### Seminari

### Prossimi Seminari

- Deep Learning meets Parametric Partial Differential Equations

Gitta Kutyniok, Institute of Mathematics, Technische Universität Berlin (DE)

giovedì 16 luglio 2020 alle ore 14:00, Online seminar: https://mox.polimi.it/elenco-seminari/?id_evento=1977&t=763724

### Seminari Passati

- Semi-implicit finite-volume integrators for all-scale atmospheric dynamics

Piotr Smolarkiewicz, European Centre for Medium-Range Weather Forecasts, Reading, Berkshire, United Kingdom

giovedì 30 maggio 2019 alle ore 14:00, Aula Consiglio VII Piano - Edificio 14, Dipartimento di Matematica POLITECNICO DI MILANOABSTRACTThis talk outlines a novel numerical approach for accurate and computationally efficient integrations of PDEs governing all-scale atmospheric dynamics. Such PDEs are not easy to handle, due to a huge disparity of spatial and temporal scales as well as a wide range of propagation speeds of natural phenomena captured by the equations. Moreover, atmospheric dynamics constitutes only a small perturbation about dominant balances that result from the Earth gravity, rotation, composition of its atmosphere and the energy input by the solar radiation. Maintaining this mean equilibrium, while accurately resolving the perturbations, conditions the design of atmospheric models and subjects their numerical procedures to stringent stability, accuracy and efficiency requirements.

The novel Finite-Volume Module of the Integrated Forecasting System (IFS) at ECMWF (hereafter IFS-FVM) solves perturbation forms of the fully compressible Euler/Navier-Stokes equations under gravity and rotation using non-oscillatory forward-in-time semi-implicit time stepping and finite-volume spatial discretisation. The IFS-FVM complements the established semi-implicit semi-Lagrangian pseudo-spectral IFS (IFS-ST) with the all-scale deep-atmosphere formulation cast in a generalised height-based vertical coordinate, fully conservative and monotone advection, flexible horizontal meshing and a predominantly local communication footprint. Yet, both dynamical cores can share the same quasi-uniform horizontal grid with co-located arrangement of variables, geospherical longitude-latitude coordinates and physics parametrisations, thus facilitating their synergetic relation.

The focus of the talk is on the mathematical/numerical formulation of the IFS-FVM with the emphasis on the design of semi-implicit integrators and the associated elliptic Helmholtz problem. Relevant benchmark results and comparisons with corresponding IFS-ST results attest that IFS-FVM offers highly competitive solution quality and computational performance.

Contact: luca.bonaventura@polimi.it

- Comunicare il progetto. Storytelling e tecniche di rappresentazione

Francesca Piredda, Politecnico di Milano

mercoledì 29 maggio 2019 alle ore 12:15, Politecnico di Milano Campus Bonardi Edificio14 aula B21 - Game-theoretical models of debt and bankruptcy

Alberto Bressan, Penn State University

martedì 28 maggio 2019 alle ore 17:00, Aula seminari del terzo pianoABSTRACTThe talk will be concerned with problems of optimal debt management. In a basic model, the interest rate as well as the bankruptcy risk are given a priori. In this case the borrower faces a standard problem of optimal control.

In alternative, debt management can be modeled as a noncooperative game between a borrower and a pool of lenders, in infinite time horizon with exponential discount. The yearly income of the borrower is governed by a stochastic process. When the debt-to-income ratio surpasses a given threshold, bankruptcy occurs.

The interest rate charged by the risk-neutral lenders is precisely determined in order to compensate for this possible loss of part of their investment.

Existence and properties of optimal feedback strategies for the borrower will be discussed, in a stochastic framework as well as in the limit deterministic setting. - Multiple solutions for the 2-dimensional Euler equations

Alberto Bressan, Pennsylvania State University

lunedì 27 maggio 2019 alle ore 16:00 precise, Aula U5-3014 (Edificio 5 terzo piano), Dip. Matematica e Applicazioni, Via Cozzi 55, MilanoABSTRACTIn one space dimension, it is well known that hyperbolic conservation

laws have unique entropy-admissible solutions, depending continuously on

the initial data. Moreover, these solutions can be obtained as limits of

vanishing viscosity approximations.

