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Applied Mathematics: an Introduction to Scientific Computing

Luca Heltai - Gianluigi Rozza
Course Type: 
PhD Course
Master Course
Anno (LM): 
First Year
Second Year
Academic Year: 
Oct. - Jan.
60 h
CFU (LM): 
  • Frontal Lectures (about 30h), Interleaved with Laboratories (about 30h): total 60h
  •  This course is shared between the PhD in Mathematical Analysis, Modeling, and Applications, the Master in High Performance Computing ( and the Laurea Magistrale in Matematica


This term we will be using Piazza for class discussion. The system is highly catered to getting you help fast and efficiently from classmates, the TA, and myself. Rather than emailing questions to the teaching staff, I encourage you to post your questions on Piazza. If you have any problems or feedback for the developers, email

Find our class page at: 

Frontal Lectures 

Review Lectures

  • Basic concepts of Vector spaces and norms
  • 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
  • High order methods/high continuity 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


Advanced Numerical Methods and Models

  • Non conforming Finite Element Methods
  • Mixed Finite Element Methods
  • Darcy's equation
  • Stokes



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


Advanced lectures

  • Object oriented programming in numerical analysis
  • Review of best practices in programming for numerical analysis
  • Working project: ePICURE (Python Isogeometric CUrve REconstruction)
  • Solution of one dimensional PDEs using Finite Elements
  • From one dimensional FEM to N-dimensional exploiting tensor structure of certain finite elements


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.


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