10 Maggio, 2021 16:00

Sezione di Fisica Matematica

The Schrödinger equation as inspiration of beautiful mathematics

Gigliola Staffilani, MIT Mathematics, Cambridge, MA, US

online

abstract

abstract

In the last two decades great progress has been made in the study of dispersive and wave equations. Over the years the toolbox used in order to attack highly nontrivial problems related to these equations has developed to include a collection of techniques: Fourier and harmonic analysis, analytic number theory, math physics, dynamical systems, probability and symplectic geometry. In this talk I will introduce a variety of results using as model problem mainly the periodic 2D cubic nonlinear Schrodinger equation. I will start by giving
a physical derivation of the equation from a quantum many-particles
system, I will introduce periodic Strichartz estimates along with some remarkable connections to analytic number theory, I will move on the concept of energy transfer and its connection to dynamical systems, and I will end with some results on the derivation of a wave kinetic equation.

10 Maggio, 2021 17:30 oclock

Sezione di Probabilità e Statistica Matematica

Malliavin derivatives of hyperbolic Anderson model with applications to its absolute continuity and spatial averages.

Guangqu Zheng, University of Kansas

abstract

abstract

In this talk, we present recent work on the hyperbolic Anderson model driven by a space-time colored Gaussian homogeneous noise with spatial dimension one and two. Under mild assumptions, we provide Lp-estimates of the iterated Malliavin derivatives of the solution in terms of the fundamental solution of the wave solution. We present two applications:
(1) We present quantitative central limit theorems for spatial averages of the solution to the hyperbolic Anderson model, where the rates of convergence are described by the total variation distance. These quantitative results have been elusive so far due to the temporal correlation of the noise blocking us from using the Itô calculus.
(2) We establish the absolute continuity of the law for the hyperbolic Anderson model. The Lp-estimates of Malliavin derivatives are crucial ingredients to verify a local version of Bouleau-Hirsch criterion for absolute continuity. Our approach substantially simplifies the arguments for the one-dimensional case, which has been studied in the recent work by Balan, Quer-Sardanyons and Song (2019).
This talk is based on the joint work (arXiv:2101.10957) with R. Balan, D. Nualart and L. Quer-Sardanyons (2021).

5 Maggio, 2021 12:15

Seminari di Cultura Matematica

La delta di Dirac: storia di un mostro

Marco Bramanti, Politecnico di Milano

online: tiny.cc/fdsCM2021

In this talk, we present recent work on the hyperbolic Anderson model driven by a space-time colored Gaussian homogeneous noise with spatial dimension one and two. Under mild assumptions, we provide Lp-estimates of the iterated Malliavin derivatives of the solution in terms of the fundamental solution of the wave solution. We present two applications:
(1) We present quantitative central limit theorems for spatial averages of the solution to the hyperbolic Anderson model, where the rates of convergence are described by the total variation distance. These quantitative results have been elusive so far due to the temporal correlation of the noise blocking us from using the Itô calculus.
(2) We establish the absolute continuity of the law for the hyperbolic Anderson model. The Lp-estimates of Malliavin derivatives are crucial ingredients to verify a local version of Bouleau-Hirsch criterion for absolute continuity. Our approach substantially simplifies the arguments for the one-dimensional case, which has been studied in the recent work by Balan, Quer-Sardanyons and Song (2019).
This talk is based on the joint work (arXiv:2101.10957) with R. Balan, D. Nualart and L. Quer-Sardanyons (2021).

5 Maggio, 2021 09:15

MOX Seminar

Robust model-based classification: advances in learning from contaminated datasets

Andrea Cappozzo, MOX - Politecnico di Milano

Online seminar: mox.polimi.it/elenco-seminari/?id_evento=2055&t=763721

abstract

abstract

At the time of writing, an ever-increasing amount of data is collected every day, with the volume of such generated records estimated to be doubling every two years. Datasets are becoming massive in terms of size and substantially more complex in nature. Nevertheless, this abundance of "raw information" does come at a price: wrong measurements, data-entry errors, breakdowns of automatic collection systems and several other causes may ultimately undermine the overall data quality. The talk will present novel methodologies for performing reliable inference, within the model-based classification and clustering framework, in presence of contaminated data. First a discriminant analysis method for anomaly and novelty detection will be introduced, with the final aim of discovering label noise, outliers and unobserved classes in a semi-supervised context. Secondly, two robust variable selection methods, effectively performing high-dimensional discrimination within an adulterated scenario, will be discussed.
Joint work with Francesca Greselin and Thomas Brendan Murphy

Contact: laura.sangalli@polimi.it

3 Maggio, 2021 16:00

Seminario Matematico e Fisico di Milano

Machine Learning as an inverse problem

Ernesto De Vito, Università di Genova

abstract

abstract

The talk is devoted to a mathematical introduction to Machine Learning
from the point of view of inverse problems. In the presentation, we focus on
supervised learning problems in the framework of kernel methods.

