4 Maggio, 2009 17:00

Sezione di Calcolo delle Variazioni ed Equazioni Differenziali

VARIATIONAL ANALYSIS VIA DISCRETE APPROXIMATIONS IN OPTIMAL CONTROL

BORIS S. MORDUKHOVICH, Wayne State University

Aula interna 3 piano

abstract

abstract

In this talk we discuss recent advances in variational analysis and its applications in
dynamic optimization models governed by nonconvex differential inclusions. Our approach
is based on discrete approximations of differential systems and thus related to both numerical
and theoretical issues in optimization and control. The main results justify stability of
discrete approximations and establish necessary optimality conditions of the Euler and Hamiltonian types.

28 Aprile, 2009 16:00

Sezione di Calcolo delle Variazioni ed Equazioni Differenziali

Shape optimization under convexity constraint

Jimmy LAMBOLEY, ENS Cachan, antenne de Bretagne

aula seminari VI piano

abstract

abstract

The shape optimization is the study of optimization problems whose unknown is a domain of $ R^d$. I will focus on the case where admissibles shapes are required to be convex sets of $ R2$. Under this constraint, it is hard to write optimality conditions. In a first part, I will show how we can write such conditions (first and second order), and I will use these ones to exhibit a class of functionals which leads to polygonal optimal shapes (work with A. Novruzi). In a second part, I will focus on the minimization of the second eigenvalue for the Laplace operator (Dirichlet conditions), model problem which shows difficulties linked to convexity constraint, and also difficulties due to the regularity of optimal shapes. We particularly show that optimal shapes are C^{1,1/2} and no more, for this problem. I end with some links with partially overdetermined problems.

24 Aprile, 2009 15:00

Sezione di Calcolo delle Variazioni ed Equazioni Differenziali

Diffusion limits for MCMC paths

Alexandros Beskos, University College of London

Aula F. Saleri VI Piano Dipartimento di Matematica

abstract

abstract

We examine complexity and optimal tuning of MCMC algorithms in high dimensions. Earlier works in the literature have shown that the step of Random-Walk Metropolis should be scaled as $1/n$ ($n$ being the dimensionality index); for the (Metropolised) Langevin algorithm the scaling is $1/n^{1/3}$. Both results have been theoretically justified in the simplified scenario of iid targets. Under these scalings MCMC trajectories converge to analytically identifiable diffusions. Such a limit allows for a complete characterisation of the algorithms and the straightforward specification of optimality criteria, via the identification of the best acceptance probability that practitioners should try to achieve.

In recent research, we have extended this framework to complex, realistic models, including conditioned diffusion models and inverse problems for the Navier-Stokes equation. We have established optimality criteria for MCMC algorithms applied on such structures and identified algorithmic complexity and best acceptance probability.
In these cases, MCMC trajectories are shown to converge to infinite dimensional Stochastic Differential Equations. We have also consider the so-called hybrid Monte-Carlo MCMC algorithm used by physicists in molecular dynamics and other applications and studied its performance in high dimensions.

21 Aprile, 2009 16:30

Sezione di Calcolo delle Variazioni ed Equazioni Differenziali

Morfogenesi in Natura : modelli di interazione fra meccanica e biologia

Pasquale Ciarletta, Ecole Normale Superieure (Parigi)

Aula Seminari F. Saleri VI Piano Dipartimento di Matematica - Politecnico di Milano

abstract

abstract

In questo seminario, presentera due esempi di modellizzazione biomeccanica per descrivere alcuni meccanismi che regolano i processi morfogenetici in Natura.
Il primo esempio concerne la morfogenesi epiteliale dell embrione di C.elegans (verme nematode di grande importanza in biologia dello sviluppo), in cui un embrione inizialmente di forma ovoidale (20x60 micron) si deforma all incirca di quattro volte la sua lunghezza iniziale in meno di tre ore. L anatomia dell embrione e rappresentata attraverso la meccanica dei continui, considerando le cellule embrionali come materiali aventi un comportamento attivo (dovuto all attivazione del complesso actina/miosina non muscolare) ed una risposta viscoelastica passiva (legata all anisotropia della rete dei microtubuli nel citoscheletro). La soluzione analitica del problema elastico a deformazioni finite descrive il rapporto fra la distribuzione del campo degli sforzi nell embrione e le proprieta biofisiche dei filamenti citoscheletrici. Tale conoscenza e fondamentale per studiare i processi biologici e chimici di meccanotrasduzione dell embrione attraverso esperimenti farmacologici ed embrioni mutati geneticamente.
Il secondo esempio ha come scopo quello di descrivere i campi di deformazione e di sforzo indotti dalla produzione e dalla riorganizzazione della materia molle, utilizzando gli strumenti della fisica non-lineare e della teoria delle biforcazioni. La crescita volumetrica di superfici sottili ed inizialmente piane e stata studiata per materiali iperelastici attraverso una decomposizione moltiplicativa del tensore gradiente di deformazione. Utilizzando il piccolo spessore della superficie come parametro di espansione, si dimostra che le equazioni di equilibrio sono del tipo Foppl-Von Karman, con la crescita che agisce come una sorgente di curvature media e gaussiana. Tale analisi prende in considerazione l anisotropia dei processi di crescita e le sue disomogeneita spaziali, spiegando come semplici espressioni del tensore di crescita possano descrivere delle forme complesse osservabili in Natura (alghe, foglie, etc.).

