4 Marzo, 2010 11:00 in punto

Gamma-type operators and the Black-Scholes semigroup

Antonio Attalienti, Rosa Maria Mininni, Universita di Bari

Aula Seminari VI piano

3 Marzo, 2010 11:00 in punto

Mathematical modelling of geothermal reservoirs

Alessandro Speranza - Maurizio Ceseri, Innovazione Industriale Tramite Trasferimento Tecnologico Onlus - Dip. di Matematica Uni Firenze

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

abstract

abstract

High enthalpy geothermal reservoirs consists essentially of three components, namely, a strong underground heat source, the presence of a
water reservoir, and a highly permeable porous medium or fractured rocks. As a physical system, such a complex liquid-solid interaction
can be modelled, once some basic assumptions on the physics, such as thermodynamic equilibrium, are made.
We show the process leading to the mathematical modelling of a geothermal reservoir, focussing in particular on the mass, momentum and energy
balance equations, on the coupling between balance equations and thermodynamics. Some further insights on the main aspects of the
thermodynamics of the geothermal fluids will also be given.
Finally, we show how boundary conditions turn out to be crucial on such systems, when applied to simplified 1 and 2D models.

2 Marzo, 2010 15:00 in punto

Numerical approximation of the flux identification problem for a scalar conservation law

Carlos Castro, Universidad Politecnica de Madrid, Madrid, Spain

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

abstract

abstract

We consider the problem of flux identification for 1-d scalar conservation laws stated as an optimal control problem.

We first prove the existence of minimizers and the convergence of discrete minima obtained by means of numerical approximation schemes. Then we address the problem of developing efficient
descent algorithms. We compare the existing two
possible approaches. The first one, the so-called discrete approach, based on a direct computation of gradients in the discrete problem and the so-called continuous one, where the discrete descent direction is obtained as a discrete copy of the continuous one. We show that, in this case, the latter is more efficient than the
discrete one in presence of discontinuities.

1 Marzo, 2010 17:00

Seminario Matematico e Fisico di Milano

THE MATHEMATICS OF DISTORTION

DIETMAR HOEMBERG, Weierstrass Institute for Applied Analysis and Stochastics

Dipartimento di Matematica, Università di Milano, Via Saldini, Sala di Rappresentanza

abstract

abstract

In mechanical engineering distortion means undesired alterations of size and/or shape of a workpiece. A typical cause for distortion are solid-solid phase transitions as they occur during the heat treatment of steel.
In my talk I will present a thermomechanical model of phase transitions in steel, show some applications in the heat treatment of steel and discuss possible optimal control approaches to compensate distortion.

26 Febbraio, 2010 10:00 in punto

Seminario Matematico e Fisico di Milano

Closure relations of the one dimensional two fluid model for the simulation of slug flows

Marco Montini, Department of Mechanical Engineering Imperial College London

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

abstract

abstract

This research is a development of an existing numerical methodology (Issa and Kempf, 2003,
based on a one-dimensional two-fluid model.
This model is capable of simulating two-phase stratified and slug flow for horizontal and nearly horizontal pipes, using a novel "slug capturing" technique. However, the governing
system of equations is conditionally well-posed only.The main objective of the research is to extend the applicability of the model to a wider range of cases.
This can be achieved through the introduction of improved closure models, which aim to better describe the physics of the flow. The approach to
the problem starts from a mathematical evaluation of the additional closure relations in order to study their contribution to the well-posedness of the system.
On one hand a characteristics analysis stablishes whether the characteristics of the problem are real, hence the system is hyperbolic and well-posed. On the other hand a stability analysis provides information on the growth rate of the instabilities: if this growth rate is
bounded for short wavelength instabilities the system can be considered well-posed. Among the
closure relations studied and tested are those relating to the effects of axial diffusion,
surface tension, virtual mass (due to relative acceleration) and shape of velocity profiles.
Suitable closure relationships are then implemented and tested in the numerical code and the results are validated against available experimental data.
For a successful improved model, numerical results must be both in good agreement with
experiments and grid independent, which is a clear manifestation of the well-posedness of the system. These factors were used to evaluate the performance of the new models.

25 Febbraio, 2010 14:30 in punto

Seminario Matematico e Fisico di Milano

On the weak order of the Euler Schemes for stochastic Partial Differential Equations

Arnaud Debussche, ENS Cachan, Antenne de Bretagne, Rennes

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

abstract

abstract

In this talk, we present recent results on the order of convergence of the Euler scheme for a Stochastic Partial Differential Equation. The strong order of convergence has been studied by many authors. However, very few results are available for the weak order of convergence.

It is well known that the Euler scheme is of strong order $1/2$ and weak order $1$ in the case of a stochastic differential equation. Two
methods are available to prove this result. The first one uses the Kolmogorov equation associated to the stochastic equation and was first used by D. Talay. A second one has been recently discovered by A. Kohatsu-Higa and is based on Malliavin calculus.

