Hamiltonian Methods in Dispersive and Wave Evolution Equations
Institute for Computational and Experimental Research in Mathematics (ICERM)
September 8, 2021  December 10, 2021
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Wednesday, September 8, 2021
Hamiltonian Methods in Dispersive and Wave Evolution Equations

3:00  3:30 pm EDTCoffee Break11th Floor Collaborative Space
Thursday, September 9, 2021
Hamiltonian Methods in Dispersive and Wave Evolution Equations

9:00  10:30 am EDTLocal wellposedness for dispersive equationsTutorial  11th Floor Lecture Hall
 Alexandru Ionescu, Princeton University
 Benoit Pausader, Brown University
Abstract
This is a short introduction to the classical techniques for dispersive equations. We will present various equations and some methods to obtain local and global wellposedness and study the asymptotics.

11:00 am  12:30 pm EDTErgodicity of Markov processes: theory and computationTutorial  11th Floor Lecture Hall
 Yao Li, University of Massachusetts Amherst
Abstract
In this short course, I’ll cover the ergodicity of Markov processes on measurable state spaces. Both theoretical results and computational methods are based on the coupling technique. The following topics will be covered. 1, Markov process, transition kernel, and coupling. 2, Renewal theory with focusing on simultaneous renewal time. 3, Lyapunov criterion for geometric/polynomial ergodicity. 4, How to construct a Lyapunov function? 5, Numerical estimation of geometric/polynomial ergodicity. 6, Numerical estimation of invariant probability measure (if time permits).

1:30  2:00 pm EDTICERM WelcomeWelcome  11th Floor Lecture Hall
 Brendan Hassett, ICERM/Brown University

2:00  3:00 pm EDTGrad Student/ PostDoc IntroductionsIntroductions  11th Floor Lecture Hall
 Yvonne Alama Bronsard, Sorbonne Université
 Nicolas Camps, Université Paris Saclay
 Patrick Flynn, Brown
 Louise Gassot, Laboratoire de Mathématiques d'Orsay  Université ParisSaclay
 Dean Katsaros, UMass amherst
 Sudipta Kolay, ICERM
 Kyle Liss, University of Maryland, College Park
 Jaemin Park, Universitat de Barcelona
 Nancy Scherich, University of Toronto

3:00  3:30 pm EDTCoffee Break11th Floor Collaborative Space
Friday, September 10, 2021
Hamiltonian Methods in Dispersive and Wave Evolution Equations

9:00  10:30 am EDTLocal wellposedness for dispersive equationsTutorial  11th Floor Lecture Hall
 Alexandru Ionescu, Princeton University
 Benoit Pausader, Brown University
Abstract
This is a short introduction to the classical techniques for dispersive equations. We will present various equations and some methods to obtain local and global wellposedness and study the asymptotics.

11:00 am  12:30 pm EDTErgodicity of Markov processes: theory and computationTutorial  11th Floor Lecture Hall
 Yao Li, University of Massachusetts Amherst
Abstract
In this short course, I’ll cover the ergodicity of Markov processes on measurable state spaces. Both theoretical results and computational methods are based on the coupling technique. The following topics will be covered. 1, Markov process, transition kernel, and coupling. 2, Renewal theory with focusing on simultaneous renewal time. 3, Lyapunov criterion for geometric/polynomial ergodicity. 4, How to construct a Lyapunov function? 5, Numerical estimation of geometric/polynomial ergodicity. 6, Numerical estimation of invariant probability measure (if time permits).

2:00  3:30 pm EDTGrad Student/ PostDoc IntroductionsIntroductions  11th Floor Lecture Hall
 Bjoern Bringmann, Institute for Advanced Study
 Stefan Czimek, Brown University (ICERM)
 Daniel Eceizabarrena, University of Massachusetts Amherst
 Eduardo GarciaJuarez, Universitat de Barcelona
 Claudia García, Universitat de Barcelona
 Susanna Haziot, Brown University Mathematics
 Anastassiya Semenova, ICERM, Brown University
 Annalaura Stingo, University of California Davis
 Jiaqi Yang, ICERM
 Xueying Yu, University of Washington
 Haitian Yue, University of Southern California

