7 Giugno, 2004 11:30

Multidomain simulation of flow in porous media

Vincent Martin, MOX-Dip. Mate. Polimi

Aula Seminari MOX-6° piano. Dip. di Matematica

abstract

abstract

I will present during this seminar the main results of my PhD thesis
performed at Inria under the direction of Jean Roberts.

It is mainly concerned with the multidomain simulations
of flow in porous media. Three different themes are considered.
First, I present a domain decomposition method with non-matching meshes
using Robin type interface conditions, for the mixed finite
elements. Second, this method was implemented in parallel
using the parallel system OcamlP3l, written in Ocaml by
computer scientists. In OcamlP3l, the user develops and
debugs sequentially, and obtains the parallel code
with a mere recompilation. A realistic 3D simulation is
given to validate the procedure.
Finally, we present a new model for flow in a porous medium containing
large fractures that may have very large and/or very small permeabilities.
In this model, the fractures are treated as interfaces between subdomains.
Existence and uniqueness of the solution is proved and an error estimate is
obtained. Some numerical experiments confirm the theorical results.

4 Giugno, 2004 12:30

Seminari di Cultura Matematica

Chiacchierata sull'analisi non standard

S. Salsa, Dip. di Matematica

aula T2.2

I will present during this seminar the main results of my PhD thesis
performed at Inria under the direction of Jean Roberts.

It is mainly concerned with the multidomain simulations
of flow in porous media. Three different themes are considered.
First, I present a domain decomposition method with non-matching meshes
using Robin type interface conditions, for the mixed finite
elements. Second, this method was implemented in parallel
using the parallel system OcamlP3l, written in Ocaml by
computer scientists. In OcamlP3l, the user develops and
debugs sequentially, and obtains the parallel code
with a mere recompilation. A realistic 3D simulation is
given to validate the procedure.
Finally, we present a new model for flow in a porous medium containing
large fractures that may have very large and/or very small permeabilities.
In this model, the fractures are treated as interfaces between subdomains.
Existence and uniqueness of the solution is proved and an error estimate is
obtained. Some numerical experiments confirm the theorical results.

28 Maggio, 2004 12:30

Seminari di Cultura Matematica

Sistemi periodici di controllo e previsione

S. Bittanti, Dip. Elettronica e Informazione

Aula B.21

I will present during this seminar the main results of my PhD thesis
performed at Inria under the direction of Jean Roberts.

It is mainly concerned with the multidomain simulations
of flow in porous media. Three different themes are considered.
First, I present a domain decomposition method with non-matching meshes
using Robin type interface conditions, for the mixed finite
elements. Second, this method was implemented in parallel
using the parallel system OcamlP3l, written in Ocaml by
computer scientists. In OcamlP3l, the user develops and
debugs sequentially, and obtains the parallel code
with a mere recompilation. A realistic 3D simulation is
given to validate the procedure.
Finally, we present a new model for flow in a porous medium containing
large fractures that may have very large and/or very small permeabilities.
In this model, the fractures are treated as interfaces between subdomains.
Existence and uniqueness of the solution is proved and an error estimate is
obtained. Some numerical experiments confirm the theorical results.

27 Maggio, 2004 16:30

Sezione di Probabilità e Statistica Matematica

Modelli d'urna con rinforzo aleatorio

Dr. Anna Maria Paganoni, Politecnico di Milano

Aula Seminari, VI piano

abstract

abstract

Si affronta lo studio di modelli d'urna con rinforzo aleatorio, ed in particolare si concentra l'attenzione su un'urna a due colori che ad ogni estrazione viene rinforzata con un numero aleatorio di palline dello stesso
colore della pallina estratta: la legge del numero di palline che vengono reinserite nell'urna dipende dal colore estratto. Si espongono risultati asintotici sul processo dei colori generati dall'urna e sul processo delle
proporzioni di palline in essa contenuta. Questi schemi d'urna trovano applicazione nella formulazione di disegni clinici sequenziali e
connessioni con la modellizzazione di disegni adattivi degli esperimenti in ambito Bayesiano.

26 Maggio, 2004 17:00

Seminario Matematico e Fisico di Milano

Three body problems in quantum mechanics

Wu-Yi Hsiang, Hong Kong University of Science and Technology (Hong Kong, Cina)

