2 Luglio, 2014 16:00

Sezione di Calcolo delle Variazioni ed Equazioni Differenziali

Traveling waves for reaction-diffusion equations involving the curvature operator

Maurizio Garrione, Università di Milano Bicocca

Aula seminari III piano

abstract

abstract

We are concerned with the existence of monotone traveling waves which connect two equilibria for a reaction-diffusion PDE where the spatial diffusion is of mean-curvature type. We study the set of the admissible speeds (i.e., the numbers c for which there exists a traveling wave as above with speed c) for different kinds of reaction terms, showing analogies and differences with respect to the linear diffusion case. The technique used relies on a change of variables leading to a suitable first order reduction.

23 Giugno, 2014 16:00 in punto

Sezione di Calcolo delle Variazioni ed Equazioni Differenziali

Anisotropic mesh adaptation for quasistatic crack propagation in brittle materials

MARCO ARTINA, TUM (Technische Universität München), Germany

Aula seminari MOX

abstract

abstract

The numerical simulation of quasistatic fracturing of brittle material, where no predefined crack path is imposed, is a challenging problem. In particular, we deal with the Francfort-Marigo model which requires the minimization of the well-known nonconvex and nonsmooth Mumford-Shah functional. To deal with a smoother problem which eases the minimization process of the energy, we consider the I-approximation via the Ambrosio-Tortorelli variational model. Moreover, from a modeling viewpoint, we induce the quasistatic crack evolution via an applied displacement slowly changing in time.
It is well established that the numerical simulation of crack propagation is often biased by an inappropriate numerical discretization, leading to non-physical results. In particular, a non-optimal or a fixed computational mesh can affect the fracture evolution, driving it along non-realistic directions. This issue can be successfully tackled by resorting to anisotropic adapted meshes, which have been employed to model phenomena exhibiting strongly directional features.
To drive mesh adaptation, we devise an a-posteriori anisotropic error estimator of the residual associated with the gradient of the energy functional. Through anisotropic meshes we are able to locate, size and orient the mesh elements in order to follow the intrinsic directionalities of the crack. Thanks to this grid adaptation process, we obtain solutions reliable from a physical viewpoint and with a relatively small computational cost.
In this talk, we describe the Francfort-Marigo model and its I-approximation in case of plane and antiplane displacements. Then, we focus on the anisotropic mesh adaptation strategy. We finally assess the reliability, robustness, and efficiency of the proposed approach on some benchmark tests.

23 Giugno, 2014 15:00 in punto

Sezione di Calcolo delle Variazioni ed Equazioni Differenziali

Analisi dello strato limite per una classe di flussi incomprimibili non lineari

Anna L. Mazzucato, Penn State University

Aula seminari III piano

abstract

abstract

Si discutono alcuni risultati recenti sull'analisi rigorosa del limite per viscosita' nulla e sullo stato limite associato per una certa classe di fluidi incomprimibili non lineari tridimensionali.

18 Giugno, 2014 16:30 in punto

Sezione di Calcolo delle Variazioni ed Equazioni Differenziali

Blow-up and global existence results for fourth order hyperbolic equations

Yongda Wang, Politecnico di Milano

Aula seminari III piano

abstract

abstract

In this talk, we consider a class of fourth order hyperbolic equations, which is viewed as a mathematical model for suspension bridges. For this kind of problems, there exists a unique local solution. By the potential well theory, we show finite time blow-up and global existence results of the problems for different initial data.

17 Giugno, 2014 15:00 in punto

Sezione di Calcolo delle Variazioni ed Equazioni Differenziali

OODA of TreeStructured Data Objects Using Persistent Homology

J. S. MARRON, Dept. of Statistics and Operations Research University of North Carolina

Aula seminari MOX

abstract

abstract

The field of Object Oriented Data Analysis has made a lot of progress on the statistical analysis of the variation in populations of complex objects. A particularly challenging example of this type is populations of tree-structured objects. Deep challenges arise, whose solutions involve a marriage of ideas from statistics, geometry, and numerical analysis, because the space of trees is strongly non-Euclidean in nature. Here these challenges are addressed using the approach of persistent homologies from topological data analysis. The benefits of this data object representation are illustrated using a real data set, where each data point is the tree of blood arteries in one person's brain. Persistent homologies give much better results than those obtained in previous studies.