3:00  3:30 pm EDTCoffee Break11th Floor Collaborative Space
Monday, September 13, 2021
Hamiltonian Methods in Dispersive and Wave Evolution Equations

9:00  10:30 am EDTComputerassisted proofs in PDEsTutorial  11th Floor Lecture Hall
 Javier Gomez Serrano, Princeton University
Abstract
In this minicourse we will present some recent results concerning computerassisted proofs in partial differential equations, starting from the basics of interval arithmetics. Particular emphasis will be put on the techniques, as opposed to the results themselves. There will be focus both on theory (lectures) and implementation (tutorial by Joel Dahne).

11:00 am  12:30 pm EDTErgodicity of Markov processes: theory and computationTutorial  11th Floor Lecture Hall
 Yao Li, University of Massachusetts Amherst
Abstract
In this short course, I’ll cover the ergodicity of Markov processes on measurable state spaces. Both theoretical results and computational methods are based on the coupling technique. The following topics will be covered. 1, Markov process, transition kernel, and coupling. 2, Renewal theory with focusing on simultaneous renewal time. 3, Lyapunov criterion for geometric/polynomial ergodicity. 4, How to construct a Lyapunov function? 5, Numerical estimation of geometric/polynomial ergodicity. 6, Numerical estimation of invariant probability measure (if time permits).

3:00  4:30 pm EDTWelcoming ReceptionReception  Hemenway's patio
Tuesday, September 14, 2021
Hamiltonian Methods in Dispersive and Wave Evolution Equations

9:00  10:30 am EDTLocal wellposedness for dispersive equationsTutorial  11th Floor Lecture Hall
 Alexandru Ionescu, Princeton University
 Benoit Pausader, Brown University
Abstract
This is a short introduction to the classical techniques for dispersive equations. We will present various equations and some methods to obtain local and global wellposedness and study the asymptotics.

11:00 am  12:30 pm EDTComputerassisted proofs in PDEsTutorial  11th Floor Lecture Hall
 Javier Gomez Serrano, Princeton University
Abstract
In this minicourse we will present some recent results concerning computerassisted proofs in partial differential equations, starting from the basics of interval arithmetics. Particular emphasis will be put on the techniques, as opposed to the results themselves. There will be focus both on theory (lectures) and implementation (tutorial by Joel Dahne).

3:00  3:30 pm EDTCoffee Break11th Floor Collaborative Space
Wednesday, September 15, 2021
Hamiltonian Methods in Dispersive and Wave Evolution Equations

9:00  10:30 am EDTLocal wellposedness for dispersive equationsTutorial  11th Floor Lecture Hall
 Alexandru Ionescu, Princeton University
 Benoit Pausader, Brown University
Abstract
This is a short introduction to the classical techniques for dispersive equations. We will present various equations and some methods to obtain local and global wellposedness and study the asymptotics.

11:00 am  12:30 pm EDTComputerassisted proofs in PDEsTutorial  11th Floor Lecture Hall
 Javier Gomez Serrano, Princeton University
Abstract
In this minicourse we will present some recent results concerning computerassisted proofs in partial differential equations, starting from the basics of interval arithmetics. Particular emphasis will be put on the techniques, as opposed to the results themselves. There will be focus both on theory (lectures) and implementation (tutorial by Joel Dahne).

3:00  3:30 pm EDTCoffee Break11th Floor Collaborative Space

3:15  4:15 pm EDTGrads/Postdocs Meet with ICERM Directorate11th Floor Lecture Hall
 Brendan Hassett, ICERM/Brown University
 Misha Kilmer, Tufts University
Thursday, September 16, 2021
Hamiltonian Methods in Dispersive and Wave Evolution Equations

3:00  3:30 pm EDTCoffee Break11th Floor Collaborative Space
Friday, September 17, 2021
Hamiltonian Methods in Dispersive and Wave Evolution Equations

9:00  10:30 am EDTLocal wellposedness for dispersive equationsTutorial  11th Floor Lecture Hall
 Alexandru Ionescu, Princeton University
 Benoit Pausader, Brown University
Abstract
This is a short introduction to the classical techniques for dispersive equations. We will present various equations and some methods to obtain local and global wellposedness and study the asymptotics.