Dipartimento di Matematica - Università degli Studi di Milano - Via Saldini 50 - Milano - Sala di Rappresentanza

abstract

abstract

In this talk, I shall describe the geometric approach to solve the Schrödinger equation for various physically meaningful three body systems such as He, H2+, H-, three bosons in R2 with d-function potential etc. The configuration space of the three body system in R3(resp. R2) (with center of gravity fixed at the origin) is an R6 (resp. R4) equipped with an SO(3) (resp. SO(2)) symmetric kinematic metric, while the potential function U is also SO(3) (resp. SO(2)) invariant. The first step is to fully utilize the SO(3) (resp. SO(2)) symmetry to reduce the Schrödinger equation to an equation solely defined at the level of the orbit space (i.e. R6/SO(3) (resp. R4/SO(2))) equipped with the orbital distance metric. One needs to make effective use of both group representation theory and equivariant differential geometry to achieve such a reduction. The orbit space of a three body system in R3 (resp. R2) equipped with the orbital distance metric is always isometric to the Riemannian cone over S2+ (1/2) (resp. S2(1/2))), namely the Euclidean hemisphere (resp. sphere) of radius 1/2. This remarkable fact (i.e. sphericality) enables us to bring in the spherical harmonics and their generalizations (namely, Jacobi polynomials and monopole harmonics) to greatly simplify the analysis of the angular part of the reduced equation. I will use the simpler case of the boson system to illustrate this step which enables us to further reduce the Schrödinger equation to an ODE solely in the radial direction. Such an ODE can be thoroughly analyzed and I will discuss the physical significance of these solutions so obtained for the three boson system. Bibliography Wu-Yi Hsiang. Kinematic geometry of mass-triangles and reduction of Schr¨odinger’s equation of three-body systems to partial differential equations solely defined on triangular parameters. Proc. Nat. Acad. Sci. U.S.A., 94(17):8936–8938, 1997. Wu-Yi Hsiang. On the kinematic geometry of many body systems. Chinese Ann. Math. Ser. B, 20(1):11–28, 1999. A Chinese summary appears in Chinese Ann. Math. Ser. A 20 (1999), no. 1, 141.

26 Maggio, 2004 

Sezione di Calcolo delle Variazioni ed Equazioni Differenziali

Existence Results for a New Variational Problem in One Dimensional Segmentation Theory

Tommaso Boccellari, Politecnico di Milano

In this talk, I shall describe the geometric approach to solve the Schrödinger equation for various physically meaningful three body systems such as He, H2+, H-, three bosons in R2 with d-function potential etc. The configuration space of the three body system in R3(resp. R2) (with center of gravity fixed at the origin) is an R6 (resp. R4) equipped with an SO(3) (resp. SO(2)) symmetric kinematic metric, while the potential function U is also SO(3) (resp. SO(2)) invariant. The first step is to fully utilize the SO(3) (resp. SO(2)) symmetry to reduce the Schrödinger equation to an equation solely defined at the level of the orbit space (i.e. R6/SO(3) (resp. R4/SO(2))) equipped with the orbital distance metric. One needs to make effective use of both group representation theory and equivariant differential geometry to achieve such a reduction. The orbit space of a three body system in R3 (resp. R2) equipped with the orbital distance metric is always isometric to the Riemannian cone over S2+ (1/2) (resp. S2(1/2))), namely the Euclidean hemisphere (resp. sphere) of radius 1/2. This remarkable fact (i.e. sphericality) enables us to bring in the spherical harmonics and their generalizations (namely, Jacobi polynomials and monopole harmonics) to greatly simplify the analysis of the angular part of the reduced equation. I will use the simpler case of the boson system to illustrate this step which enables us to further reduce the Schrödinger equation to an ODE solely in the radial direction. Such an ODE can be thoroughly analyzed and I will discuss the physical significance of these solutions so obtained for the three boson system. Bibliography Wu-Yi Hsiang. Kinematic geometry of mass-triangles and reduction of Schr¨odinger’s equation of three-body systems to partial differential equations solely defined on triangular parameters. Proc. Nat. Acad. Sci. U.S.A., 94(17):8936–8938, 1997. Wu-Yi Hsiang. On the kinematic geometry of many body systems. Chinese Ann. Math. Ser. B, 20(1):11–28, 1999. A Chinese summary appears in Chinese Ann. Math. Ser. A 20 (1999), no. 1, 141.

21 Maggio, 2004 17:00

Seminario Matematico e Fisico di Milano

Nonextensive statistical mechanics - Introduction and dynamical foundations

Constantino Tsallis, Centro Brasileiro de Pesquisas Físicas (Rio de Janeiro, Brasile)

Dipartimento di Matematica - Università degli Studi di Milano - Via Saldini 50 - Milano - Sala di Rappresentanza

abstract

abstract

"Nonlinear dynamical systems that satisfy hypothesis such as ergodicity and exponentially quick mixing are well known to be adequately studied in terms of the Boltzmann-Gibbs entropy and its corresponding statistical mechanics. These simplifying hypothesis are however NOT satisfied in vast classes of systems such as the so called ""complex systems"", ubiquitously emerging in physics, mathematics, economics, linguistics, chemistry, astrophysics, geophysics, biology, computer networks, engineering and elsewhere. A nonextensive entropy (characterized by an entropic index q, which reproduces the Boltzmann-Gibbs expression for q = 1) and its corresponding statistical mechanics provide an answer for at least part of such anomalous systems. A brief introduction will be given to the subject, followed by a survey on its dynamical foundations, which enable in particular the calculation, from first principles, of the index q associated with specific systems. Recent bibliography: ""Nonextensive Entropy - Interdisciplinary Applications"", M. Gell-Mann and C. Tsallis, eds. (Oxford University Press, New York, 2004) Full bibliography"

