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[Creative Commons License](http://creativecommons.org/ licenses/by-nc-nd/4.0/)

Applied Mathematics: an Introduction to Scientific Computing

Syllabus 2017-2018

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

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

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
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.

Instructions for git aware students (and MHPC students)

This repository contains, assignements, workspaces, and other material for the course P1.4

New material will be uploaded frequently,

Remember to set a second remote, either to our private seed

git remote add P1.4_seed https://github.com/sissa/P1.4_seed.git

or (if using ssh keys in your github account)

git remote add P1.4_seed [email protected]:sissa/P1.4_seed.git

and to update before the lectures:

git pull P1.4_seed master

Please consider contributing pull requests to correct typos, or better document the material in this repository!

Licencing

The content of this repository is distributed with a Creative Common licence. See the file LICENCE.md in this directory for more information.

Attribution

Some of the material in this repository was adapted from the python-lectures by Robert Johansson. Take a look at his repository for additional material and lectures.