Numerical Methods

by Nicola Guglielmi (GSSI)

Wednesday 1 Nov 2023, 10:00 → Wednesday 21 Feb 2024, 12:00 Europe/Rome

Ex-ISEF/Building-Room D (GSSI)

Lecturers
Nicola Guglielmi, nicola.guglielmi@gssi.it
Francesco Paolo Maiale

with a contribution of Francesco Tudisco

Timetable and workload
Lectures: 60 hours
Labs: 20 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 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 (such as MATLAB or Julia). 

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

 

Preliminary part

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.

Elements of numerical optimization

• III.1. The main methods of convex optimization.
• III.2. Univariate optimization. Multivariate optimization.

Numerical optimization

Gradient methods

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

Iterative solution of nonlinear systems

• V.1. Forward and Newton’s methods 
• V.2. Broyed and BFGS methods 
• V.3. Monotone operators
• V.4. Linear multistep methods

Constrained optimization

• VI.1. Penalty methods. 
• VI.2. Projection methods.

Spectral optimization

• VII. Eigenvalue optimization
• VII.1. Variational properties of eigenvalues.
• VII.2. Structured spectral optimization.
• VII.3. Applications to data science (e.g. spectral graph clustering).

 
Books of reference
E. Hairer, G. Wanner, S. P. Nørsett; Solving Ordinary Differential Equations I
E. Hairer, G. Wanner; Solving Ordinary Differential Equations II
Y. Saad; Iterative methods for Sparse Linear Systems (Free Online Version)
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
 

Slides Tudisco