12 Giugno, 2014 12:30 in punto

Sezione di Finanza Quantitativa

2013 Bruti Liberati Prize Winner: Moral hazard in dynamic risk management

Dylan Possamai, Universite Paris Dauphine

Aula Consiglio - Settimo piano

The field of Object Oriented Data Analysis has made a lot of progress on the statistical analysis of the variation in populations of complex objects. A particularly challenging example of this type is populations of tree-structured objects. Deep challenges arise, whose solutions involve a marriage of ideas from statistics, geometry, and numerical analysis, because the space of trees is strongly non-Euclidean in nature. Here these challenges are addressed using the approach of persistent homologies from topological data analysis. The benefits of this data object representation are illustrated using a real data set, where each data point is the tree of blood arteries in one person's brain. Persistent homologies give much better results than those obtained in previous studies.

9 Giugno, 2014 15:00 in punto

Sezione di Calcolo delle Variazioni ed Equazioni Differenziali

Spatio-temporal imaging of ruptures and the discrete-time dependent inverse source problem for the wave equation

Maarten V. De Hoop, Purdue University

Aula seminari VI piano

abstract

abstract

We first introduce and present an analysis of seismic waves starting
from the system of elastic-gravitational equations describing the free
oscillations of the earth. We establish an existence and uniqueness
result to a weak formulation under minimal regularity assumptions. We
then briefly describe the extraction of surface waves and body waves
using techniques from semi-classical analysis. We finally discuss the
discrete-time dependent inverse source problem and present an explicit
reconstruction of microseisms and ruptures from body-wave data under
certain conditions derived from local energy decay.

4 Giugno, 2014 12:15

Seminari di Cultura Matematica

L’insostenibile leggerezza di Homo

Mario Tozzi, Istituto di Geologia Ambientale e Geoingegneria del C.N.R

Aula De Donato S01

We first introduce and present an analysis of seismic waves starting
from the system of elastic-gravitational equations describing the free
oscillations of the earth. We establish an existence and uniqueness
result to a weak formulation under minimal regularity assumptions. We
then briefly describe the extraction of surface waves and body waves
using techniques from semi-classical analysis. We finally discuss the
discrete-time dependent inverse source problem and present an explicit
reconstruction of microseisms and ruptures from body-wave data under
certain conditions derived from local energy decay.

28 Maggio, 2014 12:15

Seminari di Cultura Matematica

BetOnMath: la matematica civile contro l'abuso del gioco d'azzardo

ANDRA', PAROLINI, VERANI, Politecnico di Milano - Dipartimento di Matematica MOX

Aula Rogers

We first introduce and present an analysis of seismic waves starting
from the system of elastic-gravitational equations describing the free
oscillations of the earth. We establish an existence and uniqueness
result to a weak formulation under minimal regularity assumptions. We
then briefly describe the extraction of surface waves and body waves
using techniques from semi-classical analysis. We finally discuss the
discrete-time dependent inverse source problem and present an explicit
reconstruction of microseisms and ruptures from body-wave data under
certain conditions derived from local energy decay.

27 Maggio, 2014 11:00

Seminario Matematico e Fisico di Milano

DIFFUSIONE, TRASPORTO OTTIMO E CURVATURA

Giuseppe Savare, Università di Pavia

Dipartimento di Matematica, Politecnico di Milano

abstract

abstract

La teoria delle forme di Dirichlet (Fukushima 1980, Albeverio, Ma-Rockner 1992,…) permette di costruire processi di diffusione in spazi di misura molto generali.

Ad essi è possibile associare una geometria intrinseca grazie alla distanza di Biroli-Mosco (1991), un formalismo di calcolo differenziale (il cosiddetto Gamma-calcolo) e una nozione di curvatura-dimensione seguendo l’approccio introdotto da Bakry-Emery (1984), legata ad importanti disuguaglianze geometrico-funzionali.


Più recentemente, partendo da uno spazio metrico di misura invece che da una forma di energia, si è sviluppata una teoria metrica degli spazi di Sobolev (Koskela-MacManus 1998, Cheeger 1999, Shanmugalingam 2000) che porta alla costruzione di una diffusione (generalmente nonlineare) e di un calcolo non-smooth al primo ordine.

I lavori di Lott-Villani e Sturm (2006-2009) hanno poi introdotto il punto di vista del trasporto ottimo per definire una condizione di curvatura-dimensione stabile per convergenza di Gromov-Hausdorff e con importanti applicazioni geometriche.

Questi due punti di vista mettono in luce interessanti aspetti anche nel caso più semplice di una equazione di diffusione-trasporto in R^n con coefficienti sufficientemente regolari o, più in generale, in varietà Riemanniane.


Il seminario si propone di introdurre queste tematiche e di presentare alcuni recenti risultati, ottenuti in collaborazione con Ambrosio e Gigli, che mostrano come i due punti di vista sono essenzialmente equivalenti.