3 Febbraio, 2017 12:00 oclock

Sezione di Probabilità e Statistica Matematica

MIsure invarianti per l'equazione stocastica di Navier-Stokes nel piano/ Invariant measures for Navier Stokes equation in the plane

Benedetta Ferrario, Universita' di Pavia

Aula seminari del terzo piano

abstract

abstract

Si presenta un risultato di esistenza di una misura invariante per l'equazione di Navier-Stokes nel piano con forzante di tipo rumore bianco nel tempo. Poiche' si lavora in un dominio spaziale illimitato, l'immersione dello spazio di Hilbert H^1 in H^0 non e` compatta e questo richiede una tecnica diversa dal caso di dominio spaziale finito per la dimostrazione dell'esistenza di una misura invariante. Invece del metodo di Krylov e Bogoliubov si ricorre al metodo di Maslowski e Seidler
(On sequentially weakly Feller solutions to SPDE's. Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 10 (1999), no. 2, 69-78).
Questo lavoro e` in collaborazione con Z. Brzezniak (Universita` di York).

The existence of an invariant measure for 2d Navier Stokes with white noise (in time) forcing term is given. Due to the unboundedness of the domain a different technique in proving this existence result used. In fact, rather then using Krylov-Bogoliubov method we follow the approach introduced by Maslowsly and Seidler (On sequentially weakly Feller solutions to SPDE's. Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 10 (1999), no. 2, 69-78).
This is a joint paper with Z. Brzezniak, University of York.

1 Febbraio, 2017 13:15

Sezione di Analisi

Isoperimetric inequality for metric measure spaces

Fabio Cavalletti, SISSA - Scuola Internazionale Superiore di Studi Avanzati

Aula seminari 3° piano

abstract

abstract

By using an L^1 localization argument, we prove that in metric measure spaces satisfying lower Ricci curvature bounds (more precisely RCD*(K,N) or more generally essentially non branching CD*(K,N) the classical Lévy-Gromov isoperimetric inequality holds with the associated rigidity and almost rigidity statements.

1 Febbraio, 2017 15:00

Seminari FDS

Scienza, Ricerca e società: quale informazione possibile? quale informazione credibile?

Ludovica Carlesi Manusardi, Giornalista scientifico

Aula Consiglio, 7° piano, edificio 14 Nave – Via Bonardi, 9 – Milano

abstract

abstract

L'Informazione scientifica oggi non può limitarsi a una mera divulgazione amplificata di quello che avviene nel mondo della ricerca. In una società complessa e interconnessa compito dei media è quello di cercare di rendere consapevole il grande pubblico in modo onesto e documentato dei risultati e delle ricadute che la ricerca e il progresso scientifico hanno nella vita quotidiana e nelle eventuali scelte che ne conseguono.

30 Gennaio, 2017 16:00

Seminario Matematico e Fisico di Milano

Dissipative Dynamics for Large Interacting Systems

Boguslaw Zegarlinski, Imperial College London

Aula Seminari del 6 piano

abstract

abstract

I will review recent results and not so recent open problems concerning
constructions and ergodicity of dissipative dynamics for classical and noncommutative large interacting systems. In particular I will talk about dynamics defined by formally hypoelliptic generators and some generalised Dunkl type extensions, and how to control their long time behaviour (by use of coercive inequalities or generalised gradient type bounds).

30 Gennaio, 2017 11:00

MOX Seminar

Analysis of the parallel Schwarz method for growing chains of fixed-sized subdomains

Gabriele Ciaramella, Universita' di Ginevra

Aula Seminari Saleri VI Piano MOX Dipartimento di Matematica, Edificio 14 Politecnico di Milano

abstract

abstract

A new class of Schwarz methods was recently presented in the literature for the solution of solvation models, where the electrostatic energy contribution to the solvation energy can be computed by solving a system of elliptic partial differential equations [1,2]. Numerical simulations have shown an unusual convergence behaviour of Schwarz methods for the solution of this problem, where each atom corresponds to a subdomain: the convergence of the Schwarz methods is indenpendent of the number of atoms [1], even though there is no coarse grid correction. Despite the successful implementation of Schwarz methods for this solvation model, a rigorous analysis for this unusual convergence behaviour is required, since no theoretical results are given in the corresponding literature. In this talk, we analyze the behavior of the Schwarz method for the solution of a chain of atoms and show that its convergence does not depend on the number of atoms (subdomains). We use two di fferent techniques to prove this result. The first technique is based on a Fourier expansion of the error and the analysis of transfer matrices constructed for an approximate model. The second one consists in an application of the maximum-principle and allows us to analyze very general geometries.

[1] Canc es et al., Domain decomposition for implicit solvation models, J. of Chem.
P. (2013).
[2] Lipparini et al., Fast Domain Decomposition Algorithm for Continuum
Solvation Models, J. Chem. Theory Comput. (2013).

contact: marco.verani@polimi.it

25 Gennaio, 2017 17:15 oclock

Sezione di Analisi

A Bernstein-type result for the minimal surface equation

Alberto Farina, Université Picardie Jules Verne

Aula seminari 6° piano

abstract

abstract

We prove the following Bernstein-type theorem: if $u$ is a solution to the minimal surface equation over $R^N$, such that $N-1$ partial derivatives are bounded on one side (not necessarily the same), then $u$ is an affine function. Besides its novelty, our theorem also provides a new, simple and self-contained proof of celebrated results of J. Moser and of E. Bombieri e E. Giusti.

