28 Aprile, 2010 15:00 in punto

Inverse problems in the space of Radon measures

Hanna Katriina Pikkarainen, RICAM - Austrian Academy of Sciences Austria

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

abstract

abstract

In this talk, regularization of illposed inverse problems involving Radon measures is studied. The main motivation is to nd a sparse reconstruction for an unknown Radon measure based on indirect measurements. Similar questions has been previously examined in [3].

In this talk, it is shown that the minimization of the Tikhonov functional with the total
variation norm penalty is a well-posed problem for Radon measures. The minimizer of the Tikhonov functional is characterized by using the corresponding Fenchel predual problem. Convergence and convergence rate properties of the minimizer are examined under common source conditions, like in [1, 2]. A numerical algorithm promoting sparsity is proposed. The algorithm is illustrated by a numerical example.

References
[1] M. Burger and S. Osher. Convergence rates of convex variational regularization.

Inverse Problems, 20(5):1411{1421, 2004.
[2] B. Hofmann, B. Kaltenbacher, C. Poeschl, and O. Scherzer. A convergence rates result
for Tikhonov regularization in Banach spaces with non-smooth operators. Inverse
Problems, 23(3):987{1010, 2007.

[3] O. Scherzer and B. Walch. Sparsity regularization for Radon measures. In X.-C. Tai,
K. Morken, M. Lysaker, and K.-A. Lie, editors, Scale Space and Variational Meth-
ods in Computer Vision, volume 5567 of Lecture Notes in Computer Science, pages
452{463. Springer-Verlag, 2009. Proceedings of the Second International Conference,
SSVM 2009.

21 Aprile, 2010 12:15

Seminari di Cultura Matematica

Architettura del paesaggio e geometria nell’antico Egitto

G. Magli, Politecnico di Milano

Aula B21 (Edificio 14 Nave )

In this talk, regularization of illposed inverse problems involving Radon measures is studied. The main motivation is to nd a sparse reconstruction for an unknown Radon measure based on indirect measurements. Similar questions has been previously examined in [3].

In this talk, it is shown that the minimization of the Tikhonov functional with the total
variation norm penalty is a well-posed problem for Radon measures. The minimizer of the Tikhonov functional is characterized by using the corresponding Fenchel predual problem. Convergence and convergence rate properties of the minimizer are examined under common source conditions, like in [1, 2]. A numerical algorithm promoting sparsity is proposed. The algorithm is illustrated by a numerical example.

References
[1] M. Burger and S. Osher. Convergence rates of convex variational regularization.

Inverse Problems, 20(5):1411{1421, 2004.
[2] B. Hofmann, B. Kaltenbacher, C. Poeschl, and O. Scherzer. A convergence rates result
for Tikhonov regularization in Banach spaces with non-smooth operators. Inverse
Problems, 23(3):987{1010, 2007.

[3] O. Scherzer and B. Walch. Sparsity regularization for Radon measures. In X.-C. Tai,
K. Morken, M. Lysaker, and K.-A. Lie, editors, Scale Space and Variational Meth-
ods in Computer Vision, volume 5567 of Lecture Notes in Computer Science, pages
452{463. Springer-Verlag, 2009. Proceedings of the Second International Conference,
SSVM 2009.

21 Aprile, 2010 15:00 in punto

Seminari di Cultura Matematica

Fractional Cointegration and Level Shifts in the Realized and Implied Volatility Relation

Paolo Santucci de Magistris, Dipartimento di Economia Politica e Metodi Quantitativi, Università di Pavia

Aula Seminari VI Piano

abstract

abstract

The market efficiency hypothesis predicts that implied volatility is an unbiased forecast of the future realized volatility of the spot price.
The relationship realized-implied volatility can be analyzed in terms of fractional cointegration, supposing that the dynamics of the two
series are driven by the same fractional stochastic trend, providing support for a long-run unbiasedness of implied volatility as a predictor of realized volatility. In this paper, we test for the presence of fractional cointegration according both to a regression setup, as in Christensen and Nielsen (2006), and to a full parametric specification, say a fractional VECM, evaluating the respective performances in small samples with a Monte Carlo exercise. Moreover, we consider the possibility that the persistence of implied and realized volatility is induced by a common level shift process. We provide a new inference technique to deal with the estimation of the probability of occurrence of the shift, and we show that usual fractional cointegration and common level shifts are observationally equivalent.