21 Maggio, 2004 12:30

Seminari di Cultura Matematica

Sistemi dinamici ed i fondamenti della termodinamica

L. Galgani, Univ. di Milano

Aula B.21

"Nonlinear dynamical systems that satisfy hypothesis such as ergodicity and exponentially quick mixing are well known to be adequately studied in terms of the Boltzmann-Gibbs entropy and its corresponding statistical mechanics. These simplifying hypothesis are however NOT satisfied in vast classes of systems such as the so called ""complex systems"", ubiquitously emerging in physics, mathematics, economics, linguistics, chemistry, astrophysics, geophysics, biology, computer networks, engineering and elsewhere. A nonextensive entropy (characterized by an entropic index q, which reproduces the Boltzmann-Gibbs expression for q = 1) and its corresponding statistical mechanics provide an answer for at least part of such anomalous systems. A brief introduction will be given to the subject, followed by a survey on its dynamical foundations, which enable in particular the calculation, from first principles, of the index q associated with specific systems. Recent bibliography: ""Nonextensive Entropy - Interdisciplinary Applications"", M. Gell-Mann and C. Tsallis, eds. (Oxford University Press, New York, 2004) Full bibliography"

19 Maggio, 2004 11:30

Seminario Matematico e Fisico di Milano

Onde solitarie e campi elettromagnetici

Donato Fortunato, Università di Bari

Dipartimento di Matematica e Applicazioni - Università degli Studi di Milano Bicocca - Via Bicocca degli Arcimboldi, 8 - Aula Dottorato

"Nonlinear dynamical systems that satisfy hypothesis such as ergodicity and exponentially quick mixing are well known to be adequately studied in terms of the Boltzmann-Gibbs entropy and its corresponding statistical mechanics. These simplifying hypothesis are however NOT satisfied in vast classes of systems such as the so called ""complex systems"", ubiquitously emerging in physics, mathematics, economics, linguistics, chemistry, astrophysics, geophysics, biology, computer networks, engineering and elsewhere. A nonextensive entropy (characterized by an entropic index q, which reproduces the Boltzmann-Gibbs expression for q = 1) and its corresponding statistical mechanics provide an answer for at least part of such anomalous systems. A brief introduction will be given to the subject, followed by a survey on its dynamical foundations, which enable in particular the calculation, from first principles, of the index q associated with specific systems. Recent bibliography: ""Nonextensive Entropy - Interdisciplinary Applications"", M. Gell-Mann and C. Tsallis, eds. (Oxford University Press, New York, 2004) Full bibliography"

17 Maggio, 2004 14:30

Seminario Matematico e Fisico di Milano

Approximation of multi-scale elliptic problems using patches of finite elements

Joel Wagner, iacs-epfl

Aula Seminari MOX-6° piano dip di matematica

abstract

abstract

The objective of this seminar is to present a new method to solve numerically
elliptic problems such that a better precision on the solution is needed
in certain regions of the domain wherein the equations
are to be solved (C.R.Acad.Sci.Paris, Ser.I 337 (2003) 679--684).
The approximation of this type of problems with multi-scale data can be approached using
various methods. The technique we present uses multiple levels of not necessarily nested
grids. It is a Schwarz type domain decomposition method with complete overlapping.
The proposed algorithm consists in solving the problem on a domain wherein we consider
patches of elements in the regions where we would like to obtain more accuracy.
Thus we calculate successively corrections to the solution in the patches.
The discretizations of the latter are not necessarily conforming.
The method resembles the Fast Adaptive Composite grid method or possibly
a hierarchical method with a mortar method. However it is of much more flexible use
in comparison to the latter.

The motivation for developing such a method is for example founded in air quality management.
Pollution emission sources, and in particular point source plumes, give
rise to models needing careful examination of the space-scale. Getting an accurate
simulation on large scales is linked to a simulation in subregions around the
pollution sources using finer grids. Such a method can straightforwardly be applied
on boundary layer problems through the use of patches in critical regions.

In this talk we present the algorithm and illustrate its efficiency through a model problem.
We compare the speed of convergence on nested and non-nested, structured and unstructured grids.
A spectral analysis of the iteration operator enables us to give a good estimate of the convergence
rate for given grids. It also leads to a numerical method to evaluate the
constant of the Cauchy-Buniakowski-Schwarz inequality in certain cases of approximation spaces.
Finally we illustrate on several examples our \emph{a priori} estimate for
the convergence in the grid-size.