1 Ottobre, 2009 11:00

Modelli matematici e vulcani

Augusto Neri, Istituto Nazionale di Geofisica e Vulcanologia Sezione di Pisa

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

abstract

abstract

Negli ultimi 20 anni la vulcanologia ha conosciuto un significativo avanzamento delle conoscenze grazie all’introduzione di modelli matematici nello studio dei processi vulcanici. Prima l’utilizzo di semplici modelli stazionari e 1D ha permesso di riconoscere i principali meccanismi che regolano la dinamica delle eruzioni vulcaniche. Successivamente, con l’avvento di computer sempre più veloci, lo sviluppo e l’applicazione di modelli 2D/3D transienti e multifase ha consentito di realizzare un ulteriore passo avanti permettendo di descrivere sempre meglio la complessità del fenomeno naturale. In questa presentazione verranno presentati alcuni esempi di modelli matematici utilizzati nella simulazione numerica dei processi vulcanici esplosivi e nella stima della loro pericolosità. In particolare verranno presentate alcune applicazioni ai vulcani attivi italiani, come il Vesuvio e l’Etna, nonché discussi i possibili sviluppi futuri di queste ricerche.

29 Settembre, 2009 10:00

Reduced Basis Approximation and A Posteriori Error Estimation for Parametrized Partial Differential Equations

Anthony Patera, MIT - Massachusetts Institute of Technology

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

abstract

abstract

We discuss reduced basis approximation and associated a posteriori error estimation for reliable and rapid solution of parametrized partial differential equations.

The crucial ingredients are rapidly convergent Galerkin approximations over a space spanned by snapshots on the parametrically induced solution manifold; efficient POD (in time)/Greedy (in parameter) selection of quasi-optimal samples;
effective Successive Constraint Method construction of stability-constant lower bounds; rigorous and sharp a posteriori error estimators for the norms and outputs/quantities of interest; and Offline-Online computational procedures for very rapid response in the real-time and many-query contexts.

In this talk we first describe the technical ingredients in the context of linear parabolic equations. We then present illustrative
results for a broad range of elliptic, parabolic, and hyperbolic examples drawn from heat transfer and mechanics.

29 Settembre, 2009 16:30

Lezioni Leonardesche

Optimal Transportation in the 21st century

Neil Trudinger, Australian National University

Dipartimento di Matematica, Università di Milano - Via Saldini 50, Aula Chisini

We discuss reduced basis approximation and associated a posteriori error estimation for reliable and rapid solution of parametrized partial differential equations.

The crucial ingredients are rapidly convergent Galerkin approximations over a space spanned by snapshots on the parametrically induced solution manifold; efficient POD (in time)/Greedy (in parameter) selection of quasi-optimal samples;
effective Successive Constraint Method construction of stability-constant lower bounds; rigorous and sharp a posteriori error estimators for the norms and outputs/quantities of interest; and Offline-Online computational procedures for very rapid response in the real-time and many-query contexts.

In this talk we first describe the technical ingredients in the context of linear parabolic equations. We then present illustrative
results for a broad range of elliptic, parabolic, and hyperbolic examples drawn from heat transfer and mechanics.

28 Settembre, 2009 12:30

Lezioni Leonardesche

Anisotropic mesh adaptation: from continuous models to practical mesh generation. Applications in Computational Fluid dynamics (CFD)

Adrien Loseille, Computational and Data Sciences Center for computational fluid dynamics George Mason University

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

abstract

abstract

This talk addresses classical issues that arise when applying anisotropic mesh adaptation
to real-life 3D problems as the loss of anisotropy or the necessity to truncate the minimal size when discontinuities are present in the solution. These problematics are due to the complex interaction between the components involved in the adaptive loop: the flow solver, the error estimate and the mesh generator.
A solution based on a new continuous mesh framework is proposed to overcome these issues.
We show that using this strategy allows an optimal level of anisotropy to be reached and thus enjoy the full benefit of unstructured anisotropic mesh adaptation: optimal distribution of the degrees of freedom, improvement of the ratio accuracy with respect to cpu time, ...
Some simple strategies to effectively generate an anisotropic mesh will be also given.
One application will concern the generation of boundary layers grids within an anisotropic
adaptive process.

21 Settembre, 2009 17:30

Seminario Matematico e Fisico di Milano

Asymptotic analysis in thermodynamics of viscous fluids

Eduard Feireisl, Mathematical Institute, Czech Academy of Sciences

Dipartimento di Matematica, Politecnico di Milano, Aula Consiglio VII piano

abstract

abstract

We discuss some recent results in the mathematical theory of complex fluid systems, in particular, the concept of weak solution and its relevance to a proper formulation of balance laws in fluid mechanics.
A rigorous asymptotic analysis of these systems is developed and several model problems discussed.
In particular, the following topics will be addressed in detail:

functional analytic framework and the basic ideas of the mathematical theory of continuum fluid mechanics;

thermodynamic stability and equilibrium states, behavior of energetically insulated systems for large time;

low Mach number limits and propagation of acoustic waves in thermally conducting viscous fluids.

