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 the Master in High Performance Computing.

All lectures will take place live in room 128-129, will be streamed online using the Zoom platform, and will be recorded live on YouTube.

The first lecture will be on the 5th of October 2021 at 16.30 in Room A-128/A-129 and on Zoom.

The following Zoom link will be used for all lectures:

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

Meeting ID: 857 9687 3697
Passcode: NumAna

All recordings for lectures of 2021-2022 will be added to this YouTube playlist.

Lectures from the academic year 2020-2021 are available at this YouTube playlist. 

All course material is available at

https://github.com/luca-heltai/numerical-analysis-2021-2022

Syllabus 2021-2022

Four Modules of 12h each (1.5 CFU for each module), for a total of 48h, 6 CFU

Frontal Lectures

Module 1 (Basis of Numerical Analysis - Part I - Prof. Luca Heltai)

• Well posedness, condition numbers
• Polynomial based approximations (Power basis interpolation, Lagrange interpolation, Weierstrass approximation theorem)
• Interpolatory Quadrature rules
• Orthogonal polynomials and Gauss Quadrature Formulas
• L2 projection
• Review of elementary PDEs
• Introduction to Finite Difference Methods
• Introduction to Finite Element Methods

Module 2 (Basis of Numerical Analysis - Part II - Prof. Ganluigi Rozza)

• Least square methods
• Solution methods for Linear Systems: direct and iterative solvers
• Eigenvalues/Eigenvectors
• Solution methods for non-Linear systems
• Review of ODEs

Module 3 (Basis of Numerical Modeling - Prof. Gianluigi Rozza)

• 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

Module 4 (Numerical Analysis with Python - Prof. Luca Heltai)

• Introduction to Python, Numpy, Scipy
• Exercises 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

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.