Aggregatore Risorse
Lecturer: Dr. Lorenzo Tamellini
Applied Mathematics a.y. 2020/21
Programma
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
Svolgimento
Week | Date | Lecture hours | Tot h |
1 | 19.11.2020 | 09.30 – 12.00 | 2.5 |
19.11.2020 | 14.00 – 16.00 | 2 | |
20.11.2020 | 09.30 – 11.00 | 1.5 | |
20.11.2020 | 14.00 – 17.00 | 3 | |
2 | 25.11.2020 | 09.30 – 12.00 | 2.5 |
25.11.2020 | 14.00 – 16.00 | 2 | |
26.11.2020 | 09.30 – 12.00 | 2.5 | |
26.112020 | 14.00 – 16.00 | 2 | |
27.11.2020 | 09.30 – 11.00 | 1.5 | |
27.11.2020 | 14.00 – 17.00 | 3 | |
3 | 02.12.2020 | 09.30 – 12.00 | 2.5 |
02.12.2020 | 14.00 – 16.00 | 2 | |
03.12.2020 | 09.30 – 12.00 | 2.5 | |
03.12.2020 | 14.00 – 16.00 | 2 | |
04.12.2020 | 09.30 – 11.00 | 1.5 | |
04.12.2020 | 14.00 – 17.00 | 3 | |
4 | 10.12.2020 | 09.30 – 12.00 | 2.5 |
10.12.2020 | 14.00 – 16.00 | 2 | |
11.12.2020 | 09.30 – 11.00 | 1.5 | |
11.12.2020 | 14.00 – 17.00 | 3 | |
5 | 15.12.2020 | 09.30 – 12.00 | 2.5 |
15.12.2020 | 14.00 – 16.00 | 2 | |
17.12.2020 | FINAL EXAM | - | |
18.12.2020 | FINAL EXAM | - |
Bibliografia
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;
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
Esame
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
Ciclo : XXXIV, XXXV, XXXVI
Tipologia corso : Caratterizzante
Curriculum : Ingegneria Sismica e Sismologia
Periodo: Semestre I
Anno accademico: 2020-2021
Luogo : on line