24 Giugno, 2015 11:30 oclock

Sezione di Calcolo delle Variazioni ed Equazioni Differenziali

Regularity results for a class of permeability problems

Donatella Danielli, University of Purdue

Aula consiglio VII piano

abstract

abstract

In this talk we will present an overview of regularity results (both for the solution and for the free boundary) in a class of problems which arises in permeability theory. We will mostly focus on the parabolic Signorini (or thin obstacle) problem, and discuss the modern approach to this classical problem, based on several families of monotonicity formulas. In particular, we will present the optimal regularity of the solution, the classification of free boundary points, the regularity of the regular set, and the structure of the singular set. These results have been obtained in joint work with N. Garofalo, A. Petrosyan, and T. To.
We will also discuss the regularity of solutions in a related model arising in problems of semi-permeable walls and of temperature control. This is joint work with T. Backing.

23 Giugno, 2015 14:00

MOX Seminar

Detecting functional outliers: the outliergram

prof. Juan Romo, Department of Statistics Universidad Carlos III de Madrid

Aula Fausto Saleri - 6°piano

In this talk we will present an overview of regularity results (both for the solution and for the free boundary) in a class of problems which arises in permeability theory. We will mostly focus on the parabolic Signorini (or thin obstacle) problem, and discuss the modern approach to this classical problem, based on several families of monotonicity formulas. In particular, we will present the optimal regularity of the solution, the classification of free boundary points, the regularity of the regular set, and the structure of the singular set. These results have been obtained in joint work with N. Garofalo, A. Petrosyan, and T. To.
We will also discuss the regularity of solutions in a related model arising in problems of semi-permeable walls and of temperature control. This is joint work with T. Backing.

19 Giugno, 2015 14:00 oclock

Sezione di Calcolo delle Variazioni ed Equazioni Differenziali

Pointwise convergence of Fourier series and a Calderon-Zygmund decomposition for modulation invariant singular integrals

Francesco Di Plinio, Brown University Mathematics Department

Aula seminari III piano

abstract

abstract

The Calderon-Zygmund decomposition is a fundamental tool in
the analysis of singular integral operators, prime examples of which are
the Hilbert and Riesz transforms occurring, for instance, in elliptic
PDEs. Its prime purpose is the extension of L^2 boundedness results for
operators of this type to (all) other L^p spaces. The exploited
mechanism is the extra cancellation occurring when the singular integral
operator is applied to functions with mean zero.

On the other hand, pointwise convergence of Fourier series of L^p
functions is governed by L^p bounds for maximally modulated versions of
singular integrals, for which the single zero frequency of f plays no
particular role. In this talk, we describe a novel Calderon-Zygmund
decomposition adapted to the maximally modulated setting and its
applications to both pointwise convergence of Fourier series of
functions near L^1 and sharp bounds for the bilinear Hilbert transform
near the critical exponent L^{2/3}. The presentation will be
non-technical and suitable for an advanced undergraduate audience.

Partly joint work with Ciprian Demeter (IU), Christoph Thiele (Bonn),
Andrei Lerner (Bar-Ilan U, Israel) and Yumeng Ou (Brown U).

19 Giugno, 2015 15:00 oclock

Sezione di Geometria, Algebra e loro applicazioni

D-modules and local cohomology

Robin Hartshorne, U.C. Berkeley

Aula Seminari Mox del sesto piano

abstract

abstract

The concept of modules over a ring with differential operators is an old one, but its relevance to commutative algebra dates from 1993, when Lyubeznik discovered that the local cohomology module of a polynomial ring with supports in any ideal is a finitely generated D-module, even though it is rarely finitely generated over the polynomial ring. In fact these modules are what is called holonomic, i.e. of finite length over the ring of differential operators. I will explain the ideas behind this discovery, and tell how it has helped to prove new finiteness theorems about these mysterious modules.

18 Giugno, 2015 14:00

MOX Seminar

Some inverse boundary value problems in two dimensions: theory and applications

Matteo Santacesaria, University of Helsinki

Aula Saleri VI Piano Dipartimento di Matematica -MOX politecnico di Milano

abstract

abstract

In this talk we will focus on two inverse boundary value problems, the Calderón problem and the Gelfand-Calderon problem. The first concerns the reconstruction of an electrical conductivity from voltage and current measurements on the boundary of an object; its related imaging method is called Electrical Impedance Tomography and has applications from medical imaging to non destructive testing. In the Gelfand-Calderon problem one wants to reconstruct a potential in the Schrödinger equation from some information of its solutions at the boundary of a domain (Dirichlet to Neumann map). This problem can be seen as a model for acoustic tomography, namely with applications in geophysical prospecting.
We will first discuss theoretical properties of these problems, in particular their ill-posedness and stability estimates. Then we will present some recent numerical reconstruction results. First a numerical implementation for the reconstruction of a potential in the Schrödinger equation from high energy data; this new algorithm clearly shows how the resolution increases with the energy. Then we will present a new imaging technique based on electrical measurements which allows the reconstruction of inclusions inside a body. Its main feature is the ability to identify inclusions inside inclusions and the main application is a new method for early detection of brain strokes

contact:

- elena.beretta@polimi.it
- christian.vergara@polimi.it

17 Giugno, 2015 11:00 oclock

Seminario Matematico e Fisico di Milano

Learning variational regularization models for image denoising by PDE constrained optimization

Carola Bibiane Schoenlieb, DAMTP University of Cambridge

Dipartimento di Matematica del Politecnico - Aula Consiglio 7° piano

abstract

abstract

We review some recent learning approaches in variational image regularisation — in particular for variational image denoising — based on PDE constrained (also called bilevel) optimisation. Optimal solutions are typically parameters determining the type of regularisation and data discrepancy. A general type of regulariser is considered, which encompasses total variation (TV), total generalized variation (TGV) and infimal-convolution total variation (ICTV), as well as general data discrepancies encoding different noise distributions. We analyse those models for existence and structure of minimisers, as well as optimality conditions for their characterisation. Based on this information, Newton type methods are employed to solve the models numerically. The computational verification of the developed techniques will be discussed, covering instances with different type of regularisers, several noise models, spatially dependent weights and large image databases.
This is joint work with Luca Calatroni, Chung Cao, Juan Carlos De los Reyes, and Tuomo Valkonen.