For many years it was expected that similar results would hold in

several space dimensions. However, fundamental work by De Lellis,

Szekelyhidi, and other authors, has shown that multidimensional

hyperbolic Cauchy problems usually have infinitely many weak solutions.

Moreover, all known entropy criteria fail to select a single admissible one.

In the first part of this talk I shall outline this approach based on a

Baire category argument, yielding the existence of infinitely many weak

solutions.

I then wish to discuss an alternative research program,

aimed at constructing multiple solutions to some specific Cauchy

problems. Starting with some numerical simulations, here the eventual

goal is to achieve rigorous, computer-aided proofs of the existence of

two distinct self-similar solutions with the same initial data.

While solutions obtained via Baire category have turbulent nature, these

self-similar solutions are smooth, with the exception of one or two

points of singularity. They are thus much easier to visualize and

understand. - Matematica, società, economia e sviluppo

Giulia di Nunno, University di Oslo

mercoledì 22 maggio 2019 alle ore 12:15, Politecnico di Milano Campus Bonardi Edificio14 aula B21 - Mathematical Models of Markovian Dephasing

Franco Fagnola, Politecnico di Milano

venerdì 17 maggio 2019 alle ore 11:30 precise, Aula Seminari III pianoABSTRACTWe develop a notion of dephasing under the action of a quantum? Markov semigroup in terms of convergence of operators to a block-diagonal? form determined by irreducible invariant subspaces. If the latter are all ?one-dimensional, we say the dephasing is maximal. We study characterization ?of a maximally dephasing evolution in terms of unitary dilations ?with only classical noise. In particular, we introduce an intrinsic? quantity constructed from the generator which quantities the? degree of obstruction to having a classical diffusive noise model.? (Joint work with J.E. Gough, H.I. Nurdin and L. Viola)? - Forma e complessità in Natura: perché il mondo è matematico?

Pasquale Ciarletta, Politecnico di Milano

mercoledì 15 maggio 2019 alle ore 12:15, Politecnico di Milano Campus Bonardi Edificio14 aula B21 - How to hear the shape of a drum

Fabio Cipriani, Politecnico di Milano

martedì 14 maggio 2019 alle ore 15:30, Aula seminari 3° pianoABSTRACTIn a iconic 1912 paper Hermann Weyl, motivated by problems posed by the physicist H.A. Lorentz about J.H. Jeans's radiation theory, showed that the dimension and the volume of a Euclidean domain may be traced from the asymptotic distribution of the eigenvalues of its Laplace operator.

In a as much famous 1966 paper titled "Can one hear the shape of a drum" Marc Kac popularized this and related problems connecting geometry and spectrum. He noticed that the hope to characterize {\it isometrically}, Euclidean domains or compact Riemannian manifolds by the spectrum of the Laplace operator, is vain: John Milnor in 1964 had showed the existence of non isometric 16 dimensional tori sharing a common (discrete) spectrum.

The aim of the talk is to show how to recognize {\it conformal maps} between Euclidean domains as those homeomorphisms which transform multipliers of the Sobolev-Dirichlet spaces of a domain into multipliers of the other and leave invariant the {\it fundamental tone} or {\it first nonzero eigenvalue} of the Dirichlet integral with respect to the energy measures of any multiplier. Related results hold true for {\it quasiconformal and bounded distortion maps}.

In the opposite direction, we prove that the trace of the Dirichlet integral, with respect to the energy measure of a multiplier, is a Dirichlet space that only depends upon the orbit

of the conformal group of the Euclidean space on the multiplier algebra.

The methods involve potential theory of Dirichlet forms (changing of speed measure, multipliers) and the Li-Yau conformal volume of Riemannian manifolds.

This is a collaboration with Jean-Luc Sauvageot C.N.R.S. France et Universit\'e Paris 7.