Term: 1st Semester
Duration: From 28.11.2022 to 23.12.2022
INSTRUCTOR: Prof. M. Martinelli
INSTITUTION: Consiglio Nazionale delle Ricerche – Istituto di Matematica Applicata e Tecnologie Informatiche “E. Magenes” (CNR-IMATI)
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
- Basics of linear algebra
- Basics of multivariate calculus and ODE solution
- Elementary programming skills not necessary but highly welcome
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 approximation, Fourier 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;
The final grade will be composed as follows:
- 30% homework assignments (four in total) graded during the course;
- 70% oral discussion over the program of the course, as well as exercises and Matlab scripts discussed in class