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Elena Beretta
Elena Beretta
New York University, Abu Dhabi
Eric Bonnetier
Eric Bonnetier
University of Grenoble, France
Title: On the spectrum of the Neumann-Poincar e operator
Title: On the spectrum of the Neumann-Poincar e operator
Marco Cirant
Marco Cirant
Università di Padova, Italy
Title: An introduction to the theory of MFG
Title: An introduction to the theory of MFG
Abstract: The thoery of Mean Field Games is devoted to the analysis of equilibria in large population models. I will give an introduction to the topic, from a PDE perspective: I will discuss the derivation of the PDE system for model problems, some existence and uniqueness results and, if time allows, the long-time behavior
Abstract: The thoery of Mean Field Games is devoted to the analysis of equilibria in large population models. I will give an introduction to the topic, from a PDE perspective: I will discuss the derivation of the PDE system for model problems, some existence and uniqueness results and, if time allows, the long-time behavior
Nicolas Dirr
Nicolas Dirr
Cardiff, UK
Title: Learning SPDEs from particle dynamics
Title: Learning SPDEs from particle dynamics
(joint work with Celia Reina, Johannes Zimmer and students: Peter Embacher, Zequn He,
(joint work with Celia Reina, Johannes Zimmer and students: Peter Embacher, Zequn He,
Shenglin Huang, Xiaoguai Li)
Shenglin Huang, Xiaoguai Li)
Abstract: If we are interested in the behaviour of a stochastic system on an intermediate or large scale, it is often not efficient to compute the evolution of the entire microscopic particle dynamics. Instead, one would like to have a PDE or stochastic PDE suitable for the chosen scale of interest. Using additional information from the thermodynamic structure of the system for learning the PDE turns out to have advantages over going for the operator directly. In addition to stability and a selection principle in case of non-uniqueness, we get an SPDE from fluctuating hydrodynamics and large deviations rate functionals. The key is observing fluctuations for determining the operator which defines the dissipative part of the evolution in the GENERIC framework.
Abstract: If we are interested in the behaviour of a stochastic system on an intermediate or large scale, it is often not efficient to compute the evolution of the entire microscopic particle dynamics. Instead, one would like to have a PDE or stochastic PDE suitable for the chosen scale of interest. Using additional information from the thermodynamic structure of the system for learning the PDE turns out to have advantages over going for the operator directly. In addition to stability and a selection principle in case of non-uniqueness, we get an SPDE from fluctuating hydrodynamics and large deviations rate functionals. The key is observing fluctuations for determining the operator which defines the dissipative part of the evolution in the GENERIC framework.
Houssem Haddar
Houssem Haddar
Director of Research, INRIA, Center of Saclay Ile de France
Head of the Team IDEFIX
Title: Recent advances in sampling methods and spectral signatures for inverse scattering problems
Title: Recent advances in sampling methods and spectral signatures for inverse scattering problems
Abstract: The goal of this mini-course is to present several recent advances in qualitative approaches to the inverse scattering problem at fixed frequency. The central objective of such problems is to extract information about an unknown medium from wave measurements performed outside the domain.
Abstract: The goal of this mini-course is to present several recent advances in qualitative approaches to the inverse scattering problem at fixed frequency. The central objective of such problems is to extract information about an unknown medium from wave measurements performed outside the domain.
In the first part of the course, we introduce the main ideas behind the so-called sampling methods, which have been widely studied for imaging purposes. These methods aim at reconstructing the geometry of an unknown inclusion without requiring a priori knowledge of its physical properties. We will explain the underlying principles of these techniques and discuss how recent developments, including artificial intelligence tools, can help improve their practical implementation, for instance by calibrating regularization parameters and estimating noise levels in the data.
In the first part of the course, we introduce the main ideas behind the so-called sampling methods, which have been widely studied for imaging purposes. These methods aim at reconstructing the geometry of an unknown inclusion without requiring a priori knowledge of its physical properties. We will explain the underlying principles of these techniques and discuss how recent developments, including artificial intelligence tools, can help improve their practical implementation, for instance by calibrating regularization parameters and estimating noise levels in the data.
In the second part, we turn to complementary techniques designed to recover quantitative information about the medium. In particular, we present approaches based on the introduction of artificial resonators within the probed domain. By analyzing the resulting resonant frequencies, which can be reconstructed from measurement data, one can obtain macroscopic information about the physical properties of the medium. This relies on an interesting duality between interior resonances and exterior invisibility. We will illustrate these ideas through the example of the so-called averaged Steklov eigenvalues.
