Aggregatore Risorse
Applied Mathematics (a.a. 2019/20)
Programma
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 (34 hours of lectures + 20h hours of exercises/Matlab sessions).
1. Ordinary Differential Equations (ODE) (6h lectures + 3h exercises/Matlab = 9h). Scalar ODEs and system of ODEs. Analytic solutions of linear system of ODEs (exponential matrix). Equilibria of linear and non-linear systems (linearization, Liapunov function) and bifurcations.
2. Optimization of N-variate functions (5h lectures + 3h exercises/Matlab = 8h). Free and constrained optimization of N-variate functions. Lagrange multipliers and KKT conditions, duality theory. Optimization algorithms (gradient, Newton, square penalty, log-barrier, Nelder Mead).
3. Function approximation, transforms, numerical quadrature (8h lectures + 5h exercises/Matlab = 13h). Space of square-summable functions, orthonormal bases and Parseval’s identity, Fourier and Legendre expansions, least squares. Fourier transform, DFT/FFT.
4. Partial Differential Equations (PDE) (9h lectures + 5h exercises = 14h).
a. Elliptic and parabolic PDEs: separation of variables, maximum and mean principle. Fundamental solution, Dirac’s delta.
b. Hyperbolic PDEs: method of lines for 1st order hyperbolic PDEs, inflow and outflow; rarefaction and shock waves (RH condition) for nonlinear 1st order hyperbolic PDEs (Burger, traffic); D’Alambert formula and separation of variables for wave equation.
5. Numerical Methods for PDEs (6h lectures + 4h exercises/Matlab = 10h). Weak formulation of elliptic PDEs in Sobolev spaces (Lax-Milgram’s and Cea’s) Lemma. Galerkin method, Finite Element Basis and Matrix. Meshing and Quadrature.
Svolgimento
Week | Date | Lecture hours | Tutorial hours | Subject | Tot h |
1 | 25/11/2019 | 10-12 13:30-15:30 |
| Introduction to ODE and dynamical systems, analytic solution of linear system of ODEs and their stability | 4 |
26/11/2019 | 10-12 |
13:30-16:30 | Stability of non-linear systems: linearization, Liapunov function, bifurcation diagrams. Exercise session on ODE. HW 1 | 2+3 | |
27/11/2019 | --- | --- | --- | --- | |
28/11/2019 | 10-12 13:30-16:30 |
| Introduction to optimization, unconstrained optimization Constrained optimization: KKT condition, Lagrange multipliers and duality, numerical algorithms | 5 | |
29/11/2019 |
| 09:30-12:30 | Exercise session on optimization. HW 2 | 0+3 | |
2 | 02/12/2019 | 10-12 13:30-15:30 |
| L^2 spaces, Legendre expansion, least squares Fourier expansion | 4 |
03/12/2019 |
14:00-16:00 | 09:30-12:30 | Exercise session on Legendre and Fourier Expansion Fourier transform | 2+3 | |
04/12/2019 | 10-12 |
13:30-15:30 | DFT and FFT Matlab implementation Exercise session on Fourier Transform and Matlab FFT. HW 3 | 2+2 | |
05/12/2019 | --- | --- | --- | --- | |
06/12/2019 | 10-12 13:30-15:30 |
| Introduction to PDE Elliptic/parabolic PDE: mean and max properties, fundamental solution, smoothing, Green’s function | 4 | |
3 | 09/12/2019 | --- | --- | --- | --- |
10/12/2019 | 10-12
|
13:30-16:30 | Elliptic/parabolic PDE: solutions by separation of variables and Fourier transform. Linear conservation laws Exercise session on elliptic/parabolic PDE | 2+3 | |
11/12/2019 | 09:30-12:30 |
14-16 | Non-linear conservation laws: shocks, rarefaction waves, RH conditions. Wave equation Exercise session on conservation laws and wave equation. HW4 | 3+2 | |
12/12/2019 | --- | --- | --- | --- | |
13/12/2019 | 09:30-12:30 |
14-16 | Weak form of PDE, Galerkin method, Lax-Milgram lemma Exercise session on weak form | 3+2 | |
4 | 16/12/2019 | 09:30-12:30 |
14-16 | Finite Elements: Pk basis, Quadrature, FEM matrix PDE Toolbox tutorial | 3+2 |
17/12/2019 | --- | --- | --- |
| |
18/12/2019 | --- | --- | --- |
| |
19/12/2019 | 10-12 / 14-17 |
| Exam |
| |
20/12/2019 | 10-12 |
| Exam |
|
Bibliografia
Hand-outs and notes made available during the course. For backup and further readings:
1. Ordinary Differential Equations (ODE): G. Teschl, Ordinary Differential Equations and Dynamical Systems. American Mathematical Society;
2. Optimization of N-variate functions: J. Nocedal, S. Wright. Numerical Optimization. Springer;
3. Function approximation, transforms, numerical quadrature: A. Quarteroni, R. Sacco, F. Saleri. Numerical Mathematics. Springer; D. Kammler, A First Course in Fourier Analysis, Cambridge University Press;
4. Partial Differential Equations (PDE): S. Salsa, Partial Differential Equations in Action, Springer; L. Evans, Partial Differential Equations. American Mathmatical Society
5. Numerical Methods for PDEs: A. Quarteroni, Numerical Models for Differential Problems, Springer.
Italian-speaking students can also use these books:
1. Chapters 1,2,3: Analisi Matematica 2, M. Bramanti, C. Pagani, S. Salsa, Zanichelli ed.;
2. Chapter 4: Equazioni a Derivate Parziali – Metodi, modelli e applicazioni, S. Salsa, Springer;
3. Chapter 5: Modellistica Numerica per Problemi Differenziali, A. Quarteroni, Springer.
Esame
ASSESSMENT:
The final grade will be composed as follows:
- 30% homework assignments (four in total) graded during the course;
- 50% oral discussion over
- exercises and Matlab scripts discussed in class;
- one chapter of choice of the student.
- 20% oral discussion of a small additional topic (not covered during class but related to the class topics), to be preliminarily agreed upon with the teacher.
COURSE WEBSITE: https://sites.google.com/view/tamellini-applied-mathematics
Ciclo : XXXIII, XXXIV, XXXV
Tipologia corso : Caratterizzante
Curriculum: Ingegneria Sismica e Sismologia; Rischi Idrometeorologici, Geologici, Chimici ed Ambientali
Periodo: Semestre I
Anno accademico: 2019-2020