Mathematics

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

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

Notes
On 12/10, 19/10 the class will be given in the Big Meeting Room (7th floor) and on 07/11 the class will be held in room A-128.
Otherwhise in room A-005.