05 Jun 2018

• Titolo: Cheeger N-clusters

• Sommario: We will give a brief overview of a long standing conjecture due to Caffarelli and Lin and that has recently been shown to be linked to the Pólya–Szegő (conjectured) inequality. The Cheeger N-clusters represent the natural framework in which the conjecture can be stated in a particular case, where it has been proven to be true.

28 May 2018

• Titolo: Non-differentiability sets of Lipschitz maps

• Sommario: The study of differentiability properties of Lipschitz functions has a long story. It started with H. Lebesgue who proved the almost everywhere differentiability of Lipschitz functions on the real line. In 1919 H. Rademacher understood that this almost everywhere differentiability was not just a property of the line itself, but a property of Lipschitz maps themselves. Such a beautiful result left a lot of questions open one of which is: does a vice versa hold for the Rademacher theorem, i.e., is it possible to give a characterization of non-differentiability sets of Lipschitz functions? In 1990 D. Preiss built a dense set in euclidean spaces on which every real valued Lipschitz function has a differentiability point. This amazing and counterintuitive result made it clear that a converse for the Rademacher's Theorem was not a straightforward problem at all. In the talk I will present the main ideas and techniques that have been originally introduced by G. Alberti, M. Csörnyei and D. Preiss in their (still) unpublished paper to study this problem. In particular, I will describe:

  • how the width function is defined,

  • some applications of the width function in the plane,

  • how to build a lot of non differentiable functions along every line, at any point of a given compact purely unrectifiable set.

07 May 2018

• Titolo: Modelling with measures: an examples on traffic flows

• Sommario: In the last decade, there has been a vivid research on measure‐valued solutions in PDEs and several applications. In this talk, I will discuss some possibile application of transport of measures theory. In particular, I will focus on measure‐valued solutions to transport on networks both in the linear and in the nonlinear nonlocal case. This setting makes our framework suitable to deal with multiscale flows of agents on networks in applications such as traffic management and distribution optimization. I will also show possible application, such as smart traffic lights or self‐driving cars.

20 May 2018

• Titolo: Gaussian Process Emulation to Reconstruct Past Greenland Ice Sheet Morphologies

• Sommario: Gaussian Process Emulation is a probabilistic methodology often employed to understand the behavior of highly complex computer models, where the number of observed input-output relationships is very small due to the long running times of the model. Emulation allows to create fast-to-run probabilistic surrogates of the complex simulator, and to make probabilistic inference of its underlying dynamics.
  In this talk, I will introduce emulation and some of its applications within the context of the geophysical problem of reconstructing the morphology of Greenland Ice Sheet (GIS) during the Last-Interglacial period, when temperatures were warmer than today. The problem is of vital importance to climatologists and Earth scientists in that it helps assess the impact of GIS melting to world sea-level rise.
  This is joint work with Louise Sime and Irene Malmierca from the British Antarctic Survey in Cambridge.

23 Apr 2018

• Titolo: Archimedes, a dinner and a theorem. A divertissement on the monotonicity of perimeter

• Sommario: If A and B are two convex bodies in the Euclidean n-dimensional space and A is contained in B, then the perimeter of A does not exceed the perimeter of B. This monotonicity property of the perimeter dates back to the ancient Greek and Archimedes himself took it as a postulate in his celebrated work on the sphere and the cylinder. A few years ago, a couple of papers by M. Carozza, F. Giannetti, F. Leonetti, and A. Passarelli di Napoli established lower bounds on the difference of the perimeters of A and B in terms of their Hausdorff distance when n=2 and n=3. In this talk, after a brief introduction on the problem and the known results, I will generalise these lower bounds to any dimension n. Time permitting, I will show how this approach can be extended to the case of anisotropic Wulff perimeters.

• Download: note del seminario a cura del relatore.

16 Apr 2018

• Titolo: Stochastic vorticity equation: existence and uniqueness results in the flat torus and in the whole plane

• Sommario: link.

• Slides: link.

09 Apr 2018

• Titolo: Regress later Monte-Carlo for optimal control of Markov processes

• Sommario: We develop two Regression Monte Carlo algorithms (value and performance iteration) to solve general problems of optimal stochastic control of discrete-time Markov processes. We formulate our method within an innovative framework that allow us to prove the speed of convergence of our numerical schemes. We rely on the Regress Later approach unlike other attempts which employ the Regress Now technique. We exploit error bounds obtained in our proofs, along with numerical experiments, to investigate differences between the value and performance iteration approaches. Both introduced in 2001, their characteristics have gone largely unnoticed in the literature; we show however that their differences are paramount in practical solution of stochastic control problems. Finally, we provide some guidelines for the tuning of our algorithms. During the seminar I will also provide a general introduction to the numerical solution of stochastic control problems with a focus on least-squares approximation of conditional expectations.

