Mathematics

• Prof. Luca Heltai (luca.heltai@sissa.it)
• Prof. Gianluigi Rozza (gianluigi.rozza@sissa.it)

Practical Information on the course 

This is a Joint course, between SISSA PhD in Mathematical Analysis, Modeling, and Applications, Laurea Magistrale in Matematica, Laurea Magistrale in Data Science and Scientific Computing, and Master in High Performance Computing.

Due to the large number of diverse groups that will follow the course, and for security protocols, only students with a Scholarship from SISSA will be allowed to follow some of the lectures in presence (SISSA PhD students, LM scholarship holders, DSSC scholarship holders) and few other students from Mathematics. DSSC and MHPC students will be connected online this year.

All lectures will be streamed online using the Zoom platform, and recorded live on YouTube. A link with the lecture will be provided to the students at the end of every lecture. 

According to the availability of the teachers, some lectures will be delivered live in one of SISSA classrooms. 

The first lecture will be on the 6th of October 2020 at 16.00 in Room A-133 and on Zoom.

The following Zoom link will be used for the first lecture:

https://sissa-it.zoom.us/j/85796873697?pwd=cm5HNGRidndvU05UTVAvRkdOdnNrdz09

Meeting ID: 857 9687 3697
Passcode: NumAna

You will find a recording of all lectures in this YouTube playlist.

If you want to get up-to-date information about the course, please subscribe to the group:

https://groups.google.com/g/numerical-analysis-2020-2021

(sending an email to numerical-analysis-2020-2021+subscribe@googlegroups.com is sufficient)

If you are following the course, please FILL THIS FORM.

All course material is available at 

https://github.com/luca-heltai/numerical-analysis-2020-2021

Syllabus 2020-2021

Frontal Lectures (about 24h), Interleaved with Laboratories (about 24h): total 48h, 6 CFU

Frontal Lectures

Review Lectures

• Well posedness, condition numbers
• Polynomial based approximations (Power basis interpolation, Lagrange interpolation)
• Interpolatory Quadrature rules
• Orthogonal polynomials and Gauss Quadrature Formulas
• L2 projection
• Least square methods
• Solution methods for Linear Systems: direct and iterative solvers
• Eigenvalues/Eigenvectors
• Solution methods for non-Linear systems
• Review of ODEs
• Introduction to Finite Element Methods

Mathematical Modeling

• Data assimilation in biomechanics, statistics, medicine, electric signals
• Model order reduction of matrices
• Linear models for hydraulics, networks, logistics
• State equations (real gases), applied mechanics systems, grow population models, financial problems
• Applications of ODEs
• example in electric phenomena, signals and dynamics of populations (Lotke-Volterra)
• Models for prey-predator, population dynamics, automatic controls
• Applications of PDEs, the poisson problem
  • Elastic rope
  • Bar under traction
  • Heat conductivity
  • Maxwell equation

Laboratories

Introductory lectures

• Introduction to Python, Numpy, Scipy
• Exercise on Condition numbers, interpolation, quadratures
• Using numpy for polynomial approximation
• Using numpy for numerical integration
• Using numpy/scipy for ODEs
• Working with numpy arrays, matrices and nd-arrays
• Solving non-linear systems of equations
• Using numpy/scipy for simple PDEs

Students projects

• Application of the Finite Element Method to the solution of models taken from the course

Final Project

In the directory final_project of the repository, you will find the file final_project_2020-2021.ipynb.

The deadline for the assignment is one day before the oral examination (for DSSC and LM students).

For MHPC students: you have time till February 12 2021.

Further material provided during lectures by Prof. Gianluigi Rozza [https://people.sissa.it/~grozza/amnasc/]

References and Text Books:

• A. Quarteroni, R. Sacco, and F. Saleri. Numerical mathematics, volume 37 of Texts in Applied Mathe- matics. Springer-Verlag, New York, 2000. 
  [E-Book-ITA] [E-Book-ENG]
• A. Quarteroni. Modellistica Numerica per problemi differenziali. Springer, 2008. 
  [E-Book-ITA]
• A. Quarteroni. Numerical Models for Differential Problems. Springer, 2009. 
  [E-Book-ENG]
• A. Quarteroni and A. Valli. Numerical approximation of partial differential equations. Springer Verlag, 2008. 
  [E-Book-ENG]
• S. Brenner and L. Scott. The mathematical theory of finite element methods. Springer Verlag, 2008.
  [E-Book-ENG]
• D. Boffi, F. Brezzi, L. Demkowicz, R. Durán, R. Falk, and M. Fortin. Mixed finite elements, compatibility conditions, and applications. Lectures given at the C.I.M.E. Summer School held in Cetraro, Italy June 26–July 1, 2006. Springer Verlag, 2008.
  [E-Book-ENG]
• D. Arnold. A concise introduction to numerical analysis. Institute for Mathematics and its Applications, Minneapolis, 2001. 
  [E-Book-ENG]
• A. Quarteroni, F. Saleri, P. Gervasio. Scientific Computing with Matlab and Octave. Springer Verlag, 2006.   
  [E-Book-ENG]
• B. Gustaffson Fundamentals of Scientific Computing, Springer, 2011
  [E-Book-ENG]
• Tveito, A., Langtangen, H.P., Nielsen, B.F., Cai, X. Elements of Scientific Computing, Springer, 2010
  [E-Book-ENG]

Note that, when connecting from SISSA, all of the text books above are available in full text as pdf files.