Applied Mathematics

Major: HYRIS+ROSE

Term: 1st Semester

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

INSTITUTION: Consiglio Nazionale delle Ricerche – Istituto di Matematica Applicata e Tecnologie Informatiche “E. Magenes”  (CNR-IMATI)

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

  • Introduction to Matlab and basic programming  Basic Matlab commands, if/for/while instructions. Functions and @-functions. Design of basic algorithms.
  • Optimization of N-variate functions Free and constrained optimization of N-variate functions. Lagrange multipliers and KKT conditions, duality theory. Optimization algorithms (Gradient, Newton, Log-barrier).
  • Ordinary Differential Equations (ODE) Scalar ODEs and system of ODEs. Analytic solutions of linear systems of ODEs (exponential matrix). Equilibria of linear and non-linear systems (linearization, Liapunov function) and bifurcations.
  • Function approximation and Fourier transform  Space of square-summable functions, orthonormal bases and Parseval’s identity, Fourier and Legendre expansions, least squares. Fourier transform, DFT/FFT, Dirac’s delta.
  • Partial Differential Equations (PDE) Elliptic and parabolic PDEs: separation of variables, maximum and mean principle. Fundamental solution of heat equation, Dirac’s delta. Finite differences 1d. Hyperbolic PDEs: method of lines for 1st order hyperbolic PDEs, inflow and outflow; D’Alambert formula for wave equation on the line and semiline, separation of variables

REQUIREMENTS:

  • Basics of linear algebra
  • Basics of multivariate calculus and ODE solution
  • Elementary programming skills not necessary but highly welcome
 
REFERENCES:
Class notes made available during the course. For backup and further readings:

  • Optimization of N-variate functions: J. Nocedal, S. Wright. Numerical Optimization. Springer;
  • Ordinary Differential Equations (ODE):
    • G. Teschl, Ordinary Differential Equations and Dynamical Systems, American Mathematical Society;
    • Blanchard, Devaney, Hall. Differential Equations, Cengage Learning.
  • Function approximationFourier transforms:
    • A. Quarteroni, R. Sacco, F. Saleri. Numerical Mathematics. Springer;
    • D. Kammler, A First Course in Fourier Analysis, Cambridge University Press;
  • Partial Differential Equations (PDE):
    • S. Salsa, Partial Differential Equations in Action, Springer;
    • L. Evans, Partial Differential Equations. American Mathematical Society

Italian-speaking students can also use these books:

  • ODE, optimization,  Function approximation and Fourier transform: Analisi Matematica 2, M. Bramanti, C. Pagani, S. Salsa, Zanichelli ed.;
  • PDE: Equazioni a Derivate Parziali – Metodi, modelli e applicazioni, S. Salsa, Springer;

ASSESSMENT:
The final grade will be composed as follows:

  • 30% homework assignments (four in total) graded during the course;
  • 70% oral discussion over
    • exercises and Matlab scripts discussed in class;
    • one chapter of choice of the student