16 Maggio, 2007 12:30

Seminari di Cultura Matematica

Raccontare l’invisibile. Interazioni fra letteratura e matematica nel ‘900

C. Bartocci, Univ. di Genova

Aula B.21

15 Maggio, 2007 11:30

Seminari di Cultura Matematica

New developments of the devising of DG methods for incompressible fluid flow

Prof. Bernardo Cockburn, University of Minnesota

Aula Seminari MOX, 6° piano Dipartimento

abstract

abstract

We discuss two new techniques for devising discontinuous Galerkin (DG) methods for incompressible fluid flow.
Their introduction is motivated by the fact that DG methods for the incompressible Navier-Stokes equations are locally conservative and energy-stable only if the approximate velocity is exactly divergence free. How to compute exactly divergence free velocities by using low order polynomial spaces or in three-space dimension has remained an open problem from a long time.
The first technique allows us to work with completely discontinuous velocities; it is based on the use of a simple, local postprocessing to obtain a divergence-free velocity. The second technique allows us to work with exactly divergence-free velocities without having to actually construct the corresponding spaces.

14 Maggio, 2007 14:30

Seminari di Cultura Matematica

On variational data assimilation methods for 1D and 2D fluvial hydraulics

Prof. Jerome Monnier, LMC-IMAG, Grenoble

Aula Seminari VI piano Dipartimento

abstract

abstract

We study variational data assimilation methods (optimal control of
PDEs) as applied to river flows (shallow water 1D and 2D). In river
hydraulics, available observations are generally very few
measurements of elevation (quasi-continuous in time), while velocity
or discharge measurements are often not available. This lack of
observations are even more true during flood events.

In a first step, we show that standard variationnal data
assimilation methods (standard eulerian observations are available)
can be applied successfully to river hydraulics (real data, Pearl
river, China): the inflow elevation and/or discharge are retreived
when using water measurements at three gauge stations.

Then, we study the assimilation of lagrangian data. Since in practice
the eulerian observations can be not sufficient to take full
advantage of data assimilation for some identification purposes, we
use extra lagrangian data extracted from remote sensing observations
(video images). The trajectory of particles advected by the flow can
bring some information on the surface velocity using transport
equations. Numerical twin experiments demonstrate that this method
makes it possible to improve the identification of model parameters
(local topography and/or inflow discharge).

Finally, we elaborate a joint data assimilation – coupling method
based on the optimal control process. The basic idea is to superpose
locally a 2D flow model ("a local zoom")

over the 1D global flow model (river network). The coupling is done
weakly since it is based on the optimal control of a 2D-1D relaxed
model.
Then, we take advantage of the optimal control based procedure to
define the joint data assimilation – coupling algorithm.
We show the efficiency of the method by considering a toy flooding
event.
Keywords. Variationnal data assimilation, river hydraulics, parameter
identification, lagrangian data, 1D and 2D shallow-water models,
joint data assimilation – coupling procedure.

10 Maggio, 2007 14:00

Sezione di Probabilità e Statistica Matematica

Gli Algoritmi EM Stocastici e le Verosimiglianze Simulate nei Modelli Geostatistici Fattoriali

