19 Giugno, 2013 12:30 in punto

Sezione di Probabilità e Statistica Matematica

A PDE approach to large deviations for stochastic PDE with Levy noise

Andrzej Swiech, School of Mathematics, Georgia Institute of Technology, Atlanta, USA

Aula seminari III piano

abstract

abstract

We will present a large deviation principle result for solutions of abstract stochastic evolution equations with small Levy noise. The result is obtained by a combination of PDE and probabilistic techniques. The key component of this approach is the use of Hamilton-Jacobi-Bellman integro-partial differential equations in Hilbert spaces. We will discuss the notion of
viscosity solution for such infinite dimensional integro-PDE.

18 Giugno, 2013 17:00

Sezione di Probabilità e Statistica Matematica

When Micro Prudence increases Macro Risk: The Destabilizing Effects of Financial Innovation, Leverage, and Diversification

Fabrizio Lillo, Scuola Normale Superiore di Pisa, Universita di Palermo & Santa Fe Institute

Aula Seminari Terzo piano

abstract

abstract

By exploiting basic common practice accounting and risk management rules, we pro- pose a simple analytical dynamical model to investigate the effects of micro-prudential changes on macro-prudential outcomes. Specifically, we study the consequence of the introduction of a financial innovation that allow reducing the cost of portfolio diver- sification in a financial system populated by financial institutions having capital re- quirements in the form of VaR constraint and following standard mark-to-market and risk management rules. We provide a full analytical quantification of the multivariate feedback effects between investment prices and bank behavior induced by portfolio re- balancing in presence of asset illiquidity and show how changes in the constraints of the bank portfolio optimization endogenously drive the dynamics of the balance sheet aggregate of financial institutions and, thereby, the availability of bank liquidity to the economic system and systemic risk. The model shows that when financial innovation reduces the cost of diversification below a given threshold, the strength (due to higher leverage) and coordination (due to similarity of bank portfolios) of feedback effects increase, triggering a transition from a stationary dynamics of price returns to a non stationary one characterized by steep growths (bubbles) and plunges (bursts) of market prices.

17 Giugno, 2013 17:00

Seminario Matematico e Fisico di Milano

TROPICALIZATION OF CLASSICAL MODULI SPACES

BERND STURMFELS, University of California Berkeley

Università degli Studi di Milano, Dipartimento di Matematica, Via Saldini

abstract

abstract

Algebraic geometry is the study of solutions sets to polynomial equations. Solutions that depend on an infinitesimal parameter are studied combinatorially by tropical geometry.
Tropicalization works especially well for varieties that are parametrized by monomials in linear forms. Many classical moduli spaces (for curves of low genus and few points in the plane) admit such a representation, and we here explore their tropical geometry.
Examples to be discussed include the Segre cubic, the Igusa quartic, the Burkhardt quartic, and moduli of marked del Pezzo surfaces.
Matroids, hyperplane arrangements, and Weyl groups play a prominent role. Our favorites are E6, E7 and G2.

17 Giugno, 2013 11:00 in punto

Seminario Matematico e Fisico di Milano

Option prices, models and ranges of models

Martin Klimmek, Mathematical Institute, University of Oxford

Aula Consiglio - Settimo Piano

Algebraic geometry is the study of solutions sets to polynomial equations. Solutions that depend on an infinitesimal parameter are studied combinatorially by tropical geometry.
Tropicalization works especially well for varieties that are parametrized by monomials in linear forms. Many classical moduli spaces (for curves of low genus and few points in the plane) admit such a representation, and we here explore their tropical geometry.
Examples to be discussed include the Segre cubic, the Igusa quartic, the Burkhardt quartic, and moduli of marked del Pezzo surfaces.
Matroids, hyperplane arrangements, and Weyl groups play a prominent role. Our favorites are E6, E7 and G2.

14 Giugno, 2013 14:30 in punto

Seminario Matematico e Fisico di Milano

Generalized Spatial Splines Regression with Differential Penalization

Matthieu Wilhelm, EPFL , Lausanne

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

abstract

abstract

We propose a novel method for the analysis of spatially distributed data from an exponential family distribution, able to efficiently treat data occurring over irregularly shaped domains.
We consider a generalized linear framework and extend the work of Sangalli (2013) to likelihoods other than the gaussian.
In particular, we can handle all distributions within the exponential family, including binomial, Poisson and Gamma outcomes,
hence leading to a very broad applicability of the proposed models. Specifically, we maximize a penalized log-likelihood function where the roughness penalty term involves a suitable differential operator of the spatial field over the domain of interest.
This maximization is done via a penalized iterative least square approach; see Wood (2006). Space-varying covariate information can also be included in the model in a semi-parametric setting. The proposed models exploit advanced scientific computing techniques and specifically make use of the Finite Element Method, that provide a basis for piecewise polynomial surfaces and allows to impose boundary conditions on the space distribution of the probability.

referente: laura.sangalli@polimi.it

14 Giugno, 2013 11:00 in punto

Sezione di Fisica Matematica

DSMC methods for simulation of Coulomb collisions in multi-component plasma

Alexander Bobylev, Department of Mathematics, Karlstad University, Sweden

Aula seminari III piano, Dipartimento di matematica, Politecnico di Milano.

