Mathematics

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

Syllabus 2018-2019

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

Frontal Lectures

Review Lectures

• Well posedness, condition numbers, Lax Richtmyer theorem
• Polynomial based approximations (Lagrange interpolation, Bernstein polynomials, Bsplines approximations)
• Quadrature rules and orthogonal polynomials
• Solution methods for Linear Systems: direct, iterative and least square methods
• Eigenvalues/Eigenvectors
• Solution methods for non-Linear systems
• Review of ODEs
• Review of FEM/Lax Milgram Lemma/Cea's Lemma/Error estimates

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

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.