Marco Minozzo, Università degli Studi di Verona

aula seminari III piano

abstract

abstract

In ambito geostatistico l'analisi di dati paziali multivariati viene tradizionalmente condotta facendo ricorso al modello lineare di coregionalizzazione (Wackernagel, 1995; Chiles e Delfiner, 1999) e alla "factorial kriging analysis" (Matheron, 1982).
Sebbene queste analisi, e le previsioni che ne conseguono basate sul cokriging fattoriale, tralascino ogni considerazione sulla legge
distributiva dei dati e si basino solamente sulla dipendenza spaziale misurata da variogrammi e cross-variogrammi, esse sono perfettamente
adeguate, nel senso di fornire previsioni non distorte con errore quadratico medio minimo, nel caso in cui i dati possono assumersi
avere distribuzione (multivariata) gaussiana.
D'altro canto, nel caso di dati di conteggio, di dati asimmetrici, o comunque di dati non gaussiani, come in molte applicazioni
ambientali, queste analisi possono dar luogo a previsioni dei fattori latenti altamente inefficienti.
Sull'orma delle piu' interessanti proposte apparse in letteratura per la modellazione di dati spaziali univariati non gaussiani (Diggle,
Moyeed e Tawn, 1998), qui si propone, per dati spaziali multivariati, un modello fattoriale gerarchico ("generalized linear mixed
models") costruito su di una variante del classico modello lineare di coregionalizzazione.
Per la stima dei parametri di questo modello si possono adattare alcune delle tradizionali tecniche geostatistiche come, ad esempio,
l'algoritmo di Goulard e Voltz (1992), mentre mappe dei fattori spaziali latenti (non osservabili) possono essere ottenute
utilizzando algoritmi "Markov chain Monte Carlo". Metodi di stima dei parametri piu' efficienti basati sulla verosimiglianza si
possono ottenere utilizzando gli algoritmi EM stocastici (Fort e Moulines, 2003) e le verosimiglianze simulate (Geyer, 1994;
Christensen, 2004).

Bibliografia
Chiles, J.P. and Delfiner, P. (1999).
Geostatistics: Modeling Spatial Uncertainty.
Wiley, New York.

Christensen, O.F. (2004).
Monte Carlo maximum likelihood in model-based geostatistics.
Journal of Computational and Graphical Statistics, 13, 702-718.

Diggle, P.J., Moyeed, R.A. and Tawn, J.A. (1998).
Model-based geostatistics.
Applied Statistics, 47, 299-350.

Fort, G. and Moulines, E. (2003).
Convergence of the Monte Carlo expectation maximization for curved
exponential families.
The Annals of Statistics, 31, 1220-1259.

Geyer, C.J. (1994).
On the convergence of Monte Carlo maximum likelihood calculations.
Journal of the Royal Statistical Society, Series B, 56, 261-274.

Goulard, M. and Voltz, M. (1992).
Linear coregionalization model: tools for estimation and choice of
cross-variogram matrix.
Mathematical Geology, 24, 269-286.

Matheron, G. (1982).
Pour une analyse krigeante des donnees regionalisees.
Publication N-732, Centre de Geostatistique, Fontainebleau, France.

Wackernagel, H. (1995).
Multivariate Geostatistics.
Springer, Berlin.

3 Maggio, 2007 11:00

Sezione di Probabilità e Statistica Matematica

Reduced Basis Approximation and A Posteriori Error Estimation for Parametrized PDEs

Dr. Gianluigi Rozza, Post-Doc, Massachusetts Institute of Technology

Aula Seminari MOX, VI piano Dip. Matematica

abstract

abstract

Reduced basis methodology focuses on the rapid and reliable prediction of engineering outputs associated with parametrized partial differential equations.
In particular, we consider an "output of interest" — related to energies or forces, stresses or strains, flowrates or pressure drops, temperatures or fluxes — as a function of an "input parameter" — related to geometry, physical properties, boundary conditions, or loads.
The output of interest is a linear functional of the field variable. We thus arrive at an input-output statement, evaluation of which requires solution of a parametrized partial differential equation.
We consider partial differential equations for which the parametric dependence is strictly or approximately affine; "affine" dependence implies that the parametrized differential operator can be expressed as a sum of Q products of [parameter-dependent functions] x [parameter-independent operators].

As regards "rapid,'' the method minimizes the marginal cost associated with (approximate) input-output evaluation, and is thus most useful either (a) in the real-time or interactive context, or (b) in the limit of many queries. Engineering situations which satisfy these criteria include in-the-field robust parameter estimation (or inverse problems, or nondestructive evaluation), design and optimization, and control.
As regards "reliable,'' we provide certificates of fidelity with every
prediction: an estimate that rigorously bounds the error in the input-output evaluation or field variable relative to a highly accurate (and hence very expensive) "truth" finite element solution.

In many engineering situations, the certainty provided by these error bounds is crucial. For example, in the real-time context, critical decisions must be made in the field — quickly, without recourse to extensive Offline resources — that are at least feasible and safe if not optimal.

The essential components of our approach are threefold.