20 Aprile, 2009 15:00

Sezione di Calcolo delle Variazioni ed Equazioni Differenziali

Multilevel Schwarz preconditioners for cardiac Bidomain models and applications

Luca F. Pavarino, Dipartimento di Matematica, Universita di Milano

Aula F. saleri VI Piano Dipartimento di Matematica

abstract

abstract

After a brief review of the main numerical models in cardiac electrophysiology, we construct and analyze multilevel additive Schwarz preconditioners for the anisotropic cardiac Bidomain and Monodomain models in three dimensions. To date, the Bidomain system is the most complete model of the bioelectrical activity
of the heart tissue, consisting of a degenerate parabolic system of nonlinear reaction-diffusion equations coupled with a stiff system of several ordinary differential equations describing the
dynamics of ionic currents through the cellular membrane. Due to the presence of very different scales in both space and time, the numerical discretization of this system by finite elements in space and semi-implicit methods in time produces very ill-conditioned linear systems that must be solved at each time step. The proposed multilevel algorithm employs a hierarchy of
nested meshes with overlapping Schwarz preconditioners on each level and is fully additive, hence parallel, within and among
levels. Convergence estimates can be proved, showing that the convergence rate of the resulting multilevel algorithm is independent of the number of subdomains (scalability), of the
mesh sizes of each level and of the number of
levels (optimality). Parallel numerical results, using the PETSc library and run on Linux Clusters, confirm the scalability and optimality of the method, as well as its parallel efficiency for large-scale simulations of a complete cardiac cycle on both cartesian and deformed domains in three dimensions. The resulting parallel solver has been employed in several simulation studies
of ventricular repolarization in presence of M-cells and ischemic regions.

8 Aprile, 2009 12:15

Seminari di Cultura Matematica

I viaggi nello spazio e il ritorno sulla luna

Giovanni Caprara, Direttore della Pagina delle Scienze del Corriere della Sera Ricerca e comunicazione:

Aula B 2.1, Via Bonardi 9

After a brief review of the main numerical models in cardiac electrophysiology, we construct and analyze multilevel additive Schwarz preconditioners for the anisotropic cardiac Bidomain and Monodomain models in three dimensions. To date, the Bidomain system is the most complete model of the bioelectrical activity
of the heart tissue, consisting of a degenerate parabolic system of nonlinear reaction-diffusion equations coupled with a stiff system of several ordinary differential equations describing the
dynamics of ionic currents through the cellular membrane. Due to the presence of very different scales in both space and time, the numerical discretization of this system by finite elements in space and semi-implicit methods in time produces very ill-conditioned linear systems that must be solved at each time step. The proposed multilevel algorithm employs a hierarchy of
nested meshes with overlapping Schwarz preconditioners on each level and is fully additive, hence parallel, within and among
levels. Convergence estimates can be proved, showing that the convergence rate of the resulting multilevel algorithm is independent of the number of subdomains (scalability), of the
mesh sizes of each level and of the number of
levels (optimality). Parallel numerical results, using the PETSc library and run on Linux Clusters, confirm the scalability and optimality of the method, as well as its parallel efficiency for large-scale simulations of a complete cardiac cycle on both cartesian and deformed domains in three dimensions. The resulting parallel solver has been employed in several simulation studies
of ventricular repolarization in presence of M-cells and ischemic regions.

7 Aprile, 2009 15:00

Seminari di Cultura Matematica

Reusable object oriented design techniques for a high performance multi-physics and multi-method framework.

Andrea Lani, Von Karman Institute for Fluid Dynamics - Belgium

Aula F. Saleri VI Piano Dip. di Matematica

abstract

abstract

In the scientific field, it often happens that a researcher or a team of researchers start writing a software package in order to tackle some specific problems. When new problems arise or new methodologies to solve problems are developed, one has to choose between adding new features into an existing code or writing a new one from scratch. In both cases, reusability and extensibility of the software become extremely advantageous for easing the job of the developers. This seminar presents a number of object oriented design techniques which were targeted towards scientific computing applications and developed with the aim of easing the integration of new functionalities such as arbitrary mesh-based data structures, physical models and numerical algorithms (e.g. space discretizations, time stepping schemes, linear system solvers, dots). In the context of high performance computing, suitable guidelines for achieving a developer friendly parallelization will also be addressed. All those techniques have been successfully implemented in COOLFluiD, a computational platform for the high performance simulation of multi-physics phenomena on unstructured grids developed at the Von Karman Institute during the last 7 years. A whole range of different CFD applications will be discussed as a validation of the concept design ideas.

