13 Aprile, 2016 16:15 in punto

Sezione di Analisi

Symmetry results in Caffarelli-Kohn-Nirenberg interpolation inequalities

Matteo Muratori, Università degli Studi di Pavia

Aula seminari 6° piano

abstract

abstract

We investigate the structure of functions that optimize a special family of weighted interpolation inequalities of Caffarelli-Kohn-Nirenberg type (Compositio Math. 1984). The spatial dimension is greater than or equal to 3 and the weight appearing in the L^p norms is an inverse power with exponent between 0 and 2. The non-weighted case, associated with standard Gagliardo-Nirenberg inequalities, has thoroughly been investigated by M. Del Pino and J. Dolbeault in a remarkable paper (JMPA 2002), where they prove that optimal functions coincide with explicit profiles of Aubin-Talenti type. If the exponent ranges strictly between 0 and 2, suitable analogues of Aubin-Talenti-type profiles continue to exist. However, the main issue is related to radial symmetry. Indeed, as soon as optimal functions are radial, then they are necessarily of Aubin-Talenti type. Because of the weight, standard Schwarz symmetrization techniques fail: we have therefore to exploit a completely different method. First of all, by means of a concentration-compactness analysis, we prove that optimal functions converge to the Aubin-Talenti profiles as the exponent of the power of the weight tends to zero. Then we proceed by contradiction with an argument that involves angular derivatives of possibly non-radial optimal functions, which allows us to show that radial symmetry holds at least when the exponent is close to zero. This is a joint work with J. Dolbeault and B. Nazaret.

13 Aprile, 2016 16:00

Sezione di Finanza Quantitativa

Recent results on (quasiconvex) risk measures and related portfolio optimization problems

Elisa Mastrogiacomo, dipartimento di statistica e metodi quantitativi - Università di Milano Bicocca

Aula Seminari III piano

abstract

abstract

The main goal of this talk is to present some recent results concerning the study of (quasiconvex) risk measures and related portfolio optimization problems both from a theoretical and numerical point of view.
The first (theoretical) problem that we are going to present is the characterization of Pareto optimal allocations for quasiconvex risk measures. Following the approach of Jouini et al. (2008) for convex risk measures, in the quasiconvex case we investigate the notions of (weakly) Pareto optimality and of exactness of the so-called quasiconvex inf-convolution as well as existence for weakly Pareto optimal allocations. Moreover, we give a necessary condition for weakly optimal risk sharing that is also sufficient under cash-additivity of at least one between the risk measures.
The second problem concerns with the study of optimal portfolio allocation strategies under cumulative prospect theory (CPT), both from a theoretical and empirical point of view. Our aim is twofold. First, we exhibit the impact of higher-moments and CPT parameters on suitable Mean/Risk efficient frontier. On a second stage, we investigate empirically, for different portfolios, the optimal choice problem
for an investor who behaves according to the CPT, considering different parameters for this utility function. We then compare our empirical results with the Mean Variance (MV) and the Global Minimum Variance (GMV) portfolios, from an in-sample and out-of-sample perspective.
Finally we will concentrate on a portfolio optimization problem under state-dependent expected utility. Inspired by the recent results in Bjork, Murgoci and Zhou (2014), we analyze the mean–variance portfolio selection problem with state-dependent risk aversion. We consider a continuous time Lévy model consisting of two assets, one stock price with dynamics of jump-type and a bank account with constant risk-free rate. Since this problem is time inconsistent we approach it within a
game theoretic setting and look for subgame perfect Nash equilibrium strategies. We show that, under this approach an extension of the standard dynamic programming equation to a system of nonlinear PDEs is needed.

12 Aprile, 2016 15:00

MOX Seminar

MULTIFUNCTIONAL POLYMERIC NANOCONSTRUCTS: FROM IN-SILICO TO IN-VIVO

Paolo Decuzzi, Laboratory of Nanotechnology for Precision Medicine Italian Institute of Technology, Genova

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

abstract

abstract

Multifunctional nanoconstructs are particle-based nano-scale systems designed for the ‘smart’ delivery of therapeutic and imaging agents. The Laboratory of Nanotechnology for Precision Medicine synthesizes polymeric nanoconstructs with different sizes, ranging from a few tens of nanometers to a few microns; shapes, including spherical, cubical and discoidal; surface properties, with positive, negative, neutral coatings; and mechanical stiffness, varying from that of cells to rigid, inorganic materials, such as iron oxide. These are the 4S parameters – size, shape, surface, stiffness – which can be precisely tuned in the synthesis process enabling disease- and patient-specific designs of multifunctional nanoconstructs.
In this lecture, the application of these nanoconstructs to the detection and treatment of cancer lesions and cardiovascular diseases, such as thrombosis and atherosclerosis, is discussed. The contribution of the 4S parameters in modulating nanoconstruct sequestration by the mononuclear phagocyte system, organ specific accumulation, and blood longevity is also critically presented. These polymeric nanoconstructs can be loaded with a variety of therapeutic payloads – anti-cancer molecules (docetaxel, paclitaxel, doxorubicin), anti-inflammatory molecules (curcumin, diclofenac, celecoxib) and small biologicals (peptides, siRNAs, miRNAs); and imaging agents – optical probes; Gd and iron oxide nanoparticles for MR imaging; and radio-isotopes for Nuclear Imaging.
Furthermore, the lecture presents on the fabrication and testing of microfluidic chips for analyzing the vascular and extravascular mass transport, over multiple spatial and temporal scales, of molecules, nanoparticles and cells. These microfluidic-chips can help in elucidating biophysical mechanisms modulating progression and regression of various diseases, such as cancer, cardiovascular and neurodegenerative, as well as in developing organ-on-a-chip systems for rapid drug and nanomedicine screenings and tissue regeneration.

