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Applied Mathematics: numerical analysis and scientific computing

Luca Heltai - Gianluigi Rozza
Course Type: 
PhD Course
Master Course
Anno (LM): 
Second Year
Academic Year: 
48 h
CFU (LM): 

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


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. 
  • A. Quarteroni. Numerical Models for Differential Problems. Springer, 2009. 
  • A. Quarteroni and A. Valli. Numerical approximation of partial differential equations. Springer Verlag, 2008. 
  • S. Brenner and L. Scott. The mathematical theory of finite element methods. Springer Verlag, 2008.
  • 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.
  • D. Arnold. A concise introduction to numerical analysis. Institute for Mathematics and its Applications, Minneapolis, 2001. 
  • A. Quarteroni, F. Saleri, P. Gervasio. Scientific Computing with Matlab and Octave. Springer Verlag, 2006.   
  • B. Gustaffson Fundamentals of Scientific Computing, Springer, 2011
  • Tveito, A., Langtangen, H.P., Nielsen, B.F., Cai, X. Elements of Scientific Computing, Springer, 2010

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

On 9/10 and 11/10 the lectures will be held in A-004

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