12 Mar 2018

• Titolo: Weyl's law

• Sommario: In a series of papers published around 1912, H. Weyl established an asymptotic formula for the eigenvalues of the Laplacian of bounded domains in 2 and 3 dimensions. His result, which turns out to be very useful in geometric analysis, was later on extended to bounded domains in any dimension, then to compact manifolds. In this talk, I will first explain the physical motivation of this result, namely the black body radiation problem, and then present a proof involving the so-called trace of the heat kernel.

06 Mar 2018

• Titolo: Optimal control of pure jump Markov processes with noise-free partial observation

• Sommario: link.

26 Feb 2018

• Titolo: Il funzionale di Willmore e il metodo diretto dell'approccio varifold

• Sommario: L'energia di Willmore di una superficie regolare è l'integrale di superficie del modulo quadro della sua curvatura media e il funzionale di Willmore è la funzione che assegna ad una superficie la rispettiva energia di Willmore. Presenteremo alcuni risultati classici sullo studio variazionale di questa quantità geometrica, mettendo in luce le difficoltà che sorgono quando il funzionale non è definito su uno spazio di funzioni, come in questo caso. Questo ci porterà alla definizione di varifold, che useremo come indebolimento della definizione di superficie regolare, e alla nozione di convergenza di varifold. Presenteremo un importante teorema di compattezza per varifold e mostreremo come questo si applichi al problema di minimo per il Willmore.

• Download: slides.

12 Feb 2018

• Titolo: Primal-Dual Active Set Strategy per problemi di controllo ottimo

• Sommario: La Primal-Dual Active Set Strategy (PDASS) è una delle strategie impiegate per risolvere problemi di controllo ottimo di equazioni differenziali alle derivate parziali con vincoli puntuali sulle variabili di stato e sui controlli. In questo caso, viene applicata ad un modello parabolico che descrive l'evoluzione della temperatura all'interno di una stanza. In tale modello, ogni controllo rappresenta la temperatura di eventuali sorgenti di calore (per es. radiatori, riscaldamento a pavimento,...) ed è pertanto soggetto a vincoli fisici. La PDASS è, quindi, applicata con l' obiettivo di minimizzare il costo del riscaldamento mantenendo la temperatura intorno ad un valore desiderato. Simulazioni numeriche dimostrano l' efficienza di tale metodo per l' esempio considerato.
   

Referenze:

[1] K. Kunisch, A. Rösch, Primal-Dual Active Set Strategy for a General Class of Constrained Optimal Control Problems. SIAM J. Optim., 13(2), 321–334.
[2] L. Mechelli, S. Volkwein, POD-Based Economic Optimal Control of Heat-Convection Phenomena. Submitted (2017), preprint download at: http://nbn-resolving.de/urn:nbn:de:bsz:352-2--au6ei3apyzpv0

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• Download: slides.

05 Feb 2018

• Titolo: Applicazione del trasporto ottimo ad un modello di traffico congestionato

• Sommario: The presentation starts with a review of the general theory of optimal transport, in which the most important results and their consequences are recalled. Then, this theory is applied to the study of some traffic models. The starting point is a model introduced by Beckmann in the ’50s, called continuous model of transportation, which takes the form of a divergence constraint optimization problem. The Beckmann’s problem is studied in relation to a proper Monge-Kantorovich one, and it is shown that, under some hypothesis, they admit the same solution. Consequently, the model is complicated in order to take care of congestion effects, understanding if there are still connections with the optimal transport theory and if a meaningful notion of equilibrium can be found. Avoiding deep technicalities, that are only mentioned, an interesting result is presented: the solution of the congestion traffic model solves a peculiar Monge-Kantorovich one, where the cost function depends on the solution of the problem itself. Moreover, the solution also satisfies a very famous notion of non-cooperative equilibrium.

• Download: slides.

30 Jan 2018

• Titolo: A variational approach to the mean field planning problem

• Sommario: In the talk we introduce the so-called mean field planning problem: a coupled system of PDEs, a forward continuity equation and a backward Hamilton-Jacobi
  equation. The problem can be viewed as a modification of the mean field games
  system as well as a generalization of the classical optimal transportation problem in
  its dynamic formulation à la Benamou-Brenier. We concentrate on the variational
  structure of the problem, from which a notion of weak solution can be given. In
  particular, we discuss a well-posedness result in a Lp -framework, as well as optimality
  conditions at the level of minimizing paths.
  The talk is based on a joint work with A. Porretta and G. Savaré.