18 Maggio, 2016 16:15 oclock

Sezione di Analisi

Nonlinear Problems in Exterior Domains

Benedetta Pellacci, Università di Napoli

Aula seminari 3° piano

abstract

abstract

We will study the existence of a positive solution for non-linear problems in exterior domains. This topic is related to the search of non-symmetric solutions of saturable or non-linear Schrödinger equations when the domain is not symmetric. This a joint work with Liliane Maia (University of Brasilia).

17 Maggio, 2016 17:15 oclock

Sezione di Finanza Quantitativa

Pricing a Swap on the Italian Spark Spread in the Presence of Counterparty Credit Risk

Viviana Fanelli, Department of Economics, Management and Business Law, Università degli Studi di Bari

Aula Seminari - III Piano

abstract

abstract

The liberalization of energy markets led to the development of new risk management strategies, as well as to the increase of energy trading opportunities for market players. For years researchers and practition-ers have been interested in analyzing relationships among commodity prices, and studying specic price spreads by observing their behaviour. We investigate the relationship existing between the electricity forward price and the natural gas forward price in Italy, and we propose an appropriate pricing model for the spread between the two prices. In energy markets, this spread is referred to as spark spread. The spark spread mimics nancially the generation costs of electricity for a spe-cic facility because it involves the simultaneous purchase of natural gas and sale of electricity. Furthermore, we aim at evaluating nancial derivatives that oer the ability to lock in a price margin on current and future electricity generation to utilities, generators, and market actors. In particular, we focus on studying the cash ow structure of a spark spread swap and we use a reduced form modelling approach to assess the impact of counterparty risk on pricing formulas.

13 Maggio, 2016 14:00

MOX Seminar

Coupling schemes and unfitted mesh methods for fluid-structure interaction

Mikel Landajuela, MOX, MOX, Dipartimento di Matematica, Politecnico di Milano

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

abstract

abstract

This work is devoted to the development and analysis of efficient numerical algorithms for the simulation of mechanical systems involving the interaction of a deformable thin-walled structure with an internal or surrounding incompressible fluid flow.
In the first part, we introduce two new classes of explicit coupling schemes using fitted fluid and solid meshes. The methods proposed combine a certain (parameter free) Robin-consistency in the system with (i) a projection-based time-marching in the fluid or (ii) second-order time-stepping in both the fluid and the solid. The stability properties of the methods are analyzed within representative linear settings. This part includes also a comprehensive numerical study in which state-of-the-art coupling schemes (including some of the methods proposed herein) are compared and validated against the results of an experimental benchmark.
In the second part, we consider unfitted mesh formulations. These approaches are more versatile at simulating problems with large interface deflections and/or topological changes. The spatial discretization in this case is based on variants of Nitsche’s method with cut elements. Robustness with respect to arbitrary interface intersections is guaranteed through suitable stabilization. For a fictitious domain setting using overlapping meshes, we present two new classes of splitting schemes which exploit the aforementioned interface
Robin-consistency in the unfitted frame- work. The semi-implicit or explicit nature of the splitting in time is dictated by the order in which the spatial and time discretizations are performed. In the case of the coupling with immersed structures, weak and strong discontinuities across the interface are allowed for the velocity and pressure, respectively. Stability and error estimates are provided, using energy arguments within a linear setting. A series of numerical tests, involving static and moving interfaces, illustrates the performance of the different methods proposed.

contact: christian.vergara@polimi.it

13 Maggio, 2016 15:30

MOX Seminar

Spatial statistics for complex and object data

Alessandra Menafoglio e Simone Vantini, MOX - Dipartimento di Matematica, Politecnico di Milano

Aula Seminari III Piano Edificio 14 - Dipartimento di Matematica, Politecnico di Milano

abstract

abstract

This talk is the first one of a series of seminars focusing on the analysis of complex data with spatial and/or temporal dependence.