27 Aprile, 2021 11:15 oclock

Sezione di Analisi

Existence and uniqueness result for a fluid-structure-interaction evolution problem in an unbounded 2D channel

Clara Patriarca, Dipartimento di Matematica, Politecnico Milano

abstract

abstract

In an unbounded 2D channel, we consider the vertical displacement of a rectangular obstacle in a regime of small flux for the incoming flow field, modelling the interaction between the cross-section of the deck of a suspension bridge and the wind. We prove an existence and uniqueness result for a fluid-structure-interaction evolution problem set in this channel, where at infinity the velocity field of the fluid has a Poiseuille flow profile. We introduce a suitable definition of weak solutions and we make use of a penalty method. In order to prevent collisions of the obstacle with the boundary of the channel, we introduce a strong force in the differential equation governing the motion of the rigid body and we find a unique global-in-time solution.

26 Aprile, 2021 17:30 oclock

Sezione di Probabilità e Statistica Matematica

The stochastic Strichartz estimates and stochastic nonlinear Schrödinger equations driven by Levy noise

Zdzislaw Brzezniak, University of York, UK

abstract

abstract

We establish a new version of the stochastic Strichartz estimate for the stochastic convolution driven by jump noise which we apply to the stochastic nonlinear Schrodinger equation with nonlinear multiplicative jump noise in the Marcus canonical form. With the help of the deterministic Strichartz estimates, we prove the existence and uniqueness of a global solution to stochastic nonlinear Schrodinger equation in $L^2(R^n)$ with either focusing or defocusing nonlinearity in the full subcritical range of exponents as in the deterministic case. This talk will be based on a joint paper with Wei Liu and Jiahui Zhu.

21 Aprile, 2021 17:00

Sezione di Geometria, Algebra e loro applicazioni

Vieta Formula, Distributions, and Lemniscate

Ahmed Sebbar, Chapman University

On line

abstract

abstract

We give two extensions of the classical Vieta (or Viète) formula.
The first extension leads to the classical Fabius function, an infinitely differentiable function that is nowhere analytic. The second extension discusses the corresponding formula for
the elliptic curves with complex multiplication $y^2=x^3 -Dx$, $y^2=x^3-D$.

14 Aprile, 2021 15:15 oclock

Sezione di Analisi

Modal analysis of some nonlinear beam equations

Matteo Fogato, Dipartimento di Matematica, Politecnico di Milano

abstract

abstract

We consider the equation $u_{tt}+\delta u_t +\|A^{\theta/2}u\|^2 A^\theta u=g$ where $A^2$ is a diagonal, self-adjoint and positive-definite operator, $\theta\in [0,1]$ and we study some finite-dimensional approximations of the problem. First, we analyze the dynamics in the case when the forcing term $g$ is a combination of a finite number of modes. Next, we estimate the error we commit by neglecting the modes larger than a given $N$. We then prove, for a particular class of forcing terms, a theoretical result allowing to study the distribution of the energy among the modes and, with this background, we refine the results.
Some generalizations and applications to the study of the stability of suspension bridges are given.

12 Aprile, 2021 17:00

Seminario Matematico e Fisico di Milano

Local discontinuous Galerkin methods for prestrained and bilayer plates

Ricardo H. Nochetto, University of Maryland

abstract

abstract

Prestrained plates are slender materials that develop internal
stresses at rest, deform out of plane even without external forces, and
exhibit nontrivial 3d shapes. Bilayer plates are slender structures made of
two materials that react differently to environmental (thermal,
electrical or chemical) actuation. In both cases the plates can exhibit large
bending deformations that are geometrically nonlinear. We present
reduced nonconvex models, develop variational formulations, and
design local discontinuous Galerkin methods (LDGs). Moreover, we prove
Gamma-convergence of the discrete energies and analyze discrete gradient
flows for the computation of minimizers that provide control of the
metric defect. We document the performance of the LDG methods with
several insightful simulations.