Math Courses

Numerical Methods

Name: Numerical Methods
Start: 2021-11-08T10:00:00+01:00
End: 2022-02-28T12:00:00+01:00
Location: GSSI

by Francesco Tudisco (GSSI), Nicola Guglielmi (GSSI)

Monday 8 Nov 2021, 10:00 → Monday 28 Feb 2022, 12:00 Europe/Rome

Ex-ISEF/Building-Room D (GSSI)

Ex-ISEF/Building-Room D

GSSI

Description

Lecturers
Nicola Guglielmi, nicola.guglielmi@gssi.it
Francesco Tudisco, francesco.tudisco@gssi.it

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

Boundary value problems (BVP):
- Finite differences;
- Variational methods;
- Rayleigh Ritz Galerkin methods
Numerical optimization:
- Unconstrained optimization;
- Gradient descent methods;
- Conjugate directions method;
- Constrained optimization;
- Penalization methods
Iterative methods for eigenvalue problems:
- Power method;
- Subspace iteration;
- Krylov subspace methods;
- Application to spectral clustering
Methods for sparse linear systems:
- Sparse direct solvers;
- General projection methods;
- CG and GMRES;
- Preconditioning;
Numerical quadrature:
- Order conditions;
- Error analysis;
- Superconvergence;
- Orthogonal polynomials;
- Gaussian quadrature
Linear multistep methods for ODEs:
- Explicit and implicit Adams' methods;
- Local error and stability;
- Convergence;
- Variable step size multistep methods;
- General linear multistep methods
Topics in Computational Machine Learning
- Nesterov accelerated gradient
- Stochastic gradient descent
- Graph and Convolutional Neural Networks
- Training NN: forward and backward propagation
- Implicit-depth NN
Runge Kutta methods for ODEs:
- General form;
- Convergence theory;
- Order conditions;
- Stability theory;
- A stability;
- B stability;
- Stiff problems;
- Von Neumann theorem;
- Evolution PDEs

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