3 Aprile, 2009 15:00

Seminari di Cultura Matematica

A multivariate dynamic logit model for categorical panel data with latent Markov subject-specific parameters

Alessio Farcomeni, Università La Sapienza - Roma -

Aula F. Saleri VI Piano Dip. di Matematica

abstract

abstract

For the analysis of multivariate categorical longitudinal data, we propose a model based on a marginal parametrization of the conditional distribution of each vector of response variables given the covariates, the lagged response variables, and a set of subject-specific random effects for the unobserved heterogeneity. The
latter ones are assumed to follow a first-order Markov chain. For performing maximum likelihood estimation we outline an EM algorithm. A Viterbi algorithm is described for estimation of the most likely sequence of latent states. The data analysis approach based on the proposed model is illustrated by an application to a dataset which
derives from the Panel Study on Income Dynamics and concerns fertility and female participation to the labor market.

Joint work with F. Bartolucci

1 Aprile, 2009 12:15

Seminari di Cultura Matematica

L'architettura di Tiahuanaco e degli Inca: matematica nella pietra

J. Pierre Protzen, Università della California a Berkeley, USA

Aula B 2.1, Via Bonardi 9

For the analysis of multivariate categorical longitudinal data, we propose a model based on a marginal parametrization of the conditional distribution of each vector of response variables given the covariates, the lagged response variables, and a set of subject-specific random effects for the unobserved heterogeneity. The
latter ones are assumed to follow a first-order Markov chain. For performing maximum likelihood estimation we outline an EM algorithm. A Viterbi algorithm is described for estimation of the most likely sequence of latent states. The data analysis approach based on the proposed model is illustrated by an application to a dataset which
derives from the Panel Study on Income Dynamics and concerns fertility and female participation to the labor market.

Joint work with F. Bartolucci

31 Marzo, 2009 12:30

Sezione di Geometria, Algebra e loro applicazioni

Varietà di anelli *-regolari e/o ortoreticoli modulari associati alle logiche quantistiche di von Neumann.

Luca Giudici

Aula seminari MOX, VI piano

abstract

abstract

Introduzione.

Tralasciando pochissimi casi notoriamente patologici, von Neumann
stabilì una equivalenza tra tre tipi di strutture matematiche usabili
per descrivere quei sistemi fisici puramente quantistici in cui il
calcolo algebrico tra osservabili (illimitati e non commutanti fra loro)
è sempre possibile: (1) fattori finiti F; (2) geometrie continue con
probabilità di transizione (astrattamente assimatizzate o costruite
come ortoreticoli L delle proiezioni di un F); (3) l anello *-regolare
R degli operatori (lineari chiusi e densamente definiti, ma
possibilmente illimitati) affiliati ad F.

(3) fornisce il calcolo algebrico su tutti gli osservabili del sistema
fisico, (1) solo per quelli a spettro limitato, mentre (2) fornisce il
calcolo proposizionale per tali logiche quantistiche di von Neumann.

I problemi.

(1) i fattori finiti nascono come generalizzazione dei fattori
finito-dimensionali, ma già von Neumann ne diede esempi che non sono
approssimativamente finito dimensionali , e dopo Connes conosciamo e
classifichiamo una infinità continua di tali esempi tra loro non
isomorfi. C è un qualche modo in cui tutti i fattori finiti possono
essere ricondotti ai fattori finito dimensionali (cioè i normali
*-anelli di matrici con coefficienti scalari)?

(2) in un sistema classico, il calcolo proposizionale (algebra di Boole)
è decidibile (mediante tavole di verità). Cosa accade per i sistemi
quantistici?

I risultati.

L anello *-regolare R degli operatori (anche illimitati) affiliati a un
fattore finito F (tra le algebre di von Neumann) è nella varietà
generata dai fattori finito dimensionali. Analogamente generalizzando F
a una C^*-algebra finita e di Rickart. Ne consegue che il calcolo
proposizionale (l insieme delle tautologie) della logica quantistica di
von Neumann è decidibile. Inoltre, i metodi (logico-algebrici) usati
forniscono un nuovo punto di vista per il problema di immersione di
Connes (1976) e il problema di Kaplansky sulla linearità delle
(2-)quasitracce (1952).