Applied Mathematics

Term: 1st Semester – Academic Year 2019-2020

Instructor: Lorenzo Tamellini (tamellini@imati.cnr.it)

Institution: Emory University and IUSS

CFU: 6

SSD: MAT05/MAT08

Duration: 4 weeks

Schedule: TBD

Office hours: by appointment

OBJECTIVES:  To provide advanced mathematical tools that will be used throughout the rest of the program.

DESCRIPTION:

The course is divided in five chapters as follows. Each chapter includes also exercises and hands-on Matlab sessions (35 hours of lectures + 18h hours of exercises/Matlab sessions).

  1. Ordinary Differential Equations (ODE) (7h lectures + 3h exercises/Matlab = 10h). Scalar ODEs and system of ODEs. Analytic solutions of linear system of ODEs (exponential matrix) Equilibria and bifurcations. Numerical methods for ODEs (Euler, Crank-Nicholson, Runge-Kutta, …) and absolute stability issues.
  2. Optimization of N-variate functions (6h lectures + 3h exercises/Matlab = 9h). Free and constrained optimization of N-variate functions. Lagrange multipliers and KKT conditions. Optimization algorithms (gradient, Newton, square penalty, log-barrier…).
  3. Function approximation, transforms, numerical quadrature (8h lectures + 4h exercises/Matlab = 12h). Space of square-summable functions, orthonormal bases and Parseval’s identity, Fourier and Legendre expansions, least squares. Fourier transform, FFT. Numerical schemes for quadrature of univariate and multivariate functions (midpoint, trapezoidal rule, Gauss quadrature, Monte Carlo and probabilistic methods)
  4. Partial Differential Equations (PDE) (10h lectures + 3h exercises = 13h).
    1. Elliptic and parabolic PDEs: separation of variables, fundamental solution, Dirac’s delta, maximum principle.
    2. Hyperbolic PDEs: method of lines, weak solution, RH condition, entropy, systems of hyperbolic PDEs, wave PDE, D’Alambert principle.
  5. Numerical Methods for PDEs (4h lectures + 3h exercises/Matlab = 7h). Well-posedness of elliptic PDEs in Sobolev spaces by Lax-Milgram Lemma. Finite Elements method.