# Recent seminars

## 30/11/2022, Wednesday, 11:00–12:00 Room P3.10, Mathematics Building

João Pereira, Instituto de Matemática Pura e Aplicada
Method of Moments: From Sample Complexity to Efficient Implicit Computations

The focus of this talk is the multivariate method of moments for parameter estimation. First from a theoretical standpoint, we show that in problems where the noise is high, the number of observations necessary to estimate parameters is dictated by the moments of the distribution. Second from a computational standpoint, we address the curse of dimensionality: the number of entries of higher-order moments of multivariate random variables scale exponentially with the order of the moments. For Gaussian Mixture Models (GMMs), we develop numerical methods forimplicit computations; with the empirical moment tensors. This reduces the computational and storage costs, and opens the door to the competitiveness of the method of moments as compared to expectation maximization methods. Time permitting, we connect these results to symmetric $C^P$ tensor decomposition and sketch a recent algorithm which is faster than the state-of-the-art and comes with guarantees.

## 30/09/2022, Friday, 14:00–15:00 Room P3.10, Mathematics Building

Stochastic homogenisation of Gaussian fields

In this talk, we discuss the convergence of a sequence of random fields that generalise the Gaussian Free Field and bi-Laplacian field. Such fields are defined in terms of non-homogeneous elliptic operators which will be sampled at random. Under standard assumptions of stochastic homogenisation, we identify the limit fields as the usual GFF and bi-Laplacian fields up to a multiplicative constant. This is a joint work with W. Ruszel.

## 22/07/2022, Friday, 13:30–15:00 Room P3.10, Mathematics Building

Exclusion process in contact with boundary reservoirs is one of the simplest open particle systems.For symmetric exclusion, the macroscopic time evolution of the particle density is given by the heat equation with varies types of boundary conditions (Dirichlet, Robin, and Neumann). For asymmetric exclusion, the density evolves with the initial-boundary problem of the nonlinear conservation law:

\begin{align*} & \partial_t u+\partial_x [u(1-u)]=0, \\ & u(t,0)=v_-(t), u(t,1)=v_+(t), \\ & u(0,x)=v_0(x). \end{align*}

Its (entropy) solution exhibits specific discontinuous phenomenon (shock waves, boundary layers), which become the main obstacle of applying classical relative entropy method.

The goal of the mini-course is to introduce the entropy solution to the conservation law with irregular initial and boundary data, and present some new methods and results on the hydrodynamical behaviour of open ASEP.

It will include the following topics:

1. introduction to the ASEP with reservoirs, the stationary states, the hydrodynamic equation;
2. the concept of entropy solution, the viscous approximate, entropy inequality, boundary entropy;
3. main methods used to prove the hydrodynamic limit: the Young measure, the stochastic compensated compactness, the grading technique at boundaries.

## 21/07/2022, Thursday, 13:30–15:00 Room P3.10, Mathematics Building

Exclusion process in contact with boundary reservoirs is one of the simplest open particle systems.For symmetric exclusion, the macroscopic time evolution of the particle density is given by the heat equation with varies types of boundary conditions (Dirichlet, Robin, and Neumann). For asymmetric exclusion, the density evolves with the initial-boundary problem of the nonlinear conservation law:

\begin{align*} & \partial_t u+\partial_x [u(1-u)]=0, \\ & u(t,0)=v_-(t), u(t,1)=v_+(t), \\ & u(0,x)=v_0(x). \end{align*}

Its (entropy) solution exhibits specific discontinuous phenomenon (shock waves, boundary layers), which become the main obstacle of applying classical relative entropy method.

The goal of the mini-course is to introduce the entropy solution to the conservation law with irregular initial and boundary data, and present some new methods and results on the hydrodynamical behaviour of open ASEP.

It will include the following topics:

1. introduction to the ASEP with reservoirs, the stationary states, the hydrodynamic equation;
2. the concept of entropy solution, the viscous approximate, entropy inequality, boundary entropy;
3. main methods used to prove the hydrodynamic limit: the Young measure, the stochastic compensated compactness, the grading technique at boundaries.

## 20/07/2022, Wednesday, 13:30–15:00 Room P3.10, Mathematics Building

Exclusion process in contact with boundary reservoirs is one of the simplest open particle systems.For symmetric exclusion, the macroscopic time evolution of the particle density is given by the heat equation with varies types of boundary conditions (Dirichlet, Robin, and Neumann). For asymmetric exclusion, the density evolves with the initial-boundary problem of the nonlinear conservation law:

\begin{align*} & \partial_t u+\partial_x [u(1-u)]=0, \\ & u(t,0)=v_-(t), u(t,1)=v_+(t), \\ & u(0,x)=v_0(x). \end{align*}

Its (entropy) solution exhibits specific discontinuous phenomenon (shock waves, boundary layers), which become the main obstacle of applying classical relative entropy method.

The goal of the mini-course is to introduce the entropy solution to the conservation law with irregular initial and boundary data, and present some new methods and results on the hydrodynamical behaviour of open ASEP.

It will include the following topics:

1. introduction to the ASEP with reservoirs, the stationary states, the hydrodynamic equation;
2. the concept of entropy solution, the viscous approximate, entropy inequality, boundary entropy;
3. main methods used to prove the hydrodynamic limit: the Young measure, the stochastic compensated compactness, the grading technique at boundaries.