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Applied Mathematics: an Introduction to Scientific Computing

Lecturer: 
Luca Heltai - Gianluigi Rozza
Course Type: 
PhD Course
Anno (LM): 
Second Year
Academic Year: 
2016-2017
Period: 
October-January
Duration: 
48 h
CFU (LM): 
6
Description: 

Syllabus 2016-2017

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
Advanced Numerical Methods and Models

A short introduction on a selection of following topics:


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

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
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
Location: 
A-134 Frontal Lectures and A-003 Laboratories
Location: 
A-134

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