Contact: paolo.zunino@polimi.it

8 Aprile, 2016 10:30

MOX Seminar

Mathematical modelling of hydraulic fracturing and related problems

Sergey V. Golovin, Lavrentyev Institute of Hydrodynamics - Novosibirsk State Universit, Russia

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

abstract

abstract

Hydraulic fracturing is an important part of modern technologies for intensification of hydrocarbon production. Propagation of the hydraulic fracture is stipulated by a high-pressure pumping of viscous fluid, which creates pressure on fracture's walls high enough to overcome the rock closure stresses and cause the rock failure. Process of the hydraulic fracture growth is governed by several factors: flow of viscous fluid with additives in a narrow fracture's gap, elastic reaction of the fracture's walls; filtration of fluid from the fracture to the reservoir, rock failure and advance of the fracture’s tip.
In the talk we discuss mathematical models of fracture propagation in a poroelastic inhomogeneous medium, model for estimation of the productivity of a horizontal multiple fractured well, and a related model for description of cerebral hemodynamics in application to modelling of arteriovenous malformations. On the base on exact and numerical solutions, we demonstrate the influence of the inhomogeneity of physical properties of the reservoir to the dynamics of fracture propagation. For a multiple fractured horizontal well we analyze the dependence of fractures positioning and geometry to the long-term production of the well. By using the model of cerebral hemodynamics, we explain some characteristic features of experimentally observed data for fluid pressure and velocity in blood vessels.(golovin@hydro.nsc.ru)

contact: anna.scotti@polimi.it

8 Aprile, 2016 14:30

Sezione di Geometria, Algebra e loro applicazioni

Misure di Carleson e operatori di Toeplitz in domini strettamente pseudoconvessi limitati

Samuele Mongodi, Università di Pisa

Aula seminari del terzo piano

abstract

abstract

Le misure di Carleson furono introdotte da Carleson per studiare il problema della corona; una misura $\mu$ è di Carleson per uno spazio di Banach $A$ di funzioni olomorfe se A si immerge con continuità in $L^p(\mu)$. Una delle prime domande che ci si pone è come caratterizzare le misure di Carleson tramite proprietà geometriche del dominio, preferibilmente invarianti per biolomorfismo.

Per un generico limitato strettamente pseudoconvesso, questo è stato fatto da Abate, Raissy e Saracco, stabilendo un legame tra le misure di Carleson, la trasformata di Berezin e gli operatori di Toeplitz, entrambi questi ultimi definiti in termini del nucleo di Bergman del dominio.

In questo seminario, presenterò questi risultati e parlerò di alcune generalizzazioni che sto studiando attualmente.

6 Aprile, 2016 12:15

Seminari di Cultura Matematica

Il ruolo del ricercatore nella società

Carlo Bottani, Politecnico di Milano

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

Le misure di Carleson furono introdotte da Carleson per studiare il problema della corona; una misura $\mu$ è di Carleson per uno spazio di Banach $A$ di funzioni olomorfe se A si immerge con continuità in $L^p(\mu)$. Una delle prime domande che ci si pone è come caratterizzare le misure di Carleson tramite proprietà geometriche del dominio, preferibilmente invarianti per biolomorfismo.

Per un generico limitato strettamente pseudoconvesso, questo è stato fatto da Abate, Raissy e Saracco, stabilendo un legame tra le misure di Carleson, la trasformata di Berezin e gli operatori di Toeplitz, entrambi questi ultimi definiti in termini del nucleo di Bergman del dominio.

In questo seminario, presenterò questi risultati e parlerò di alcune generalizzazioni che sto studiando attualmente.