14 Aprile, 2010 12:15

Seminari di Cultura Matematica

Keplero, fra armonia del cosmo e leggi della fisica

A.M. Lombardi, Itis Primo Levi, Bollate

Aula B21 (Edificio 14 Nave )

The market efficiency hypothesis predicts that implied volatility is an unbiased forecast of the future realized volatility of the spot price.
The relationship realized-implied volatility can be analyzed in terms of fractional cointegration, supposing that the dynamics of the two
series are driven by the same fractional stochastic trend, providing support for a long-run unbiasedness of implied volatility as a predictor of realized volatility. In this paper, we test for the presence of fractional cointegration according both to a regression setup, as in Christensen and Nielsen (2006), and to a full parametric specification, say a fractional VECM, evaluating the respective performances in small samples with a Monte Carlo exercise. Moreover, we consider the possibility that the persistence of implied and realized volatility is induced by a common level shift process. We provide a new inference technique to deal with the estimation of the probability of occurrence of the shift, and we show that usual fractional cointegration and common level shifts are observationally equivalent.

14 Aprile, 2010 11:00 in punto

Seminari di Cultura Matematica

Reactive transport in porous media

Michel Kern, INRIA Paris-Rocquencourt

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

abstract

abstract

Understanding mass transfer phenomena in the subsurface is becoming increasingly important for environmental issues: predicting pollution and remediation strategies, assessing the feasibility of a nuclear underground storage site or a carbon dioxide sequestration site. These studies usually involve interaction between several phenomena, such as transport and chemistry. Under the hypothesis of local equilibrium, the model couples transport PDEs with local algebraic chemical equations. In this talk, we present numerical methods for the simulation of a coupled model folr reactive transort. We briefly
recall the methods for the individual models: flow simulations are based on Darcy s law, and require accurate methods (based on mixed
finite elements) for computing the velocity in heterogeneous situations. The velocity is the main input for the transport model, an advection-dispersion type equation, possibly with sharp fronts. This is solved using a combination of finite volume (for advection) and mixed finite element (for the dispersion--diffusion part). Chemical equations are solved by a variant of Newton s method. We give a formulation for the coupled model based on aqueous and fixed total
concentrations, and show how the usual fixed point method for the nonlinear coupled problem can be replaced by a Newton-Krylov method.
This leads to a global formulation for the coupled problem that enables keeping the transport and chemistry codes (usually written by
different groups) separate. A key point for the efficiency of the method is the design of a suitable preconditioner for the linearized (matrix free) problem. We study formulations that respect the block structure of the problem, and analyze several block preconditioners on a
simplified model, using the field of values of the Jacobian.

This is joint work with Laila Amir and Abdelaziz Taakili.

31 Marzo, 2010 12:15

Seminari di Cultura Matematica

Paradossi e verità

C. Citrini, Politecnico di Milano

Aula B21 (Edificio 14 Nave )

Understanding mass transfer phenomena in the subsurface is becoming increasingly important for environmental issues: predicting pollution and remediation strategies, assessing the feasibility of a nuclear underground storage site or a carbon dioxide sequestration site. These studies usually involve interaction between several phenomena, such as transport and chemistry. Under the hypothesis of local equilibrium, the model couples transport PDEs with local algebraic chemical equations. In this talk, we present numerical methods for the simulation of a coupled model folr reactive transort. We briefly
recall the methods for the individual models: flow simulations are based on Darcy s law, and require accurate methods (based on mixed
finite elements) for computing the velocity in heterogeneous situations. The velocity is the main input for the transport model, an advection-dispersion type equation, possibly with sharp fronts. This is solved using a combination of finite volume (for advection) and mixed finite element (for the dispersion--diffusion part). Chemical equations are solved by a variant of Newton s method. We give a formulation for the coupled model based on aqueous and fixed total
concentrations, and show how the usual fixed point method for the nonlinear coupled problem can be replaced by a Newton-Krylov method.
This leads to a global formulation for the coupled problem that enables keeping the transport and chemistry codes (usually written by
different groups) separate. A key point for the efficiency of the method is the design of a suitable preconditioner for the linearized (matrix free) problem. We study formulations that respect the block structure of the problem, and analyze several block preconditioners on a
simplified model, using the field of values of the Jacobian.

This is joint work with Laila Amir and Abdelaziz Taakili.

29 Marzo, 2010 17:00

Seminario Matematico e Fisico di Milano

FRONT PROPAGATION IN LAMINAR AND TURBULENT FLOWS

ANGELO VULPIANI, Università di Roma La Sapienza

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

abstract

abstract

The problem of front propagation in a stirred medium is addressed
in the case of laminar and turbulent flows in three different regimes:
slow reaction, fast reaction and geometrical optics limit. It is well
known that a consequence of stirring is the enhancement of front speed
with respect to the non-stirred case. By means of numerical
simulations and theoretical arguments we describe the behavior of
front speed as a function of the stirring intensity. The large scale
of the velocity field mainly rules the front speed behavior even in
the presence of smaller scales. In the unsteady (time-periodic) case,
the front speed displays a phase-locking on the flow frequency and,
albeit the Lagrangian dynamics is chaotic, chaos in front dynamics
only survives for a transient. Asymptotically the front evolves
periodically and chaos manifests only in the spatially wrinkled
structure of the front.