15 Giugno, 2015 17:00 oclock

Seminario Matematico e Fisico di Milano

Gruppo di rinormalizzaione, effetto Kondo e modelli gerarchici

Giovanni Gallavotti, Universita' di Roma

Sala di Rappresentanza, Dipartimento di Matematica, Via Saldini

abstract

abstract

L'effetto Kondo si manifesta con la non divergenza a temperatura 0 della suscettivita` magnetica di una impurita` in un gas di fermioni: la non divergenza avviene solo in presenza di una interazione antiferromagnetica e non nel caso ferromagnetico.
E` un effetto quantistico, non deducibile da una teoria perturbativa: la sua teoria, in un modello unidimensionale, fu risolta da Wilson attraverso il gruppo di rinormalizzazione. Fu uno dei primi casi in cui si pote` capire un fenomeno multiscala controllato da un punto fisso non banale: in questo seminario illustrero` il problema in un modello che pur mantiene la caratteristica di essere un problema non perturbativo a molte scale e che sembra richiedere il metodo del gruppo di rinormalizzazione per la sua soluzione.

10 Giugno, 2015 12:30

Sezione di Finanza Quantitativa

Approximating irregular SDEs via iterative Skorokhod embeddings

Thomas Kruse, University of Duisburg-Essen, Faculty of Mathematics

Aula Consiglio - VII piano

abstract

abstract

The Bachelier Finance Society, and the Dipartimento di Matematica - Politecnico di Milano, in cooperation with Springer, are proud to announce the Nicola Bruti Liberati Prize 2014 winner Thomas Kruse, who is currently a Post-doctoral researcher at the Laboratoire de Mathematiques et Modelisation d'Evry. Thomas studied for his PhD at the University of Bonn under the supervision of Prof. Stefan Ankirchner.
Thomas's thesis covered inverse optimal stopping problems, and control problems motivated by issues in illiquidity. The prize committee was very impressed with both the financial and mathematical contributions of the thesis. We would like to congratulate the winner warmly for his achievement.

On June 10, 2015, the prize winner will deliver a lecture on

Approximating irregular SDEs via iterative Skorokhod embeddings

8 Giugno, 2015 16:30 oclock

Seminario Matematico e Fisico di Milano

T.B.A.

Andrei OKOUNKOV, Columbia University

Dipartimento di Matematica, Via Saldini, Aula Chisini

The Bachelier Finance Society, and the Dipartimento di Matematica - Politecnico di Milano, in cooperation with Springer, are proud to announce the Nicola Bruti Liberati Prize 2014 winner Thomas Kruse, who is currently a Post-doctoral researcher at the Laboratoire de Mathematiques et Modelisation d'Evry. Thomas studied for his PhD at the University of Bonn under the supervision of Prof. Stefan Ankirchner.
Thomas's thesis covered inverse optimal stopping problems, and control problems motivated by issues in illiquidity. The prize committee was very impressed with both the financial and mathematical contributions of the thesis. We would like to congratulate the winner warmly for his achievement.

On June 10, 2015, the prize winner will deliver a lecture on

Approximating irregular SDEs via iterative Skorokhod embeddings

7 Giugno, 2015 11:00

MOX Seminar

Analysis of a class of two-scale finite element methods

Endre Suli, University of Oxford

Aula Seminari MOX-6° piano dip di matematica

abstract

abstract

The classical Galerkin finite element method exhibits spurious
numerical oscillations when applied to a non-self-adjoint elliptic
boundary value problem with dominant hyperbolic behaviour and the
subscales in the problem (such as boundary layers) are not properly
resolved.

The aim of this talk is to discuss the analysis of a class of
two-scale finite element methods based on the use of residual-free
bubbles to capture the subscales. In the case of general finite
element spaces consisting of continuous piecewise polynomials, the
method is shown to exhibit an optimal rate of convergence. The
analysis relies on a delicate function-space interpolation result
due to Luc Tartar which states that
(L_2(\Omega),H^1_0(\Omega)_{1/2,\infty}\supset
(L_2(\Omega),H^1(\Omega))_{1/2,1}, with continuous embedding.

The talk in based on joint work with Franco Brezzi and Donatella
Marini (University of Pavia), and with my former D.Phil. student at
Oxford, Andrea Cangiani (now also at the University of Pavia).


References

1. F. Brezzi, D. Marini, and E. Suli.
Residual-free bubbles for advection-diffusion problems: the general
error analysis.
Numerische Mathematik. 85 (2000) 1, 31-47.

2. A. Cangiani and E. Suli.
Enhanced RFB method.
Numerische Mathematik. Accepted for publication, 2005.

3. A. Cangiani and E. Suli.
Enhanced residual-free-bubble finite element method.
International Journal for Numerical Methods in Fluids. 47 (2005) 10-11, 1307-1313.