Firstly, Alessandra Menafoglio will introduce Object Oriented Geostatistics as a new branch of spatial statistics addressing the problem of analyzing georeferenced complex observations, in the form of
high- or infinite-dimensional data, possibly heterogeneous and constrained. For instance, sedimentological and/or geochemical observations are routinely available in the form of functional (e.g., curves, surfaces or images) or distributional data (e.g., cumulative distribution or probability density functions). All these diverse types of information are treated within a unifying framework, by considering the available observations as “object” data. Object-Oriented Kriging and stochastic simulation will be introduced, relying upon a flexible geostatistical methodology able to treat infinite-dimensional data provided that these can be embedded into an appropriate space of objects (i.e., a Hilbert space). The latter should properly capture all the relevant data features through its geometry. Motivated by real case studies, particular emphasis will be given to the problem of prediction (i.e., Kriging) and uncertainty assessment (via stochastic simulation) of distributional data, interpreted as points within the Hilbert space of functional compositions, endowed with the Aitchison geometry.

In the second part of the seminar, Simone Vantini will introduce a general non parametric framework for the analysis of complex data with spatial dependence based on random Voronoi tessellations of the area of investigation (i.e., Bagging Voronoi Analysis). In detail, the approach will be presented in connection to a real application dealing with the analysis of varved (annually laminated) sediment data from lake Kassjon in Northern Sweden. These data will be considered as an (6400-year long) instance of sequentially dependent functional data possibly clustered and misaligned. The general framework will be thus displayed in an algorithm named Bagging Voronoi K-Medoid Alignment. A comparison with simpler approaches will be also presented. This latter comparison will show the importance of jointly dealing with clustering, misalignment, and dependence while inferring on past environmental and climate changes from this kind of data.

contact: laura.sangalli@polimi.it

12 Maggio, 2016 15:30

Seminario Matematico e Fisico di Milano

COLLOQUIUM DIPARTIMENTO DI MATEMATICA E APPLICAZIONI, MILANO-BICOCCA con SEMINARIO MATEMATICO E FISICO DI MILANO: "High dimensional expanders: from Ramanujan graphs to Ramanujan complexes"

ALEX LUBOTSKY, Hebrew University and ETH-Zurich

Aula 3014, edificio U5 del Dipartimento di Matematica e Applicazioni, Universita' di Milano - Bicocca, Via R. Cozzi 55

abstract

abstract

Expander graphs in general, and Ramanujan graphs, in particular, have played a major role in combinatorics and computer science in the last 4 decades and more recently also in pure math. Approximately 10 years ago a theory of Ramanujan complexes was developed by Li, Lubotzky-Samuels-Vishne and others.

In recent years a high dimensional theory of expanders is emerging. The notions of geometric and topological expanders were defined by Gromov in 2010 who proved that the complete d-dimensional simplicial complexes are such. He raised the basic question of existence of such bounded degree complexes of dimension d>1. Ramanujan complexes were shown to be geometric expanders by Fox-Gromov-Lafforgue-Naor-Pach in 2013, but it was left open if they are also topological expanders.

By developing new isoperimetric methods for "locally minimal small" F_2- co-chains, it was shown recently by Kaufman- Kazdhan- Lubotzky for small dimensions and Evra-Kaufman for all dimensions that the d-skeletons of (d+1)-dimensional Ramanujan complexes provide bounded degree topological expanders. This answers Gromov's original problem, but still leaves open whether the Ramanujan complexes themselves are topological expanders.
it was shown recently by Kaufman- Kazdhan- Lubotzky for small dimensions and Evra-Kaufman for all dimensions that the d-skeletons of (d+1)-dimensional Ramanujan complexes provide bounded degree topological expanders. This answers Gromov's original problem, but still leaves open whether the Ramanujan complexes themselves are topological expanders.

We will describe these developments and the general area of high dimensional expanders and some of its open problems.

11 Maggio, 2016 12:15

Seminari di Cultura Matematica

Anelli e paraboloidi di carta: origami, arte e geometria

Paola Magrone, Università degli Studi Roma Tre

Aula B21, ed. 14 (Nave), Campus Bonardi, via Bonardi 9, Milano

Expander graphs in general, and Ramanujan graphs, in particular, have played a major role in combinatorics and computer science in the last 4 decades and more recently also in pure math. Approximately 10 years ago a theory of Ramanujan complexes was developed by Li, Lubotzky-Samuels-Vishne and others.

