15 Marzo, 2016 16:30

Seminario Matematico e Fisico di Milano

Liouville Theorems in PDE’s: old and new

Enzo Mitidieri, Università degli Studi di Trieste

Sala Consiglio, 7 piano, Dipartimento di Matematica, Via Bonardi 9, Milano

abstract

abstract

We review some Liouville type results from the classical Cauchy-Liouville theorem for harmonic functions up to some recent contributions related to quasilinear second order and semilinear higher order problems.


This seminar is organized within the PRIN 2012 Research project «Equazioni alle derivate
parziali di tipo ellittico e parabolico: aspetti geometrici, disuguaglianze collegate, e applicazioni
- Partial Differential Equations and Related Analytic-Geometric Inequalities» Grant Registration
number 2012TC7588_003, funded by MIUR – Project coordinator Prof. Filippo Gazzola

14 Marzo, 2016 16:00

Seminario Matematico e Fisico di Milano

Renormings in Banach spaces. Convexity and flatness

Stanimir Troyanski, University of Murcia and Bulgarian Academy of Sciences

Sala di Rappresentanza, Dipartimento di Matematica, Via Saldini 50

We review some Liouville type results from the classical Cauchy-Liouville theorem for harmonic functions up to some recent contributions related to quasilinear second order and semilinear higher order problems.


This seminar is organized within the PRIN 2012 Research project «Equazioni alle derivate
parziali di tipo ellittico e parabolico: aspetti geometrici, disuguaglianze collegate, e applicazioni
- Partial Differential Equations and Related Analytic-Geometric Inequalities» Grant Registration
number 2012TC7588_003, funded by MIUR – Project coordinator Prof. Filippo Gazzola

11 Marzo, 2016 09:45

MOX Seminar

How algorithm re-thinking can open the way to exascale

Cristiano Malossi, IBM Research - Zurich Foundations of Cognitive Solutions

Aula Seminari "Saleri" VI Piano MOX-Dipartimento di Matematica, Politecnico di Milano

abstract

abstract

Recent years have seen a dramatic change in core technology. The end of Dennard scaling has now placed energy as a primary design principle in parwith performance, all the way from the hardware to the application software. To continue the exponential overall improvements, technologists have turned into multi-core chips and parallelism at all scales. This trend opens up new challenges in application development: how to program more complex machines and how to scale applications to the O(10^6) cores characteristic of leadership class supercomputers, while maintaining the pace with the very fast increase in power requirements. This talk will present several examples where a careful algorithm re-engineering has opened the way to fast, scalable, and energy-efficient solutions, for a range of applications from solution of complex PDEs to big data analytics.

contact: christian.vergara@polimi.it

11 Marzo, 2016 11:00 in punto

Sezione di Analisi

Fractional Diffusion Equations. Theory on bounded domains

Juan Luis Vazquez, Universidad Autonoma de Madrid

Sala Consiglio 7° piano

abstract

abstract

In this talk I will report on some of the progress made by the author and collaborators on the topic of nonlinear diffusion equations involving long distance interactions in the form of fractional Laplacian operators. In recent work with Bonforte and Sire we address the solution of Dirichlet problems in bounded domains, where the correct definition of what is the operator has a number of possible answers leading to different qualititative and quantitative results.

This seminar is organized within the PRIN 2012 Research project «Equazioni alle derivate
parziali di tipo ellittico e parabolico: aspetti geometrici, disuguaglianze collegate, e applicazioni
- Partial Differential Equations and Related Analytic-Geometric Inequalities» Grant Registration
number 2012TC7588_003, funded by MIUR – Project coordinator Prof. Filippo Gazzola

3 Marzo, 2016 11:00

MOX Seminar

Large scale flow simulations

Barbara Wohlmuth, TU München

Aula Seminari

abstract

abstract

Large scale flow problems play an important role in many multi-physics applications. In this talk, we consider different aspects such as the role of local mass conservation in a weak or strong setting and non-linear solution techniques in case of inequality constraints. Many natural pairings of conforming discrete spaces do not satisfy a uniform inf-sup condition and thus do have either to be stabilized or the velocity space has to be enriched. Quite often these standard techniques result in discrete velocities which do not yield strongly the divergence free condition. Here we propose easy to realize local a posteriori corrections which allow us to lift the $H^1$-conforming velocity onto a $H(\text{div})$-conforming mixed finite element, e.g. of Raviart--Thomas type. By doing so we achieve a significantly improved accuracy in case of coupled transport terms where, e.g., the velocity enters into the advective part of an energy equation. Secondly we consider an example from porous media flow and discuss the role of inequality constraints on the saturation. In contrast, to simple contact laws where a non-linear complementarity condition can be defined in terms of the max-function and affine terms, non-linear constitutive relations also enters. Thus the realizations of active set strategies are more involved. Nevertheless, robust numerical results can be obtained if combined with an adaptive time stepping scheme.

contact: paola.antonietti@polimi.it

26 Febbraio, 2016 11:00

Seminario Matematico e Fisico di Milano

Regularity and singularity of area-minimizing surfaces

Camillo De Lellis, Universitaet Zuerich

Sala Consiglio, 7 piano, Dipartimento di Matematica, Via Bonardi 9, Milano

abstract

abstract

The Plateau's problem, named after the Belgian physicist J. Plateau, is a classic in the calculus of variations and regards minimizing the area among all surfaces spanning a given contour. A successful existence theory, that of integral currents, was developed by De Giorgi in the case
of hypersurfaces in the fifties and by Federer and Fleming in the general case in the sixties.
When dealing with hypersurfaces, the minimizers found in this way are rather regular: the corresponding regularity theory has been the achievement of several mathematicians in the 60es,70es and 80es (De Giorgi, Fleming, Almgren, Simons, Bombieri, Giusti, Simon among others).