25 Gennaio, 2017 15:00

Seminari FDS

Viaggi e divulgazione: incontro con Syusy Blady

Maurizia Giusti, Conduttrice televisiva e cabarettista

Aula Rogers, Via Ampere 2 – Milano

abstract

abstract

Un incontro con Syusy Blady (pseudonimo di Maurizia Giusti) la nota conduttrice televisiva delle trasmissioni della serie "Turisti per caso".
Syusy ci parlera' di comunicazione e divulgazione tramite il turismo, ma anche di un argomento affascinante che ha avuto modo di approfondire nel corso dei suoi viaggi, le mura megalitiche poligonali.

19 Gennaio, 2017 11:00

MOX Seminar

Two novel classes of Discontinuous Galerkin Finite Element Methods for Time-Dependent PDE

Michael Dumbser, University of Trento

AULA CONSIGLIO VII PIANO - Edificio 14 La Nave - Dipartimento di Matemcatica MOX, Politecnico di Milano

abstract

abstract

The topic of this talk are two new classes of high order discontinuous Galerkin (DG) finite element methods for the numerical solution of time-dependent partial differential equations (PDE) in two and three space dimensions.
The first class of schemes (semi-implicit DG methods on staggered grids) has been developed for the incompressible and the compressible Navier-Stokes equations. The key novelty is the use of staggered grids within a DG scheme. The special location of the data (pressure and velocity) on the staggered grid allows a significant simplification of the computation of the numerical fluxes across elements. Compared to standard collocated DG schemes, the linear system to be solved for the pressure has optimal properties: it is symmetric positive definite, has the minimum number of
unknowns and uses the smallest possible stencil. The second class of methods (explicit ADER-DG schemes on collocated grids) is a fully-discrete predictor-corrector scheme with piecewise polynomial basis functions in space and time. Via a weak formulation of the PDE in spacetime, the predictor constructs an approximate solution of an element-local Cauchy problem in the small, i.e. based only on element-local initial data. For that reason, time consuming MPI communication can be avoided during the arithmetically intensive predictor step. The numerical fluxes across element interfaces are then taken into account in a subsequent corrector step. The new method is therefore particularly suitable for implementation on modern supercomputers. Furthermore, it allows the use of local time steps in a very natural and easy way, while remaining fully conservative and high order accurate in both space and time. Shock waves
and other discontinuities are treated via a new a posteriori limiter, which does not rely on nonlinear data reconstruction or filtering, as usual DG limiters, but which solves the PDE again, using a more robust finite volume scheme on a locally refined subgrid. We show some examples from the fields of computational fluid dynamics, magneto-hydrodynamics (MHD) and continuum mechanics.

References:
M. Dumbser, O. Zanotti, R. Loubère and S. Diot. A Posteriori Subcell Limiting of the Discontinuous Galerkin
Finite Element Method for Hyperbolic Conservation Laws. Journal of Computational Physics, 278:47–75, 2014
O. Zanotti, F. Fambri and M. Dumbser. Solving the relativistic magnetohydrodynamics equations with ADER
discontinuous Galerkin methods, a posteriori subcell limiting and adaptive mesh refinement. Monthly Notices
of the Royal Astronomical Society (MNRAS), 452:3010-3029, 2015
M. Dumbser, I. Peshkov, E. Romenski and O. Zanotti. High order ADER schemes for a unified first order
hyperbolic formulation of continuum mechanics: viscous heat-conducting fluids and elastic solids. Journal of
Computational Physics 314:824–862, 2016
M. Dumbser and R. Loubère. A simple robust and accurate a posteriori subcell finite volume limiter for the
discontinuous Galerkin method on unstructured meshes. Journal of Computational Physics, 319:163–199, 2016
M. Tavelli and M. Dumbser. A staggered space-time discontinuous Galerkin method for the three-dimensional
incompressible Navier-Stokes equations on unstructured tetrahedral meshes. Journal of Computational Physics,
319:294–323, 2016

contact: christian.vergara@polimi.it

18 Gennaio, 2017 17:15 oclock

Sezione di Analisi

Risultati di approssimazione e loro applicazioni al Digital Image Processing per la diagnosi di patologie vascolari

Gianluca Vinti, Università di Perugia

Aula seminari 6° piano

abstract

abstract

Viene mostrato come alcuni risultati di approssimazione tramite operatori di tipo campionamento svolgono un ruolo importante nell’ambito della ricostruzione di segnali/immagini e vengono discusse alcune loro applicazioni ad immagini CT per la diagnosi di patologie vascolari dell’aorta addominale.

20 Dicembre, 2016 14:00

MOX Seminar

Recent developments on Hybrid High-Order methods for nonlinear problems

Daniele Di Pietro, University of Montpellier, Institut Montpellierain Alexander Grothendieck (FRANCIA)

Aula Seminari Saleri, VI piano Ed. 14 La Nave MOX -Dipartimento di Matematica, Politecnico di Milano

abstract

abstract

Hybrid High-Order (HHO) methods are a class of new generation numerical schemes for PDEs with several advantageous features, including: (i) support of general polytopal meshes in arbitrary space dimension, (ii) arbitrary approximation order; (iii) compliance with the physics, including robustness with respect to the variations of physical coefficients and reproduction of key continuous properties at the discrete level; (iv) reduced computational cost thanks to hybridization, static condensation, and compact stencil.
In this presentation we address some recent developments concerning the application of HHO methods to nonlinear problems

contact: christian.vergara@polimi.it