Publications associées à The Momentum Map In Poisson Geometry

Rank-adaptive structure-preserving model order reduction of Hamiltonian systems

Cecilia Pagliantini, Jan Sickmann Hesthaven, Nicolò Ripamonti

This work proposes an adaptive structure-preserving model order reduction method for finite-dimensional parametrized Hamiltonian systems modeling non-dissipative phenomena. To overcome the slowly decaying Kolmogorov width typical of transport problems, the ...

EDP SCIENCES S A2022

Structure-preserving approaches and data-driven closure modeling for model order reduction

Nicolò Ripamonti

In this thesis, we propose model order reduction techniques for high-dimensional PDEs that preserve structures of the original problems and develop a closure modeling framework leveraging the Mori-Zwanzig formalism and recurrent neural networks. Since high ...

EPFL2022

Generalised Howe duality and injectivity of induction: the symplectic case

Thomas Gerber

We study the symplectic Howe duality using two new and independent combinatorial methods: via determinantal formulae on the one hand, and via (bi)crystals on the other hand. The first approach allows us to establish a generalised version where weight multi ...

2022

Structure-Preserving Reduced Basis Methods For Poisson Systems

Cecilia Pagliantini, Jan Sickmann Hesthaven

We develop structure-preserving reduced basis methods for a large class of nondissipative problems by resorting to their formulation as Hamiltonian dynamical systems. With this perspective, the phase space is naturally endowed with a Poisson manifold struc ...

AMER MATHEMATICAL SOC2021

Scattering Map for the Vlasov–Poisson System

Klaus Martin Widmayer

We construct (modified) scattering operators for the Vlasov–Poisson system in three dimensions, mapping small asymptotic dynamics as t→−∞ to asymptotic dynamics as t→+∞. The main novelty is the construction of modified wave operators, but we also obtain a ...

2023

Stability of a Point Charge for the Vlasov–Poisson System: The Radial Case

Klaus Martin Widmayer

We consider the Vlasov–Poisson system with repulsive interactions. For initial data a small, radial, absolutely continuous perturbation of a point charge, we show that the solution is global and disperses to infinity via a modified scattering along traject ...

2021

Symplectic dynamical low rank approximation of wave equations with random parameters

Fabio Nobile, Eleonora Musharbash, Eva Vidlicková

In this paper we propose a dynamical low-rank strategy for the approximation of second order wave equations with random parameters. The governing equation is rewritten in Hamiltonian form and the approximate solution is expanded over a set of 2S dynamical ...

2020

Dynamical Reduced Basis Methods for Hamiltonian Systems

Cecilia Pagliantini

We consider model order reduction of parameterized Hamiltonian systems describing nondissipative phenomena, like wave-type and transport dominated problems. The development of reduced basis methods for such models is challenged by two main factors: the ric ...

Springer2021

Structure-Preserving Reduced Basis Methods for Hamiltonian Systems with a State-dependent Poisson Structure

Cecilia Pagliantini, Jan Sickmann Hesthaven

We develop structure-preserving reduced basis methods for a large class of nondissipative problems by resorting to their formulation as Hamiltonian dynamical systems. With this perspective, the phase space is naturally endowed with a Poisson manifold struc ...

2018

Symplectic Model-Reduction with a Weighted Inner Product

Babak Maboudi Afkham, Jan Sickmann Hesthaven

In the recent years, considerable attention has been paid to preserving structures and invariants in reduced basis methods, in order to enhance the stability and robustness of the reduced system. In the context of Hamiltonian systems, symplectic model redu ...

2018

The tropical momentum map: a classification of toric log symplectic manifolds

Tudor Ratiu

We give a generalization of toric symplectic geometry to Poisson manifolds which are symplectic away from a collection of hypersurfaces forming a normal crossing configuration. We introduce the tropical momentum map, which takes values in a generalization ...

Springer Verlag2017

Dynamical Low Rank approximation of PDEs with random parameters

Eleonora Musharbash

In this work, we focus on the Dynamical Low Rank (DLR) approximation of PDEs equations with random parameters. This can be interpreted as a reduced basis method, where the approximate solution is expanded in separable form over a set of few deterministic b ...

EPFL2017

Structure-Preserving Model-Reduction of Dissipative Hamiltonian Systems

Babak Maboudi Afkham, Jan Sickmann Hesthaven

Reduced basis methods are popular for approximately solving large and complex systems of dierential equations. However, conventional reduced basis methods do not generally preserve conservation laws and symmetries of the full order model. Here, we present ...

2019

Structure preserving model reduction of parametric Hamiltonian systems

Babak Maboudi Afkham, Jan Sickmann Hesthaven

While reduced-order models (ROMs) are popular for approximately solving large systems of differential equations, the stability of reduced models over long-time integration remains an open question. We present a greedy approach for ROM generation of paramet ...

2017

Noether Theorems and Quantum Anomalies

Tudor Ratiu

A quantum anomaly is the breaking of symmetry with respect to some transformations after the quantization of a classical Hamiltonian or Lagrangian system. It is shown that both the Noether theorems (including their infinite-dimensional versions) and the ex ...

Springer Verlag2017

Motivic and p-adic Localization Phenomena

Dimitri Stelio Wyss

In this thesis we compute motivic classes of hypertoric varieties, Nakajima quiver varieties and open de Rham spaces in a certain localization of the Grothendieck ring of varieties. Furthermore we study the

p

-adic pushforward of the Haar measure under a ...

EPFL2017

Revisiting the quest for a universal log-law and the role of pressure gradient in "canonical" wall-bounded turbulent flows

Peter Monkewitz

The trinity of so-called "canonical" wall-bounded turbulent flows, comprising the zero pressure gradient turbulent boundary layer, abbreviated ZPG TBL, turbulent pipe flow, and channel/duct flows has continued to receive intense attention as new and more r ...

Amer Physical Soc2017

Multisymplectic Variational Integrators For Nonsmooth Lagrangian Continuum Mechanics

Tudor Ratiu, François Gay-Balmaz, François Marie Alain Demoures

This paper develops the theory of multisymplectic variational integrators for nonsmooth continuum mechanics with constraints. Typical problems are the impact of an elastic body on a rigid plate or the collision of two elastic bodies. The integrators are ob ...

Cambridge Univ Press2016

Energetic ion dynamics and confinement in 3D saturated MHD configurations

David Pfefferlé

In the following theoretical and numerically oriented work, a number of findings have been assembled. The newly devised VENUS-LEVIS code, designed to accurately solve the motion of energetic particles in the presence of 3D magnetic fields, relies on a non- ...

EPFL2015

L-2-cohomology and complete Hamiltonian manifolds

Tudor Ratiu

A classical theorem of Frankel for compact Kahler manifolds states that a Kahler S-1-action is Hamiltonian if and only if it has fixed points. We prove a metatheorem which says that when the Hodge theory holds on non-compact manifolds, Frankel's theorem st ...

Elsevier2015