(i) Rapidly convergent global Reduced-Basis (RB) approximations —
(Galerkin) projection onto a space W_N spanned by solution of the governing partial differential equation at N (optimally) selected points S_N in the parameter set. Typically, N will be small, as we focus attention on the (smooth) low-dimensional parametrically-induced manifold of interest.
Our approach is premised upon a classical Finite Element (FE) method ''truth'' approximation space of (typically very large) dimension. It is the FE truth approximation upon which we build our RB approximation, and with respect to which we measure the RB error//.
(ii) Rigorous /a posteriori/ error estimation procedures — relaxations of the error-residual equation that provide inexpensive yet sharp bounds for the error in the RB field-variable approximation, and output(s) approximation.
(iii) Offline/Online computational procedures — decomposition stratagems which decouple the generation and projection stages of the RB
approximation: very extensive (parameter-independent) pre-processing performed Offline once that then prepares the way for subsequent very inexpensive calculations performed Online for each new input-output evaluation required.
The operation count for the Online stage depends only on N and the parametric complexity of the problem. The Online computational complexity and mathematical stability does not depend on the dimension of the underlying "truth" FE approximation space/; we may thus consider a highly accurate truth approximation.

In the seminar a demo of the rbMIT_System Sotware will be provided and the emphasis wil be given to the construction of lower bounds for the coercivity and inf-sup stability constants required in the a posteriori error analysis of reduced basis approximations to affinely parametrized partial differential equations. The method reduces the Online calculation to a small Linear Program: the objective is a parametric expansion of the underlying Rayleigh quotient; the constraints reflect stability information at optimally selected parameter points. The method is simple and general to implement, the Offline stage is based on standard eigenproblems that can be efficiently treated by the Lanczos method, and the Online Linear Program is typically of modest size.

2 Maggio, 2007 

Sezione di Calcolo delle Variazioni ed Equazioni Differenziali

A non smooth version of Sard theorem

Alexander Ioffe, Technion Israel Institute of Technology, Haifa

Reduced basis methodology focuses on the rapid and reliable prediction of engineering outputs associated with parametrized partial differential equations.
In particular, we consider an "output of interest" — related to energies or forces, stresses or strains, flowrates or pressure drops, temperatures or fluxes — as a function of an "input parameter" — related to geometry, physical properties, boundary conditions, or loads.
The output of interest is a linear functional of the field variable. We thus arrive at an input-output statement, evaluation of which requires solution of a parametrized partial differential equation.
We consider partial differential equations for which the parametric dependence is strictly or approximately affine; "affine" dependence implies that the parametrized differential operator can be expressed as a sum of Q products of [parameter-dependent functions] x [parameter-independent operators].

As regards "rapid,'' the method minimizes the marginal cost associated with (approximate) input-output evaluation, and is thus most useful either (a) in the real-time or interactive context, or (b) in the limit of many queries. Engineering situations which satisfy these criteria include in-the-field robust parameter estimation (or inverse problems, or nondestructive evaluation), design and optimization, and control.
As regards "reliable,'' we provide certificates of fidelity with every
prediction: an estimate that rigorously bounds the error in the input-output evaluation or field variable relative to a highly accurate (and hence very expensive) "truth" finite element solution.

In many engineering situations, the certainty provided by these error bounds is crucial. For example, in the real-time context, critical decisions must be made in the field — quickly, without recourse to extensive Offline resources — that are at least feasible and safe if not optimal.

The essential components of our approach are threefold.

(i) Rapidly convergent global Reduced-Basis (RB) approximations —
(Galerkin) projection onto a space W_N spanned by solution of the governing partial differential equation at N (optimally) selected points S_N in the parameter set. Typically, N will be small, as we focus attention on the (smooth) low-dimensional parametrically-induced manifold of interest.
Our approach is premised upon a classical Finite Element (FE) method ''truth'' approximation space of (typically very large) dimension. It is the FE truth approximation upon which we build our RB approximation, and with respect to which we measure the RB error//.
(ii) Rigorous /a posteriori/ error estimation procedures — relaxations of the error-residual equation that provide inexpensive yet sharp bounds for the error in the RB field-variable approximation, and output(s) approximation.
(iii) Offline/Online computational procedures — decomposition stratagems which decouple the generation and projection stages of the RB
approximation: very extensive (parameter-independent) pre-processing performed Offline once that then prepares the way for subsequent very inexpensive calculations performed Online for each new input-output evaluation required.
The operation count for the Online stage depends only on N and the parametric complexity of the problem. The Online computational complexity and mathematical stability does not depend on the dimension of the underlying "truth" FE approximation space/; we may thus consider a highly accurate truth approximation.

