13 Luglio, 2012 11:15 in punto

Sezione di Probabilità Quantistica

A class of Langevin equations with non Markovian gaussian noises

Rolando Rebolledo, Pontificia Universidad Catolica de Chile

Università di Pavia, aula Beltrami del Dipartimento di Matematica

abstract

abstract

This seminar proposes a generalized Langevin equation for a classical mechanical system imbedded in a reservoir. Firstly, a finite number n of particles is considered in the reservoir and the action of this reservoir on the small system is described by a memory kernel and a zero-mean gaussian process of a very general nature, based on the gaussian transform introduced in [2]. This gives raise to an integrodifferential equation for the evolution of a generic particle in the main system. The existence of a unique solution X_n of the above equation is proved: it is also a gaussian process. A thermodynamics limit when n goes to infinity is considered and it is proved that X_n converges in distribution to the solution of a non-Markovian evolution equation. The above class of equations includes as a particular case the one studied by Kupferman in [1].

This report is based on a joint research with Carlos Lizama from the University of Santiago.

[1] R. Kupferman. Fractional kinetics in Kac--Zwanzig heat bath models. Journal of Statistical Physics, Jan 2004.

[2] C. Lizama and R. Rebolledo. Comm. on Stochastic Analysis, 4(4):541--551, 2010.

12 Luglio, 2012 14:30 in punto

Sezione di Probabilità Quantistica

Nonlinear complementarity constraints for hydrogen-water flow with hydrogen dissolution in porous media

Ibtihel Ben Gharbia, INRIA-Rocquencourt

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

abstract

abstract

The modeling of migration of hydrogen produced by the corrosion of the nuclear waste packages in an underground storage including the dissolution
of hydrogen involves a set of nonlinear partial differential equations with nonlinear complementarity constraints. A modern
and efficient solution strategy, the Newton-min method, is applied to this geoscientific
problem. We investigate its applicability and efficiency from both theoretical and numerical point of view for solving linear and nonlinear complementarity problems. Numerical and theorical results will be presented to show the ability of our solver.

10 Luglio, 2012 14:30 in punto

Sezione di Probabilità Quantistica

BOUNDARY OPTIMAL CONTROL PROBLEMS FOR THE INCOMPRESSIBLE MHD EQUATIONS: A THEORETICAL AND COMPUTATIONAL PERSPECTIVE

Giorgio Bornia, Universita di Bologna

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

abstract

abstract

Optimal control problems for the incompressible Magnetohydrodynamics (MHD) equations are of great interest due to a wide range of possible applications involving electrically conductive fluids,such as aluminum casting in metallurgy
and crystal growth in semiconductor industry.
In such applications it may be a great improvement
to achieve the control on the fluid state variables through the action of the magnetic Lorentz force. In this talk we consider a class of MHD boundary optimal control problems,in which the flow is controlled through the boundary conditions of the magnetic field.

We propose a new approach for the formulation of these problems based on lifting functions of inhomogeneous boundary conditions.
This approach yields both theoretical and numerical advantages which will be discussed.

The optimal control problems under consideration are formulated in terms of the minimization
of a cost functional with the nonlinear constraints of the Navier-Stokes and MHD equations. The Lagrange multiplier principle is used to derive an optimality system, whose numerical solution is a challenging task.
We describe the algorithms and computational methodologies and we illustrate some numerical results in order to show that optimal solutions can be computed in an effective manner.

4 Luglio, 2012 14:30 in punto

Sezione di Probabilità Quantistica

Assessment of Mortgage Default Risk via Bayesian State Space Models

Refik Soyer, Department of Decision Sciences, Geroge Washington University, Washington DC

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

abstract

abstract

Managing risk at the aggregate level is crucial for banks and financial
institutions as required by the Basel II framework. In modeling mortgage
default risk at the aggregate level, issues such identifying the dynamic
behavior of default rate as well as the effect of macroeconomic variables
on default risk are of concern to both practitioners and researchers. In
addressing these issues, we propose discrete time Bayesian state space
models with Poisson observations and a stochastic default rate. In doing so
we discuss parameter updating and estimation using Markov chain Monte Carlo
methods. In assessing the dynamic nature of the mortgage default rate, we
compare the forecasting performance of the proposed models with a Bayesian
Poisson regression model used as a benchmark. We illustrate the use of the
proposed models using actual U.S. residential mortgage data and discuss
insights gained from Bayesian analysis. Furthermore, we discuss
multivariate extensions of the proposed model.

