Math Courses

Numerical Methods

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

Europe/Rome
Ex-ISEF-Building-Room D (GSSI)

Ex-ISEF-Building-Room D

GSSI

20
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
 

Notes from the lectures by FT
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