In the second part, we turn to complementary techniques designed to recover quantitative information about the medium. In particular, we present approaches based on the introduction of artificial resonators within the probed domain. By analyzing the resulting resonant frequencies, which can be reconstructed from measurement data, one can obtain macroscopic information about the physical properties of the medium. This relies on an interesting duality between interior resonances and exterior invisibility. We will illustrate these ideas through the example of the so-called averaged Steklov eigenvalues.
Ziad Kobeissi
Ziad Kobeissi
Title : A Mean-Field-Game Approach to Overfishing
Title : A Mean-Field-Game Approach to Overfishing
Abstract : In this presentation, we propose a model for managing fisheries, described by a system of three coupled partial differential equations. The first is a reaction-diffusion equation representing the dynamics of the fish population, which follows standard approaches in the mathematical literature on spatial ecology. The other two equations are derived using a mean-field-game framework to model a large population of fishermen, where the number of fishermen is assumed to be large enough to be treated as infinite. Each fisherman aims to maximize their individual profit, calculated as the revenue from selling fish minus the cost of moving their boat. Under two different structural assumptions about the nonlinearities in the fish dynamics, we prove theoretical results illustrating the tragedy of the commons. Specifically, we show that a lack of coordination among fishermen can significantly harm, or even lead to the extinction of, the fish population. Our findings are supported by several numerical simulations.
Abstract : In this presentation, we propose a model for managing fisheries, described by a system of three coupled partial differential equations. The first is a reaction-diffusion equation representing the dynamics of the fish population, which follows standard approaches in the mathematical literature on spatial ecology. The other two equations are derived using a mean-field-game framework to model a large population of fishermen, where the number of fishermen is assumed to be large enough to be treated as infinite. Each fisherman aims to maximize their individual profit, calculated as the revenue from selling fish minus the cost of moving their boat. Under two different structural assumptions about the nonlinearities in the fish dynamics, we prove theoretical results illustrating the tragedy of the commons. Specifically, we show that a lack of coordination among fishermen can significantly harm, or even lead to the extinction of, the fish population. Our findings are supported by several numerical simulations.
Shari Moskow
Shari Moskow
Drexel University, USA
https://drexel.edu/coas/faculty-research/faculty-directory/mathematics/ShariMoskow/
moskow@math.drexel.edu
The Inverse Born Series for reconstructions of coefficients of a nonlinear Helmholtz equation.
The Inverse Born Series for reconstructions of coefficients of a nonlinear Helmholtz equation.
N. Defilippis, S. Moskow and J. Schotland
N. Defilippis, S. Moskow and J. Schotland
We investigate the Born and inverse Born series for a scalar wave equation with linear and nonlinear terms, the nonlinearity being cubic of Kerr type. We show conditions which guarantee convergence of the inverse Born series, enabling recovery of the coefficients of the linear and nonlinear terms. Furthermore, we show that if the unknown perturbation is only in the nonlinear term, an arbitrarily strong nonlinearity can be reconstructed for sufficiently small data. Similar convergence results hold for general polynomial nonlinearities. Our results are illustrated with numerical examples.
We investigate the Born and inverse Born series for a scalar wave equation with linear and nonlinear terms, the nonlinearity being cubic of Kerr type. We show conditions which guarantee convergence of the inverse Born series, enabling recovery of the coefficients of the linear and nonlinear terms. Furthermore, we show that if the unknown perturbation is only in the nonlinear term, an arbitrarily strong nonlinearity can be reconstructed for sufficiently small data. Similar convergence results hold for general polynomial nonlinearities. Our results are illustrated with numerical examples.
Michael Vogelius
Michael Vogelius
Rutgers University, USA
Title: Scattering vs. non-scattering
Title: Scattering vs. non-scattering
Abstract: I shall review some recent work through which we have determined quite general conditions on boundary (ir)regularity, geometry and non-degeneracy that guarantee scattering from transmitting inhomogeneities in the context of the Helmholtz equation. This work ties the scattering problem to well known problems of Schiffer and Pompeiu. The work is jointly with Fioralba Cakoni and Narek Hovsepyan.
Abstract: I shall review some recent work through which we have determined quite general conditions on boundary (ir)regularity, geometry and non-degeneracy that guarantee scattering from transmitting inhomogeneities in the context of the Helmholtz equation. This work ties the scattering problem to well known problems of Schiffer and Pompeiu. The work is jointly with Fioralba Cakoni and Narek Hovsepyan.
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