
Course unit
MATHEMATICAL METHODS FOR INFORMATION ENGINEERING
INP9087776, A.A. 2019/20
Information concerning the students who enrolled in A.Y. 2019/20
ECTS: details
Type 
ScientificDisciplinary Sector 
Credits allocated 
Educational activities in elective or integrative disciplines 
MAT/05 
Mathematical Analysis 
6.0 
Core courses 
INGINF/04 
Automatics 
3.0 
Course unit organization
Period 
Second semester 
Year 
1st Year 
Teaching method 
frontal 
Type of hours 
Credits 
Teaching hours 
Hours of Individual study 
Shifts 
Lecture 
9.0 
72 
153.0 
No turn 
Prerequisites:

The course requires basic knowledge from analysis (sequences, series, Riemann integration), linear algebra, probability, statistics and signals and systems (convolution, Fourier series, Fourier transform, power spectral density) 
Target skills and knowledge:

This course, addressed to the most theoretical part of Information Engineering, aims at leading the students to comprehend and employ the key mathematical tools of the field. It is expected that, at the end of the course, students understand the following concepts and use the following tools:
1.Complete and separable metric spaces;
2.Normed vector spaces;
3. Linear and continuous functionals and operators on infinite dimensional vector spaces;
4. Hilbert spaces;
5. Abstract integration;
6. Various types of convergence;
7.Calculus of variations, EulerLagrange equations;
8. Elements of convex analysis and abstract optimization;
9. Maximum entropy problems;
10. Elements of Markov chains.
At the end of the course, students are also expected to be able to apply the above listed concepts and tools to problems of optimal control, identification, spectral estimation and inverse problems which are central in Information Engineering. 
Examination methods:

The knowledge and acquired skills are checked through written tests. There are two different ways modes:
a) pass a midterm and a final test, each of two hours. Such tests occur half way trough and at the end of the semester. They feature five questions each worth six points. The final grade is computed on the basis of the average of the scores obtained in the two tests;
b) a unique threehour written test. These tests occur during the exam sessions. Such tests feature six questions each worht five points.
In both cases, the majority of the questions is taken from those proposed or even solved in class. 
Assessment criteria:

Knowledge and acquired skills will be checked along the following evaluation criteria:
1. Completeness of acquired knowledge;
2. Comprehension of the concepts introduced in the course;
3. Capability to exemplify various properties;
4. Capability to correctly employ the acquired tools in specific cases;
5. Capability to employ the tools in relevant applications in the field of Information Engineering. 
Course unit contents:

Metric Spaces. Completeness. Separability. Normed vector spaces. Banach spaces. Bounded linear operators. Continuous linear functionals. Dual space. Hilbert Spaces. Subspaces. Orthogonality. Inverse Problems. Bounded Operators on Hilbert Space. Elements of Abstract Integration. Measure spaces. Measurable functions. Simple functions. Abstract integral. Beppo Levi's theorem. Dominated convergence. Lebesgue spaces. BIBO stability in continuous time. Differentiability and abstract integral. Square integrable functions. Different types of convergence and their relation. Harmonic analysis: ReiszFischer theorem.Convergence of Fourier series. Fourier transform and DTFT. Reproducing Kernel Hilbert Spaces. Smoothness and decay of Fourier coefficients. Kernel methods in function estimation and identification. Representer theorem. Elements of Calculus of Variations. EulerLagrange equations. Hamilton principle in classical mechanics. Convex sets, convex functions. Jensen's inequality. Fundamental result of convex optimization. Application to the NeymanPearson Lemma. Constrained optimization: Lagrange Lemma. KarushKuhnTucker conditions. Optimal Control Problems. HamiltonJacobiBellman equations. Existence and uniqueness for ordinary differential equations. Maximum entropy problems: Boltzmann, Dempster, Burg. Frechet differential. Geometry of maximum entropy problems. Subgradients and subdifferentials. Jensen's inequality in general. Relative entropy (KullbackLeibler index). Relative entropy rate. Elements of Markov Chains theory. PerronFrobenius theorem. 
Planned learning activities and teaching methods:

The educational activities consist mainly in frontal lectures at the blackboard. These allow students to follow the presented material and readily take notes. The theoretical parts are continuously illustrated through significant examples and applications. Moreover, exercises to be solved autonomously are often proposed. Most of these exercises are then solved in class during the following lectures. There are also computer presentations on topics which require complex graphics. 
Additional notes about suggested reading:

All teacher lecture notes are made available to the students through the moodle platform. Further material such as journal articles or lecture notes are also made available in the same fashion. Finally, indications are provided on URL sites where supplementary interesting material is available. 
Textbooks (and optional supplementary readings) 

Bramanti, Marco, Metodi di Analisi Matematica per l'Ingegneria. Bologna: Esculapio, 2017.

Rudin, Walter, Real and complex analysis. New York: McGrawHill, 1974.

Boyd, Stephen; Vandenberghe, Lieven, Convex optimizationStephen Boyd, Lieven Vandenberghe. Cambridge: Cambridge University press, 2004.