8 Settembre, 2009 16:30

Lezioni Leonardesche

Topology and Geometry of the Space of Closed Curves in a Manifold

Dennis Sullivan, CUNY Graduate Center and SUNY-Stony Brook

Dipartimento di Matematica - Università degli Studi di Milano, Via Saldini 50 - Aula Chisini

We discuss some recent results in the mathematical theory of complex fluid systems, in particular, the concept of weak solution and its relevance to a proper formulation of balance laws in fluid mechanics.
A rigorous asymptotic analysis of these systems is developed and several model problems discussed.
In particular, the following topics will be addressed in detail:

functional analytic framework and the basic ideas of the mathematical theory of continuum fluid mechanics;

thermodynamic stability and equilibrium states, behavior of energetically insulated systems for large time;

low Mach number limits and propagation of acoustic waves in thermally conducting viscous fluids.

22 Luglio, 2009 15:00

Lezioni Leonardesche

Variational Problems in Weighted Sobolev Spaces with Applications to Computational Fluid Dynamics

Ana Maria Soane, Department of Mathematics and Statistics, University of Maryland, Baltimore County (UMBC)

Aula Seminari F. Saleri VI Piano MOX- Dipartimento di Matematica, Politecnico di milano

abstract

abstract

We study variational problems in weighted Sobolev spaces on bounded domains with corners. The specific forms of these variational formulations are motivated by, and applied to, a finite element scheme for the time-dependent Navier-Stokes equations. Specifically, we introduce new variational formulations for the Poisson and Helmholtz problems in what would be a weighted counterpart of H^2 intersection H^1_0, show the existence and uniqueness of solutions, and establish the relationship with the traditional H^1 formulations. We formulate a conforming
C^1 finite element method for solving these variational problems and establish optimal convergence rates. We apply our methods to the time-dependent Navier-Stokes equations on non-convex polygonal domains by adapting an iterative algorithm due to Liu, Liu, and Pego (Comm. Pure Appl. Math., 60 (2007)). The well-posedness of the modified algorithm follows from our analysis of the Poisson and Helmholtz problems. Numerical results for several benchmark
problems are provided.

8 Luglio, 2009 11:00

Lezioni Leonardesche

A mathematical model for bioremediation processes of polluted soils.

Antonio Fasano, Universita di Firenze

Aula Seminari F. Saleri VI Piano Mox - Dipartimento di Matematica, Politecnico di Milano

abstract

abstract

Bioremediation of soils exploits selected species of bacteria able to metabolize a specific pollutant (typically a hydrocarbon). Bacteria are injected and proliferate creating a biofilm over the soil scheleton where they feed on the pollutant. We consider in particular the case in which bacteria are transported by an unsaturated flow, which, despite its importance, has not been treated extensively in the literature. The mathematical problem consists of a system of coupled nonlinear partial differential equations for which an existence theorem is proved using a technique of nested fixed points.

2 Luglio, 2009 14:00

Sezione di Geometria, Algebra e loro applicazioni

Geometry of moduli spaces of foliations in projective spaces

Fernando Cukierman, Universidad de Buenos Aires

aula seminari del III piano

Bioremediation of soils exploits selected species of bacteria able to metabolize a specific pollutant (typically a hydrocarbon). Bacteria are injected and proliferate creating a biofilm over the soil scheleton where they feed on the pollutant. We consider in particular the case in which bacteria are transported by an unsaturated flow, which, despite its importance, has not been treated extensively in the literature. The mathematical problem consists of a system of coupled nonlinear partial differential equations for which an existence theorem is proved using a technique of nested fixed points.

22 Giugno, 2009 17:00

Seminario Matematico e Fisico di Milano

Numerical calculus with infinite and infinitesimal numbers and the Infinity computer

Yaroslav D. Sergeyev, Università della Calabria

Dipartimento di Matematica, Università di Milano, Sala di Rappresentanza

abstract

abstract

The lecture introduces a new methodology allowing one to execute numerical computations with finite, infinite, and infinitesimal numbers. It is based on the principle The part is less than the whole introduced by Ancient Greeks and applied to all numbers (finite, infinite, and infinitesimal) and to all sets and processes (finite and infinite). It is shown that it becomes possible to write down finite, infinite, and infinitesimal numbers by a finite number of symbols as particular cases of a unique framework different from those of the non-standard analysis and surreal numbers. The new approach gives possibility to introduce a new type of a computer - the Infinity Computer - able to operate numerically not only with finite numbers but also with infinite and infinitesimal ones (European Patent has been granted recently). The Infinity Calculator using the Infinity Computer technology is presented during the talk. An additional information can be downloaded from the page www.theinfinitycomputer.com