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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

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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

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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

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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

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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

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18/12/2019

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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