Dipartimento di Matematica

Abstract

We give a complete classification of the Markovian self-adjoint extensions of the minimal realization of a second order elliptic differential operator on a bounded n-dimensional domain by providing an explicit one-to-one correspondence between such extensions and the class of Dirichlet forms on the boundary which are additively decomposable by the bilinear form of the Dirichlet-to-Neumann operator plus a Markovian form. By such a result two further equivalent classifications are provided: the first one is expressed in terms of a decomposition of the bilinear forms associated to the extensions, the second one uses the decomposition of the resolvents provided by the Krein formula. The Markovian part of the decomposition allows to characterize the operator domain of the corresponding extension in terms of Wentzell-type boundary conditions. Some analogous results hold also in a nonlinear setting.

Abstract

I describe some conjectures on smoothing toric Fano 3-folds
motivated by mirror symmetry.

Abstract

The image quality of ground based astronomical telescopes suffers from turbulences in the atmosphere. Adaptive Optics (AO) systems use wavefront sensor measurements of incoming light from guide stars to determine an optimal shape of deformable mirrors (DM) such that the image of the scientific object is corrected after reflection on the DM(s). The operation of the AO systems of the new generation of large astronomical telescopes under construction requires new mathematical methods that achieve reconstruction in real time. In the talk we will in particular focus on methods for wavefront reconstruction from pyramid sensor data and an analysis of the atmospheric tomography operator.

Contact christian.vergara@polimi.it

Abstract

Some aspects of the global regularity of solutions to boundary value problems for nonlinear elliptic equations and systems of p-Laplacian type will be discussed. In particular, second-order regularity properties of solutions, and the boundedness of their gradient will be focused. The results to be presented are optimal as far as the regularity of the right-hand sides of the equations and of the boundary of the ground domains are concerned. The talk is based on joint researches with V.Maz'ya.

Abstract

In this talk, I will first discuss relationship between some deep learning models and traditional algorithms such as finite element and multigrid methods. Such relationships can be used to study, explain and improve the model structures, mathematical properties and relevant training algorithms for deep neural networks. I will report a class of new training algorithms that can be used to improve the efficiency of convolutional neural networks (CNN) by significantly reducing the redundancy of the model without losing accuracy. By combining multigrid and deep learning methodologies, I will present a unified model, known as MgNet, that simultaneously recovers some CNNs for image classification and multigrid methods for solving discretized PDEs. MgNet can also be used to derive a new class of CNNs that mathematically unify many existing CNN models and practically prove to be computationally competitive.

Contact: alfio.quarteroni@polimi.it

Abstract

Da qualche anno a questa parte nel mondo matematico sono in corso molte ricerche
per cercare di descrivere in modo soddisfacente e predittivo il comportamento
di aggregati cellulari a vari livelli di evoluzione e organizzazione.
Questi modelli possono aiutare biologi e medici a validare meglio la loro ricerca,
esplorando ipotesi ed alternative altrimenti impraticabili.
In questo seminario esporrò alcuni risultati che ho ottenuto negli ultimi anni su alcuni di questi problemi: crescita di biofilm, movimento di protisti su reti, morfogenesi.
I modelli matematici sono tutti di tipo differenziale e hanno in comune il movimento a velocità finita delle cellule, pur in contesti matematici abbastanza diversi.

Abstract

Soluzioni a valori misure si presentano in modo naturale per importanti classi di equazioni di evoluzione nonlineari (equazione dei mezzi porosi, equazioni "forward-backward", leggi di conservazione). Nel seminario saranno esposti alcuni recenti risultati di esistenza, unicità e comportamento qualitativo di soluzioni entropiche
a valori misure di Radon di leggi di conservazione iperboliche in una dimensione spaziale, con flusso limitato e lipschitziano. Tempo permettendo, sarà discusso il legame fra tali soluzioni e soluzioni viscose discontinue di equazioni di Hamilton-Jacobi. I risultati presentati sono contenuti in alcuni lavori con M. Bertsch, F. Smarrazzo e A. Terracina.

Abstract

In 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.

Abstract

This 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