In codimension higher than one, a phenomenon which is absent for hypersurfaces, namely that of branching, causes very serious problems: a famous theorem of Wirtinger and Federer shows that any holomorphic subvariety in $\mathbb C^n$ is indeed an area-minimizing current. A celebrated monograph of Almgren solved the issue at the beginning of the 80es, proving that the singular set of a general area-minimizing (integral) current has (real) codimension at least 2.
However, his original (typewritten) manuscript was more than 1700 pages long. In a recent series of works with Emanuele Spadaro we have given a substantially shorter and simpler version of Almgren's theory, building upon large portions of his program but also bringing some new ideas from partial differential equations, metric analysis and metric geometry.

26 Febbraio, 2016 10:00 in punto

Sezione di Analisi

Pullback attractors for random and non-autonomous dynamical systems: an introduction with applications

Tomás Caraballo, Universidad de Sevilla

Aula seminari III piano

abstract

abstract

The theory of pullback attractors for Random/Non-autonomous Dynamical Systems has been extensively developed over the last three decades, mainly due to the fact that it provides a suitable framework to analyze the global behavior of dynamical systems containing some kind of stochasticity or randomness as well as some time dependent forcing. In this talk we will introduce the basic tools of the theory of pullback attractors for handling both non-autonomous and random dynamical systems. We will show how both problems can be analyzed in a unified formulation thanks to the concept of cocycle. We will also emphasize on the different effects that different kind of noise can produce on the asymptotic behaviour of the solutions. Our results will be illustrated with some basic academic examples as well as some other interesting models appearing in the applied sciences (for example, reaction-diffusion equations and chemostast models).

25 Febbraio, 2016 15:30

Sezione di Geometria, Algebra e loro applicazioni

SEMILINEAR ELLIPTIC PDES ON COMPLETE MANIFOLDS WITH BOUNDARY

Guglielmo Albanese, Università degli Studi di Milano

Aula seminari del 6 piano, Edificio La Nave, Dipartimento di Matematica, Via Bonardi 9, Milano

abstract

abstract

(Il seminario del 18 febbraio è cancellato per indisposizione del relatore)

25 Febbraio, 2016 12:00

MOX Seminar

Finite element modeling of the mitral valve from clinical imaging: a tool to support the optimization of surgical treatment

Emiliano Votta, DEIB, Politecnico di Milano

Aula Seminari "Saleri", VI piano MOX Dipartimento di Matematica, Politecnico di Milano

abstract

abstract

The mitral valve is a particularly complex apparatus that separates the left atrium from the left ventricle and guarantees unidirectional blood flow from the former to the latter.
Mitral valve patholgies have a high prevalence and usually require surgical treatment, repair being preferred to substitution whenever possible. At the same time, though, mitral repair can be complex and the corresponding clinicl outcomes are not always satisfactory. For this reason the improvement of repair techniques and devices remains a focal clinical and research activity.
In this process, numerical modelling has been used since the '90s to gain insight into mitral valve biomechanics, so to better understand mitral valve pathophysiology and support the development of more effective surgical approaches. In the last decade in particular, numerical models based on medical imaging have been developed, exploting the detailed anatomical information yielded by medical images. This evolution of the modelling approach has paved the way towards the use of numerical models as a support tool for the patient-specific tailoring of mitral valve surgical repair.
In the seminar, the modeling activity carried out in this field by the Biomechanics Research Group at DEIB, Politecnico di Milano, will be presented; the potential of the developed modeling approach will be exemplified through recent clinical applications, and emphasis will focused on current and future developments.

contact:christian.vergara@polimi.it

24 Febbraio, 2016 17:00

Seminario Matematico e Fisico di Milano

Divergent series: from Thomas Bayes’s bewilderment to today’s resurgence via the rainbow

Sir Michael Berry, H H Wills Physics Laboratory, Bristol, UK

Sala Consiglio, 7 piano, Dipartimento di Matematica, Via Bonardi 9, Milano

abstract

abstract

Following the discovery by Bayes in 1747 that Stirling’s series for the factorial is divergent, the study of asymptotic series has today reached the stage of enabling summation of the divergent tails of many series with an accuracy far beyond that of the smallest term. Several of these advances sprang from developments of Airy’s theory of waves near optical caustics such as the rainbow. Key understandings by Euler, Stokes, Dingle and Écalle unify the different series corresponding to different parameter domains, culminating in the concept of resurgence: quantifying the way in which the low orders of such series reappear in the high orders.