Dipartimento di Scienze di Base e Applicate per l'Ingegneria

The courses will be delivered at Dipartimento di Scienze di Base e Applicate per l’Ingegneria, via Antonio Scarpa.

Room 1B1 is located in PAL RM002 (ex Pal. B)

Time	Starting day 05-11-2014	Final day 3-12-2014
10.00 12.00	Course [2] Tue 1B1 ( 5 wed first lesson)	Course [2] Tue 1B1
10.00 12.00	Course [1] Wed 1B1	Course [1] Wed 1B1

 

• Course [1] Lorenzo Giacomelli

An introduction to Sobolev spaces and differential equations

Tentative program (details and level will be discussed with the students)

- Lebesgue measure, measurable functions. - Lp spaces: Hoelder inequality,reflexivity, separability, duality; strong and weak convergence. - Sobolev spaces: weak derivatives, definition and properties; approximation and density; extensions and traces. - Inequalities: Sobolev, Poincar´e,Gagliardo-Niremberg. - Bilinear forms: continuity, coercivity, Lax-Milgram theorem. Applications to linear differential equations. - Monotone operators, Browder-Minty theorem. Applications to nonlinear differential equations.

References

H. Brezis. Functional Analysis, Sobolev Spaces, and Partial Differntial Equations. Universitext, Springer.

L.C. Evans. Partial Differential Equations. Graduate Text in Mathematics 19, AMS.

http://www.dmmm.uniroma1.it/~lorenzo.giacomelli/phd1415/index.html

 

• Course [2] Virginia De Cicco

An introduction to the functions of bounded variation.

Programme:

- Functions of bounded variation of one variable: definition, examples, characterization, properties, history. - Functions of bounded variation of several variables: definition, pointwise properties, main theorems, applications.

 

• Course [3] 20 hours March-April - Micol Amar

Tecniche di omogeneizzazione e applicazioni ai tessuti biologici

Homogenization techniques and applications to biological tissues

Richiami di analisi funzionale Tecniche di omogeneizzazione: espansioni asintotiche, convergenza 2-scale Applicazioni a problemi di conduzione elettrica nei tessuti biologici

Brief survey of functional analysis Homogenization techniques: asymptotic expansions, two-scale convergence Application to problems of electrical conduction in biological tissues

 

• Course [4 ] 20 hours 22-01-2015 - 12-03-2015–Tue. - Thu. 15:00- 17:00-Erkki Somersalo

Introduction to Computational Inverse Problems

Abstract: Inverse problems constitute an important and active field of research of applied mathematics, with application areas in medical imaging, geophysics, engineering, remote sensing and environmental sciences, life sciences and many more. In a nutshell, mathematical modeling aims at building models that explain and predict consequences of given causes; in inverse problems, the objective is to identify and quantify the causes of an observed consequence. Characteristic to inverse problems is that the observations can be explained by many different causes (non-uniqueness), and that small errors in observations may propagate to huge errors in the solution of the problem (instability). To address these problems, sophisticated techniques have been developed, including classical regularization methods and Bayesian statistical techniques, both of which are covered in this introductory course. As prerequisites, the students should be familiar with basic linear algebra and numerical methods; familiarity with probability and statistics can be helpful. Examples and exercises involve the use of Matlab.

The course consists of 20 hours of lectures plus 10 hours of special topics and/or a workshop. Below is an outline of the possible topics that constitute the material. Topics that are marked with an asterisk (*) provide material for the special topics.

- Introduction (2 h): Introductory examples of inverse problems; Illposedness: non-uniqueness, instability.

- Regularization methods (8 +4* h): Linear problems: Truncated singular value decomposition (TSVD) method (2 h); Tikhonov regularization (2 h); Different formulas: Sherman-morrison-Woodbury formula (2 h); Selection of the regularization parameter (2 h); Truncated iterative methods * (4 h).

- Bayesian framework (10 + 2* h): Stochastic interpretation (2 h); Bayes’ formula, update of beliefs (2 h); Gaussian models, reinterpretation of regularization (4 h); Informative priors * (2 h).

- Sampling methods* (6* h): Markov chain monte Carlo (MCMC) (2 h); Dynamic problems: Particle filters (4 h).

• Course [5] March-May 2015 18 hours -Daniele Andreucci

PARTIAL DIFFERENTIAL EQUATIONS

I Galerkin’s Method

1. Orthonormal bases in separable Hilbert spaces.

2. Compact operators.

3. Eigenfunctions of compact operators.

4. Fourier’s method for elliptic equations.

5. Galerkin’s method for elliptic and parabolic equations.

II Optimal estimates for diffusion problems in unbounded domains

1. Heat equation: the Cauchy problem.

2. Non linear equations: the Cauchy problem.

3. Qualitative behavior of solutions and its dependence on the geometry of the domain

 

Vedi anche le pagine su corsi di dottorato svolti a Sapienza, Tor Vergata e Roma 3