In the seminar a demo of the rbMIT_System Sotware will be provided and the emphasis wil be given to the construction of lower bounds for the coercivity and inf-sup stability constants required in the a posteriori error analysis of reduced basis approximations to affinely parametrized partial differential equations. The method reduces the Online calculation to a small Linear Program: the objective is a parametric expansion of the underlying Rayleigh quotient; the constraints reflect stability information at optimally selected parameter points. The method is simple and general to implement, the Offline stage is based on standard eigenproblems that can be efficiently treated by the Lanczos method, and the Online Linear Program is typically of modest size.

27 Aprile, 2007 11:00

Sezione di Geometria, Algebra e loro applicazioni

A few short combinatorial proofs in Geometry

Joseph Zaks, University of Haifa , Haifa (Israel)

Aula Seminari del III piano del Dipartimento di Matematica Francesco Brioschi , Politecnico di Milano, via Bonardi n. 9 (edificio La Nave )

abstract

abstract

We will treat the unit-distance graph over the real (or rational) points in Ed, discuss their connectivity and chromatic numbers, and use it in connection with the Beckman-Quarels Theorem ( Every mapping from Ed to itself, d >= 2, that preserves distance one is an isometry ). We will show that every closed curve on the unit sphere in E^d that meets all the d major hyperplanes has length of at least pi.

23 Aprile, 2007 17:00

Seminario Matematico e Fisico di Milano

Combinatorial and geometric methods in topology

Carlo Petronio, Università di Pisa

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

abstract

abstract

"How many different objects can be obtained by gluing together in pairs the faces of an octahedron? After deciding what ""object"" and ""different"" mean, this is an apparently very elementary question, but the answer is not quite so. Before facing it I will go one dimension down and consider the gluings of the edges of a polygon, discussing surface topology and showing that there are extremely few surfaces one can get from a given polygon compared to the number of inequivalent gluing patterns. Then I will introduce the notions of curvature and hyperbolic geometry in two and three dimensions, I will discuss rigidity and I will sketch how this applies to the original question, yielding the fact that the number of different results is indeed quite big."

18 Aprile, 2007 12:30

Seminari di Cultura Matematica

Nello spazio in tre. Le missioni spaziali e il problema dei tre corpi

A. Finzi, Dip. di Ing. Aerospaziale

Aula B.21

"How many different objects can be obtained by gluing together in pairs the faces of an octahedron? After deciding what ""object"" and ""different"" mean, this is an apparently very elementary question, but the answer is not quite so. Before facing it I will go one dimension down and consider the gluings of the edges of a polygon, discussing surface topology and showing that there are extremely few surfaces one can get from a given polygon compared to the number of inequivalent gluing patterns. Then I will introduce the notions of curvature and hyperbolic geometry in two and three dimensions, I will discuss rigidity and I will sketch how this applies to the original question, yielding the fact that the number of different results is indeed quite big."

18 Aprile, 2007 

Sezione di Calcolo delle Variazioni ed Equazioni Differenziali

Behavior of some triply nonlinear equations

Alain Miranville, Università de Poitiers Asymptotic

"How many different objects can be obtained by gluing together in pairs the faces of an octahedron? After deciding what ""object"" and ""different"" mean, this is an apparently very elementary question, but the answer is not quite so. Before facing it I will go one dimension down and consider the gluings of the edges of a polygon, discussing surface topology and showing that there are extremely few surfaces one can get from a given polygon compared to the number of inequivalent gluing patterns. Then I will introduce the notions of curvature and hyperbolic geometry in two and three dimensions, I will discuss rigidity and I will sketch how this applies to the original question, yielding the fact that the number of different results is indeed quite big."