3 Luglio, 2012 16:30

Lezioni Leonardesche

DOES IT PAY TO BE GREEDY ?

RONALD DEVORE, Texas A & M University

Università di Milano, Dipartimento di Matematica, Via Saldini

Managing risk at the aggregate level is crucial for banks and financial
institutions as required by the Basel II framework. In modeling mortgage
default risk at the aggregate level, issues such identifying the dynamic
behavior of default rate as well as the effect of macroeconomic variables
on default risk are of concern to both practitioners and researchers. In
addressing these issues, we propose discrete time Bayesian state space
models with Poisson observations and a stochastic default rate. In doing so
we discuss parameter updating and estimation using Markov chain Monte Carlo
methods. In assessing the dynamic nature of the mortgage default rate, we
compare the forecasting performance of the proposed models with a Bayesian
Poisson regression model used as a benchmark. We illustrate the use of the
proposed models using actual U.S. residential mortgage data and discuss
insights gained from Bayesian analysis. Furthermore, we discuss
multivariate extensions of the proposed model.

2 Luglio, 2012 14:30 in punto

Lezioni Leonardesche

Trust-Region Based Nonlinear Solver for Multiphase Flow in Heterogeneous Porous Media

Hamdi Tchelepi, School of Earth Sciences, Stanford University

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

abstract

abstract

We describe a nonlinear solver for immiscible two-phase transport in heterogeneous porous media, where viscous, buoyancy, and capillary
forces are all present and significant. The flux (phase velocity) is a nonlinear function of saturation and typically has an inflection
point. With strong buoyancy effects, the flux can be non-monotonic with additional inflection points. The non-convexity and non-monotonicity of the flux function are major sources of difficulty
for nonlinear solvers of transport in natural porous media. We present a modified-Newton algorithm for the numerical simulation of two-phase flow that employs trust-regions of the flux function to guide the Newton iterations. The flux function is divided into saturation trust
regions. The delineation of these regions is dictated by the inflection, unit-flux, and end points. The updates are performed such that two successive iterations cannot cross any trust-region boundary. If a crossing is detected, we chop the saturation value back the
appropriate trust-region boundary.

We analyze the nonlinear transport equation with finite-volume discretization and phase-based upstream weighting. Then, we prove
convergence of the trust-region based Newton method for arbitrary timestep sizes for single-cell problems. Numerical results across the
full range of the parameter space of viscous, gravity and capillary forces indicate that our trust-region based scheme is unconditionally
convergent for 1D transport. That is, for a given choice of timestep size, the unique saturation solution is found independently of the initial guess. For problems dominated by buoyancy and capillarity, the trust-region based Newton solver overcomes the often severe limits on
timestep size associated with existing methods. To validate the effectiveness of the new nonlinear solver for large-scale,heterogeneous reservoir models, we compare it with the current
state-of-the-art nonlinear solvers for two-phase flow and transport using the SPE 10 model problem (highly heterogeneous 3D model with more than 1 million cells). Compared with the existing nonlinear solvers, our trust-region based nonlinear solver results in superior
convergence performance and achieves significant reduction in the total Newton iterations by more than an order of magnitude along with a corresponding reduction in the overall computational cost.

28 Giugno, 2012 17:00

Seminario Matematico e Fisico di Milano

HIGHER ORDER ELLIPTIC PROBLEMS IN NON-SMOOTH DOMAINS

VLADIMIR MAZ YA, University of Liverpool, Department of Mathematical Sciences

Politecnico di Milano, Dipartimento di Matematica, Aula VII piano

abstract

abstract

We discuss sharp continuity and regularity results for solutions of the polyharmonic equation in an arbitrary open set. The absence of information about geometry of the domain puts the question of regularity properties beyond the scope of applicability of the methods devised previously, which typically rely on specific geometric assumptions.
Positive results have been available only when the domain is sufficiently smooth, Lipschitz or diffeomorphic to a polyhedron.