In recent years a high dimensional theory of expanders is emerging. The notions of geometric and topological expanders were defined by Gromov in 2010 who proved that the complete d-dimensional simplicial complexes are such. He raised the basic question of existence of such bounded degree complexes of dimension d>1. Ramanujan complexes were shown to be geometric expanders by Fox-Gromov-Lafforgue-Naor-Pach in 2013, but it was left open if they are also topological expanders.

By developing new isoperimetric methods for "locally minimal small" F_2- co-chains, it was shown recently by Kaufman- Kazdhan- Lubotzky for small dimensions and Evra-Kaufman for all dimensions that the d-skeletons of (d+1)-dimensional Ramanujan complexes provide bounded degree topological expanders. This answers Gromov's original problem, but still leaves open whether the Ramanujan complexes themselves are topological expanders.
it was shown recently by Kaufman- Kazdhan- Lubotzky for small dimensions and Evra-Kaufman for all dimensions that the d-skeletons of (d+1)-dimensional Ramanujan complexes provide bounded degree topological expanders. This answers Gromov's original problem, but still leaves open whether the Ramanujan complexes themselves are topological expanders.

We will describe these developments and the general area of high dimensional expanders and some of its open problems.

11 Maggio, 2016 14:30

Sezione di Geometria, Algebra e loro applicazioni

Uni- and multivariate discrete orthogonal polynomials using Dirac operators

Hendrik De Bie, Ghent University

Aula seminari del 3 piano

abstract

abstract

In the second talk, we will reverse the question of the previous talk. Given a family of discrete orthogonal polynomials, can we construct a Dirac model so that they arise as expansion coefficients?
We will answer this question affirmatively for the Racah polynomials and their generalization the Bannai-Ito polynomials. In the univariate case we will use a generalized Dirac operator in 3D, while for the multivariate case we have to resort to nD. Along the way, we will construct a new symmetry algebra that can be interpreted as a higher rank Bannai-Ito algebra.

11 Maggio, 2016 16:15 oclock

Sezione di Analisi

Some inverse boundary value problems for PDEs: theory and applications

Matteo Santacesaria, Politecnico di Milano

Aula seminari III piano

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. In particular we will review some classical strategy to attack these problems, based on the so-called complex geometrical optics solutions and inverse scattering theory. Then we will present a new reconstruction method able to detect singularities of a conductivity from the Dirichlet-to-Neumann map: this is based on some microlocal properties of our PDE. Numerical results will be presented as well. The latter is an ongoing project in collaboration with A. Greenleaf, M. Lassas, S. Siltanen and G. Uhlmann.

9 Maggio, 2016 10:30

Sezione di Geometria, Algebra e loro applicazioni

Spherical monogenics in dimension 3 and discrete orthogonal polynomials

Hendrik De Bie, Ghent University

Aula seminari del 3 piano

abstract

abstract

In this educational talk, my main goal is to show how a family of discrete orthogonal polynomials from the Askey scheme naturally arises in the study of spherical monogenics in dimension 3.
This connection will be revealed by making explicit the action of the so(3) Lie algebra that preserves the monogenics on a basis constructed using a tower of CK extensions.
It will then be shown that different towers of CK extensions lead to different orthogonal bases. The expansion coefficients between two such bases are subsequently expressed using the Krawtchouk discrete orthogonal polynomials.

5 Maggio, 2016 11:30

Sezione di Probabilità e Statistica Matematica

Control of stochastic McKean-Vlasov equation and applications

Huyên Pham, Université Paris Diderot

Aula seminari III piano, Dipartimento di Matematica, Politecnico di Milano (via Bonardi 9, Milano, Edificio 14 “La Nave”)

In this educational talk, my main goal is to show how a family of discrete orthogonal polynomials from the Askey scheme naturally arises in the study of spherical monogenics in dimension 3.
This connection will be revealed by making explicit the action of the so(3) Lie algebra that preserves the monogenics on a basis constructed using a tower of CK extensions.
It will then be shown that different towers of CK extensions lead to different orthogonal bases. The expansion coefficients between two such bases are subsequently expressed using the Krawtchouk discrete orthogonal polynomials.