2013/2014

2014/2015

DM270

MAT/05 (MATHEMATICAL ANALYSIS)

DEPARTMENT OF ELECTRICAL,COMPUTER AND BIOMEDICAL ENGINEERING

INDUSTRIAL ENGINEERING

ENERGIA

2°

1st semester (29/09/2014 - 16/01/2015)

6

68 lesson hours

ITALIAN

WRITTEN AND ORAL TEST

SAVARE' GIUSEPPE (titolare) - 6 ECTS
COLLI FRANZONE PIERO - 2 ECTS
SEGATTI ANTONIO GIOVANNI - 1 ECTS

Differential and integral calculus for scalar and vector functions, matrices and linear transformations, sequences and series, power series in the real line, complex numbers, polar coordinates.

Students will be introduced to the basic mathematical tools for signal theory and optimization. To this aim, the course is divided in two parts. In the first part, MATHEMATICAL METHODS (6CFU), they will learn how to work in the complex framework, evaluate integrals of olomorphic functions, manipulate power and Fourier series, adopt the point of view of signal theory, calculate and operate with Z, Fourier and Laplace transforms, solve simple ordinary differential equations with constant coefficients, understand convolutions. The second part, OPTIMIZATION AND DISCRETE TRANSFORMS (3CFU), will be devoted to the elementary notions of free and constraint optimization and to the basic techniques of the mathematical theory of discrete signals (DFT, FFT, convolutions) with simple applications to difference equations and numerical approximations.

Complex functions
Complex functions
Manipulation of complex numbers
Rational, exponential, and trigonometric functions, logarithms
Power series
Conplex derivatives, holomorphic functions, Cauchy-Riemann conditions
Line integrals, Cauchy theorem, analyticity of holomorphic functions
Singularities, Laurent series, residue formula
Evaluation of integrals, Jordan lemma
Signals
Discrete and continuous signals
Elementary manipulation of signals: sum, linear combination, shift and rescaling.
Scalar products and norms
Z transform
Definition, simple properties, examples
Applications to linear difference equations
Fourier series
Periodic signals, trigonometric and exponential functions, Fourier series.
Pointwise and energy convergence, Gibbs phenomenon.
Parseval identity
Applications
Fourier transform
Definition of Fourier transform, relationships with Fourier series, elementary properties
Riemann-Lebesgue lemma
Inversion theorem for piecewise regular functions
Plancherel identity, Fourier transform for L^2 functions
Laplace transform
Definition, links with the Fourier transform, main properties
Inversion of Laplace transform, residue and Heaviside formula
Application to simple ordinary differential equations
Convolution
Definition and simple example of convolutions
Links with Fourier and Laplace transform
Simple applications to differential equations
Optimization
Unconstrained Optimization Problems
- Gradient methods and line-searches
- Newtonian methods: trust-regions, quasi-Newton and Gauss-Newton for least-squares problems
Constrained Optimization Problems
- Optimality conditions, penalization and SQP methods
Discrete transforms
Discrete Fourier transform (DFT)
The algorithm of Fast Fourier Transform (FFT)
Discrete convolution
Applications to difference and approximation problems, stability

Lectures (hours/year in lecture theatre): 49
Practical class (hours/year in lecture theatre): 45
Practicals / Workshops (hours/year in lecture theatre): 0

M. Codegone. Metodi matematici per l'Ingegneria. Zanichelli. .

G. Savaré. Lecture notes. The pdf file can be downloaded from the web site of the course. .

M. Giaquinta, G. Modica. Note di Metodi Matematici per Ingegneria Informatica. Pitagora, Bologna.

F. Tomarelli. Esercizi di Metodi Matematici per l'Ingegneria. CLU.

Matlab Optimization and Signal Proccessing Toolbox. User's guide. The MathWorks Inc..

F.J. Bonnan, C.J. Gilbert, C. Lemarechal C, C.A. Sagastizabal. Numerical Optimization. Theoretical and practical aspects. Springer Verlag (Universitext), 2006. Second edition.

A written and a computer lab test.

A written and a computer lab test.