6 Aprile, 2016 16:00

Dipartimento di Matematica del Politecnico di Milano

Nonlinear poro-elastic and poro-visco-elastic models: analysis and applications

Giovanna Guidoboni, PhD, Associate Professor, Mathematical Sciences, Indiana University and Purdue University at Indiana

Sala Consiglio - 7° piano - Dipartimento di Matematica

abstract

abstract

We consider the initial and boundary value problem for a system of partial differential equations describing the motion
of a fluid-solid mixture under the assumption of full saturation. The ability of the fluid phase to flow within the solid
skeleton is described by the permeability tensor, which is assumed here to be a multiple of the identity and to depend
nonlinearly on the volumetric solid strain.
In particular, we study the problem of existence of weak solutions in bounded domains, accounting for non-zero volumetric
and boundary forcing terms. We investigate the influence of viscoelasticity on the solution functional setting and on the
regularity requirements for the forcing terms. The theoretical analysis shows that different time regularity requirements
are needed for the volumetric source of linear momentum and the boundary source of traction depending on whether or not
viscoelasticity is present.
The theoretical results are further investigated via numerical simulations based on a novel dual mixed hybridized finite
element discretization. When the data are sufficiently regular, the simulations show that the solutions satisfy the energy
estimates predicted by the theoretical analysis.
Interestingly, the simulations also show that, in the purely elastic case, the Darcy velocity and the related fluid energy
might become unbounded if indeed the data do not enjoy the time regularity required by the theory.
Applications will be discussed in the case of ocular perfusion.

6 Aprile, 2016 17:00

Seminario Matematico e Fisico di Milano

On the Tate and Mumford-Tate conjectures for varieties with h^{2,0}=1

Ben Moonen, Universita'di Nijmegen

Sala di rappresentanza, Dipartimento di Matematica, Università degli Studi, Via C. Saldini 50

abstract

abstract

I will explain my proof of the Tate and Mumford-Tate conjecture for divisor classes on a variety with h^{2,0}=1 (e.g., surfaces with p_g=1), under a mild assumption on their moduli. The proof involves a combination of several new techniques, including a generalization of the Kuga-Satake construction and a new interpretation of 'half-twists' of CM-Hodge structures. I will start by explaining the method of Deligne-Andre for K3 surfaces, which is the main source of inspiration for the proof.

5 Aprile, 2016 14:30

MOX Seminar

A hybrid RANS/LES technique based on Reynolds stress tensor reconstruction

Antonella Abba', Dipartimento di scienze e tecnologie aerospaziali, Politecnico di Milano

Aula Seminari III Piano Dipartimento di Matematica, POLITECNICO DI MILANO

abstract

abstract

Both from industrial and research point of view, very often is required a good description of turbulent flows and their unsteady features. LES is able to represent an instantaneous picture of the flow, ensuring good accuracy, but it is very demanding in term of cost. In a fully resolved LES simulation of a wall bounded flow, the number of the grid points scales with the Reynolds number as N ? Re 1.76 , while for RANS the dependency concerns only the wall normal direction and the number of grid points scales as N ? lnRe. Nevertheless, we remark that RANS can only provide a description of the mean quantities. Therefore, we can identify a sort of chiasmus: LES is sufficiently detailed but too expensive, RANS is sufficiently cheap but too superficial. For these reasons, seems natural trying to combine the two approach, obtaining hybrid RANS/LES methods.
Applying the hybid filter to the Navier-Stokes equations it is possible to obtain a new set of formally correct hybrid equations. A new approach for the Reynolds stress tensor has been proposed in which no additional equations are required. This results in a very simple and cheap hybrid method, in which the RANS contribution is used to integrate, and not to substitute, LES in the context of coarse grid.

contact: nicola.parolini@polimi.it

4 Aprile, 2016 10:30

Dipartimento di Matematica del Politecnico di Milano

CANCELLATO causa motivi familiari --- Generalized parametric solutions for optimal design and patient-specific simulations

P. Diez, Universitat Politecnica de Catalunya BarcelonaTech, Spain

Aula Consiglio VII Piano Dipartimento di Matematica, Politecnico di Milano

abstract

abstract

SEMINARIO TENUTO NELL'AMBITO DEL CORSO DI DOTTORATO: "METODI NUMERICI AVANZATI NEL CALCOLO SCIENTIFICO"

Reduced order models are nowadays a standard approach to represent a solution with less degrees of freedom than in the original model. The large majority of available approaches requires first an offline calculation and, second, solving a problem in an online phase. A particular and singular feature of the Proper Generalized Decomposition (PGD) is that the offline computation provides a generalized parametric solution, also referred as computational vademecum, precluding the need of solving any problem in the second phase. This results from taking the parameters as if they were standard physical coordinates (space and/or time). Thus, user-prescribed parameters like material constants, external loads, geometry, etc. have a similar treatment as the usual coordinates. Consequently, the online phase consists merely in a functional evaluation and therefore is very efficient from a computational viewpoint. Moreover, the generalized parametric solution can be further exploited by making use of the explicit form of the parametric dependence (viz. Fourier transform, gradients for sensibilities, …). The separation in spatial dimensions (and time in some cases), material and geometrical parameters allows for a drastic reduction in computational costs.
This approach is extremely efficient in different applications. We discuss two cases in which the geometrical parameterization of the problem is accounted for in the computational vademecum: 1) optimal automotive shape design parameterizing the aerodynamic external flow, and 2) medical diagnosis and support for decision-making via real-time simulations based on patient-specific data.

contact: simona.perotto@polimi.it