The techniques developed recently allow to establish the boundedness of derivatives of solutions to the Dirichlet problem for the polyharmonic equation under no restrictions on the underlying domain and to show that the order of the derivatives is maximal. An appropriate
notion of polyharmonic capacity is introduced which allows one to describe the precise correlation between the smoothness of solutions and the geometry of the domain.

22 Giugno, 2012 14:00

Seminari PhD di Equazioni Differenziali

Condizioni di Keller-Osserman per disequazioni differenziali con termini di gradiente.

Marco Magliaro, Università di Bergamo

aula seminari VI piano

abstract

abstract

Condizioni di Keller-Osserman per disequazioni differenziali con termini di gradiente.
Il seminario riguarda risultati di tipo Liouville per disequazioni differenziali della forma $ Delta_ varphi u ge f(u)l(| nabla u|)$ nello
spazio Euclideo, nel gruppo di Heisenberg e, più in generale, nei gruppi di Carnot. L operatore $ Delta_ varphi$, detto $ varphi$-laplaciano, è un
operatore differenziale non lineare che generalizza in modod naturale il
$p$-laplaciano. Verrà dimostrato un risultato di non esistenza di soluzioni intere non negative sotto un opportuna condizione di non integrabilità all infinito riguardante le funzioni $ varphi$, $f$ ed $l$, che estende la classica condizione di Keller-Osserman introdotta per la disequazione $ delta u ge f(u)$.
Verrà anche mostrato che i risultati ottenuti sono sharp.

21 Giugno, 2012 14:30 in punto

Sezione di Geometria, Algebra e loro applicazioni

Operatori di Toeplitz e misure di Carleson in domini fortemente pseudoconvessi

Jasmin Raissy, Universita di Milano Bicocca

Aula Seminari 3 piano

abstract

abstract

Descriverò le proprietà degli operatori di Toeplitz associati ad una misura di Borel positiva finita definita su un dominio limitato fortemente pseudoconvesso $D subset subset C^n$. In particolare, otterremo condizioni ottimali sulla misura che assicurano che l operatore di Toeplitz associato mandi lo spazio di Bergman $A^p(D)$ in $A^r(D)$ con $r>p$, generalizzando e rendendo più precisi i risultati di Cuckovic e McNeal. Per ottenere tali condizioni, darò una caratterizzazione geometrica delle misure di Carleson e delle misure di Carleson vanishing su spazi di Bergman pesati in termini della geometria di Kobayashi intrinseca del dominio, generalizzando in questo ambito i risultati ottenuti da Kaptanoglu per la palla unitaria. (Lavoro in collaborazione con M. Abate e A. Saracco).

15 Giugno, 2012 11:00

Sezione di Fisica Matematica

Geometry, topology, and out-of-equilibrium dynamics in epithelial morphogenesis.

Andrea Gamba, Politecnico di Torino

Aula Seminari MOX, VI piano

abstract

abstract

The self-organization of epithelial tissues is a crucial event in the development of multicellular organisms. The process can be studied experimentally by embedding single epithelial cells in protein gels and observing their spontaneous growth. The cells divide and self-assemble in highly symmetric multicellular structures (cysts) surrounding a hollow spherical space (lumen), thus mimicking the formation of cysts and acini in a living organism during the development of organs such as the kidneys or
lungs. The orderly growth of experimental cysts is driven by cell-cell adhesive forces, cell membrane elasticity, and the self-organized
partitioning of the cell membrane in domains characterized by different chemical and mechanical properties (cell polarity). Genetic engineering and pharmacological inhibitors allow to interfere with these processes, leading
to the production of aberrant, disordered cell aggregates similar to those observed in several types of cancers. We have studied the process of
experimental cyst growth by theoretical modeling and accurate dynamic quantitation of cyst morphology during the growth process. Our analysis
shows that cell-cell contact statistics bears a clear signature of the strongly non-equilibrium character of cyst morphogenesis.
Out-of-equilibrium dynamics drives the appearance of aberrant multi-lumen phenotypes, unless strict control of cell division geometry is enforced.
The comparison of theoretical modeling and experimental results unveils the presence of multiple local equilibria associated with geometrically frustrated configurations of the cell aggregate. We show that a quasi-equilibrium state can be recovered upon inducing tissue fluidization by the use of pharmacological inhibitors of cell membrane tension.