In this talk, we generalize such results to the infinite dimensional case. We show how to adapt Talay s method. The main difficulty is due
to the presence of unbounded operators in the Kolmogorov equation. A tricky change of unknown allows to treat the case of a linear equation. It also works for an equation whose linear part defines a group, the nonlinear Schr odinger equation for instance. The case of a semilinear equation of parabolic type is more difficult and we use Malliavin calculus, but not in the same way as in Kohatsu-Higa s method. We prove for instance that, in the case of a nonlinear heat
equation in dimension one with a space time white noise, the Euler scheme has weak order $1/2$, it is well known that the strong order is $1/4$.

25 Febbraio, 2010 11:30

Seminario Matematico e Fisico di Milano

NON-COOPERATIVE DIFFERENTIAL GAMES

ALBERTO BRESSAN, Pennsylvania State University

Dipartimento di Matematica, Politecnico di Milano, Sala Consiglio VII piano

abstract

abstract

In a non-cooperative differential game, two or more players can influence the time evolution of a system. Each player chooses a feedback control $u=u(x)$ in order to minimize his own cost functional.

If a Nash equilibrium solution exists, under suitable regularity conditions the value functions satisfy a (highly nonlinear) system of Hamilton-Jacobi equations.

After a general overview, the talk will present a
new analytical approach to the study of these PDEs, based on a homotopy method.

Examples show that this approach is also naturally motivated by some economic models.

23 Febbraio, 2010 14:30 in punto

Sezione di Fisica Matematica

Particelle inerziali in flussi turbolenti: questioni aperte da un punto di vista fisico e numerico.

Alessandra Lanotte, CNR ISAC, Roma

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

abstract

abstract

Saranno discussi alcuni sviluppi recenti riguardo le proprietà
dinamiche e statistiche di particelle inerziali disperse in flussi
turbolenti. Tali particelle sono tipicamente piccole, ma di taglia
finita, e differiscono in densità dal fluido che le trasporta. Esempi
di particelle inerziali pesanti - con densità molto maggiore del
fluido-, sono le goccioline in nubi liquide, o polveri e sostanze
chimiche disperse in atmosfera. Lo studio delle particelle pesanti è
rilevante anche per molte applicazioni industriali. Saranno presentati
risultati ottenuti tramite simulazioni numeriche dirette di turbolenza
statisticamente omogenea ed isotropa, con fase dispersa costituita da
particelle pesanti di diversa inerzia. Infine, saranno discussi i limiti
di una descrizione in termine di particelle puntiformi, e le questioni
aperte per la descrizione fisica di particelle inerziali di taglia
finita, non trascurabile.


ABSTRACT IN INGLESE:

Some recent advancements about the dynamical and statistical properties
of inertial particles in turbulent flows will be reviewed. These are
small but finite size particles, with a density contrast with respect of
the advecting flow. Examples of heavy inertial particles - with density
much higher of the fluid-, are droplets in liquid water clouds, or dust
and chemicals dispersed in the atmosphere. Heavy particles are relevant
for industrial applications also. I will present results obtained by
direct numerical simulations of homogeneous and isotropic turbulence,
seeded with heavy particles of different inertia. The limit of the
description in terms of point-particles will finally be mentioned, as
starting point for tackling the physics of inertial particles with a
finite, large size.

15 Febbraio, 2010 15:00 in punto

Sezione di Fisica Matematica

Some recent results on the convergence of Semi-Lagrangian schemes

Roberto Ferretti, Dip. di Matematica, Università di Roma III

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

abstract

abstract

We review some recent constributions to the theory of convergence for SL schemes. A first result concerns the equivalence between SL and exact Lagrange-Galerkin schemes, which allows to recover $L2$ stability of SL schemes in a variety of cases, under both small and large Courant numbers. A second result is related to an adaptation of the classical Barles-Souganidis theorem for the approximation of viscosity solutions. In this situation we show that, under an inverse Courant condition, SL schemes behave like "asymptotically monotone" schemes, and this implies convergence to continuous viscosolutions.

15 Febbraio, 2010 15:00

Seminario Matematico e Fisico di Milano

50 FAKE PLANES

TIM STEGER, Università di Sassari

Aula seminari 3014, Dipartimento di Matematica, Università di Milano Bicocca

abstract

abstract

Un piano proiettivo finto è una varietà complessa, compatta, liscia, e senza frontiera, di dimensione complessa ~2, la quale ha gli stessi numeri di Betti del piano proiettivo complesso.

Il primo piano finto è stato costruito da [Mumford, 1979]. Cartwright e Steger sono riusciti a portare a termine il lavoro
cominciato in [Prasad, Yeung, 2007], e hanno elencato tutti i piani finti. A meno di equivalenza olomorfa e antiolomorfa, ce ne sono ~50. Il lavoro dipende fondamentalmente da diversi calcoli numerici e simbolici a proposito di certi sottogruppi di matrici di $U(2,1)$, calcoli eseguiti con l aiuto degli elaboratori elettronici.