29 Marzo, 2010 15:00

Seminario Matematico e Fisico di Milano

METRIC PROPERTIES OF CALABI-YAU MANIFOLDS

P.M.H. WILSON, Department of Pure Mathematics and Mathematical Statistics, University of Cambridge

Dipartimento di Matematica, Università di Milano

abstract

abstract

This talk aims to introduce some general ideas relating properties from algebraic
geometry to concepts from metric geometry, in particular that of Gromov-Hausdorff limits
of metric spaces.
Let $X$ be a Calabi-Yau manifold of dimension $n$, that is a complex projective manifold
which admits a nowhere vanishing holomorphic $n$-form, and no holomorphic $i$-forms for
$0 < i < n$. By a famous theorem of Yau, for each K"ahler class in the real second cohomology,
there exists a unique Ricci
at K"ahler metric on X with K"ahler form in the given
class, the Calabi Yau metric; hence there is a well-defined metric space structure on $X$.
A natural question then arises: if we degenerate either the complex or Kähler structures
on X in the sense of algebraic geometry, obtaining a singular projective variety, what
can be said about the metric limits (in the sense of Gromov-Hausdorff) of the corresponding Ricci
at K"ahler manifolds? We will suggest some answers to this question and explain
their relevance for the geometry of Calabi-Yau manifolds.

25 Marzo, 2010 08:30 in punto

Sezione di Calcolo delle Variazioni ed Equazioni Differenziali

Stime analitiche e numeriche per la frequenza fondamentale di membrane poligonali: questioni facili, meno facili, e ancora aperte.

Simone Pezzuto, Dipartimento di Matematica, Politecnico

Aula Seminari III piano

abstract

abstract

Nel 1877 Lord Reyleigh congetturò che, tra tutti i tamburi di area fissata, quello circolare producesse la nota più grave possibile.
Una prima dimostrazione arrivò solo nel 1923-24 indipendentemente da G. Faber e E. Krahn.
Nel 1945, infine, G. Pólya e G. Szegö provarono un
Teorema di più ampio respiro, che permise di rispondere alla congettura di Reyleigh così come molte alte.
Il loro lavoro venne coronato dal libro Isoperimetrical Inequalities in Mathematical Physis , pubblicato nel 1952, divenuto oramai un classico sull argomento.
Proprio ispirandomi a quest ultimo cercherò, nel seminario, di descrivere in termini matematici la congettura di Reyleigh, legata al
problema agli autovalori per l operatore di Laplace, tracciando i punti cardine della dimostrazione (basata sulla simmetrizzazione di
Steiner). Più in generale, mostrerò alcune stime più fini per poligoni regolari, e congetture ancora aperte a riguardo.
Verranno inoltre discussi alcuni interessanti aspetti riguardo il problema agli autovalori, come la soluzione sul triangolo equilatero e
la regolarità fino al bordo delle autofunzioni.

web.mate.polimi.it/cdv/

25 Marzo, 2010 10:00 in punto

Sezione di Calcolo delle Variazioni ed Equazioni Differenziali

Un introduzione al posizionamento ottimo

Ilaria Lucardesi, Dipartimento di Matematica, Politecnico

Aula Seminari III piano

abstract

abstract

Introduzione ai problemi di posizionamento ottimo.

Il problema tipo è il seguente: data una regione geografica $ Omega$ identificata con un aperto di $R^n$ e abitata da una popolazione distribuita con densità $ varphi,$ posizionare in $ Omega$ un insieme $ Sigma$ da scegliere in una classe opportuna di insiemi ammissibili, in modo tale da minimizzare la distanza media della popolazione da esso.
Con questo modello si può rappresentare il problema di localizzare un assegnato numero di scuole, uffici postali o reti ferroviarie, in modo da rendere minimo il costo totale di trasporto dei residenti verso i servizi.
L approccio al problema è variazionale: una configurazione ottima $ Sigma$ è ottenuta come minimo di un opportuno funzionale.

Il seminario si sviluppa in 2 parti.
Nella prima vengono introdotti gli strumenti teorici: la teoria geometrica della misura (in particolare misure e distanza di Hausdorff), la teoria della $ Gamma$-convergenza e la teoria del trasporto ottimo.

Nella seconda parte vengono presentati i risultati classici di posizionamento e irrigazione , nel caso in cui $ Omega$ sia limitato e la densità di popolazione $ varphi$ sia non negativa.

Si presenta un problema nuovo: il posizionamento ottimo su un aperto $ Omega,$ ammettendo che la densità di popolazione possa avere segno qualsiasi.

web.mate.polimi.it/cdv/