abstract

abstract

The general DSMC method for solving Boltzmann equation for long-range potentials and Landau-Fokker-Planck equation was proposed by Bobylev and Nanbu in 2000 [1] (partly as a development of earlier appoach of Nanbu [2] to Coulomb collisions).  The methods of [1,2] were later applied to various model problems of plasma physics, discussed in detail and further developed by several authors (see, for example, [3,4] and references in [4]). However the general method of [1] was not clearly understood and therefore many authors still use a more complicated original scheme of [2] with reference to [1] just for the formal proof of consistency with the Landau-Fokker-Planck equation. The reason is that the first presentation of the method was done in [1] in too formal and general way. We present in this paper a completely different approach, which leads to basically the same general method, but makes its essence absolutely clear and transparent. The method is explained for the general case of multi-component plasma. We also present some rigorous estimates for accuracy of the method. Finally some numerical results on typical problems of physics of collisional plasma are presented and discussed.      
 
References
 
1. A.V.Bobylev and K.Nanbu, Theory of collision algorithms for gases and plasmas based on the Boltzmann equation and the Landau-Fokker-Planck equation, Phys.Rev.E 61, 4576-4586 (2000).
 
2. K.Nanbu, Theory of cumulative small-angle collisions in plasmas, Phys.Rev. E 55, 4642-4652 (1997).
 
3. A.V.Bobylev, E.Mossberg and I.F.Potapenko, A DSMC method for the Landau-Fokker-Planck equation, in the book Rarefied Gas Dynamics (Proc. of 25th RGD Symposium, St. Petersburg, July 2006), Eds. M.S.Ivanov and A.K.Rebrov, 479-483, Novosibirsk 2007.
 
4. G.Dimarco, R.Caflisch, L.Pareschi, Direct Simulation Monte Carlo schemes for Coulomb interactions in plasmas, Commun. Appl. Indust. Math. 1, 72-91 (2010).

13 Giugno, 2013 11:00 in punto

Sezione di Fisica Matematica

Quasi oscillation-free high order spectral elements using a numerical subgrid approach

Simone Marras, Naval Postgraduate School

Aula Seminari VI PIANO Dipartimento di Matematica, Politecnico di Milano

abstract

abstract

A major issue in the accurate numerical approximation of transport dominated problems
by high-order methods is the ability of the solver to limit the spurious oscillations that affect the solution. Classical filter-based stabilization used with spectral elements may suffer in this respect, hence suggesting the search for suitable alternatives. In this talk, we will show how the Variational Multiscale Stabilization (VMS) method originally derived for finite elements is built for high-order spectral elements. We will demonstrate how VMS greatly improves the solution of the transport-diffusion equation by reducing over- and under-shoots while preserving spectral accuracy, and can therefore be considered an alternative to filter-based schemes. To further suppress oscillations that may occur in the proximity of internal and boundary layers, VMS is coupled to a properly designed discontinuity capturing scheme. Results are shown for 2D, and pseudo-3D problems using spectral elements up to order 16.

7 Giugno, 2013 14:30

Seminari PhD di Equazioni Differenziali

Existence of solutions for non level-convex problems in the supremal form

ANA RIBEIRO, Faculdade de Ciencias e Tecnologia-Universidade Nova de Lisboa

Aula seminari VI piano

abstract

abstract

It is well known that lower semicontinuity of functionals in
the supremal form F(u)=esssupx¸ƒ¶ f(Þu(x))is related to the
level-convexity of the supremand f . We adress the problem of
existence of solutions for the Dirichlet boundary problem in the lack
of this convexity condition relating it with some differential
inclusion problem. This is a joint work with E. Zappale.

6 Giugno, 2013 12:00

Sezione di Calcolo delle Variazioni ed Equazioni Differenziali

Quantitative stratification and critical sets of harmonic functions

Daniele Valtorta, Università degli Studi dell Insubria

Aula seminari III piano

abstract

abstract

Given a harmonic function defined on the unit ball of R^n, we discuss techniques to
obtain effective volume estimates on the tubular neighborhood of its critical sets.


We use a technique recently introduced by proff. Jeff Cheeger and Aaron Naber,
called quantitative stratification technique. It is based on approximate symmetries
of the function u at different scales. Studying how these approximate symmetries
interact with each other, we obtain the effective volume estimates.

These results
are described in a preprint available on arXiv.

We also discuss possible improvements
of the results using a refined quantitative differentiation argument and packing
estimates for semi-algebraic sets.

5 Giugno, 2013 12:15

Seminari di Cultura Matematica

Marte a quasi un anno dall'arrivo di Curiosity

Luigi Bignami, Giornalista scientifico

Aula B.2.1 (edificio 14 - Nave ) via Bonardi 9, Milano.

Given a harmonic function defined on the unit ball of R^n, we discuss techniques to
obtain effective volume estimates on the tubular neighborhood of its critical sets.


We use a technique recently introduced by proff. Jeff Cheeger and Aaron Naber,
called quantitative stratification technique. It is based on approximate symmetries
of the function u at different scales. Studying how these approximate symmetries
interact with each other, we obtain the effective volume estimates.

These results
are described in a preprint available on arXiv.

We also discuss possible improvements
of the results using a refined quantitative differentiation argument and packing
estimates for semi-algebraic sets.