


Politecnico di Torino  
Academic Year 2016/17  
01RLMNG Variational and multiscale methods 

Master of sciencelevel of the Bologna process in Mathematical Engineering  Torino 





Esclusioni: 01RMQ; 03JNY 
Subject fundamentals
For what concerns variational methods, some classical tools will be presented, for the correct statement of rather general minimization problems. It will also be shown how several problems, initially not related to the minimization of any functional, can indeed be framed in a variational setting, thus presenting the Calculus of Variations as a useful tool in the solution of problems of quite different nature. In addition to general principles and techniques, numerous examples shall be discussed, some of which will possibly be studied in greater detail.
For what concerns multiscale methods, focus will be on the mathematical principles of signal analysis, from discrete Fourier analysis to timefrequency and timescale analysis (wavelets). From a modern point of view, signals are modeled by vectors in Hilbert spaces (of sequences or functions), and the problems of approximation, compression, etc., reduce to the representation of the signal with respect to a suitable basis. One is then interested in looking for optimal and structured bases, often constructed by operations such as translations, dilations and modulations, starting from a suitable window. The classical results from signal processing (sampling, interpolation, filters, filter banks, etc.) can be reinterpreted by this geometricanalytic language from Functional Analysis, which is illuminating in its own right and is also the language currently used in timefrequency analysis and mathematical signal processing. 
Expected learning outcomes
Ability to properly state a typical problem of the Calculus of Variations, in a suitable function space, taking into account the prescribed boundary conditions.
Ability to apply the general theory to specific problems, in order to prove the existence of a solution and obtain its optimality conditions. Ability to properly state the typical problems of signal analysis in suitable function spaces, and to investigate these problems using appropriate mathematical tools. 
Prerequisites / Assumed knowledge
Students are expected to be familiar with the topics of the following courses: Mathematical Analysis I and II, Geometry, Functional Analysis.

Contents
For what concerns variational methods:
Introduction and statement of some classical problems: isoperimeters, brachistochrone, catenary, revolution surfaces of minimal area. Historical background: the Dirichlet Principle and the counterexample of Weierstrass. The modern approach via compactness and semicontinuity: the "direct method" and the role of convexity. The Rayleigh quotient and applications to eigenvalues. SturmLiouville problems. Different kinds of boundaryvalue problems. Frechet and Gateuax derivatives. Optimality conditions and EulerLagrange equations. Applications to differential equations. Sobolev spaces as a natural environment. Classical and weak solutions. Regularity in the onedimensional case, and return to a classical solution. Applications to mechanics and the Principle of Minimal Action. Some applications to partial differential equations. Laplace and Poisson equations. Constrained problems and Lagrange multipliers. Further examples and applications, and detailed study of some specific examples. For what concerns multiscale methods: Review on Hilbert spaces, orthonormal bases, Riesz bases. Discrete Fourier Analysis (discretetime Fourier transform, z transform, discrete Fourier transform). Sampling and interpolations. Localization (shorttime Fourier transform) and uncertainty principle. Filters and filter banks. Wavelets bases of sequences. Wevelets bases of functions. 
Delivery modes
The course consists of lessons and exercise classes. For what concerns multiscale methods, exercise classes will also take place in computer labs.

Texts, readings, handouts and other learning resources
Notes of the course will be made available on the web portal.

Assessment and grading criteria
The exam, which will be oral, will test the knowledge of the topics of the course, and the ability to apply the general theory to the solution of exercises, along the lines of the exercises and the examples illustrated during classes.

