. Hairer, G. Wanner; Solving Ordinary Differential Equations II, Springer
R. Fletcher;  Practical methods of optimization. Wiley, 2001.
Y. Saad;  Numerical methods for Large Eigenvalue Problems (Free Online Version)


 
Examination and grading
Students will be evaluated on the basis of a written exam and computational assessment to be taken at the end of the course. Both tests are graded based on the ECTS grading scale