# Asset Publisher Asset Publisher

## Programme

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.

## Execution

 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

## Bibliography

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.

## Exam

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.