11:00 am  12:30 pm EDTComputerassisted proofs in PDEsTutorial  11th Floor Lecture Hall
 Javier Gomez Serrano, Princeton University
Abstract
In this minicourse we will present some recent results concerning computerassisted proofs in partial differential equations, starting from the basics of interval arithmetics. Particular emphasis will be put on the techniques, as opposed to the results themselves. There will be focus both on theory (lectures) and implementation (tutorial by Joel Dahne).

3:00  3:30 pm EDTCoffee Break11th Floor Collaborative Space
Monday, September 20, 2021

9:50  10:00 am EDTWelcome11th Floor Lecture Hall
 Brendan Hassett, ICERM/Brown University

10:00  10:40 am EDTQuasilinear Diffusion of magnetized fast electrons in a mean field of quasiparticle waves packets11th Floor Lecture Hall
 Speaker
 Irene Gamba, University of Texas at Austin
 Session Chair
 Luis Vega, Basque Center for Applied Mathematics (BCAM)
Abstract
Quasilinear diffusion of magnetized fast electrons in momentum space results from stimulated emission and absorption of waves packets via waveparticle resonances. Such model consists in solving the dynamics of a system of classical kinetic diffusion processes described by the balance equations for electron probability density functions (electron pdf) coupled to the time dynamics waves (quasiparticles) in a quantum process of their resonant interaction. Such description results in a Mean Field model where diffusion coefficients are determined by the local spectral energy density of excited waves whose perturbations depend on flux averages of the electron pdf.
We will discuss the model and a mean field iteration scheme that simulates the dynamics of the space average model, where the energy spectrum of the excited wave time dynamics is calculated with a coefficient that depends on the electron pdf flux at a previous time step; while the time dynamics of the quasilinear model for the electron pdf is calculated by the spectral average of the quasiparticle wave under a classical resonant condition where the plasma wave frequencies couples the spectral energy to the momentum variable of the electron pdf. Recent numerical simulations will be presented showing a strong hot tail anisotropy formation and stabilization for the iteration in a 3 dimensional cylindrical model.
This is work in collaboration with Kun Huang, Michael Abdelmalik at UT Austin. 
10:55  11:15 am EDTCoffee Break11th Floor Collaborative Space

11:15 am  12:00 pm EDTModeling inviscid water waves11th Floor Lecture Hall
 Speaker
 Christophe Lacave, Universite Grenoble Alpes
 Session Chair
 Luis Vega, Basque Center for Applied Mathematics (BCAM)
Abstract
We consider numerical strategies to handle twodimensional water waves in a fully nonlinear regime. The freesurface is discretized via lagrangian tracers and the numerical strategy is constructed carefully to include desingularizations, but no artificial regularizations. We approach the formation of singularities in the wave breaking problem and also model solitary waves and the effect of an abruptly changing bottom. We present a rigorous analysis of the singular kernel operators involved in these methods.

12:15  1:15 pm EDTLunch/Free Time

1:15  1:55 pm EDTAnomalous conduction in one dimensional chains: a wave turbulence approach.11th Floor Lecture Hall
 Miguel Onorato, Università di Torino
Abstract
Heat conduction in 3D macroscopic solids is in general well described by the Fourier's law. However, low dimensional systems, like for example nanotubes, may be characterized by a conductivity that is sizedependent. This phenomena, known as anomalous conduction, has been widely studied in one dimensional chains like FPUT, mostly using deterministic simulations of the microscopic model. Here, I will present a mesoscopic approach based on the wave turbulence theory and give the evidence, through extensive numerical simulations and theoretical arguments, that the anomalous conduction is the result of the presence of long waves that rapidly propagate from one thermostat to the other without interacting with other modes. I will also show that the scaling of the conductivity with the length of the chain obtained from the mesoscopic approach is consistent with the one obtained from microscopic simulations.

2:10  2:50 pm EDTOn the asymptotic stability of shear flows and vortices11th Floor Lecture Hall
 Alexandru Ionescu, Princeton University
Abstract
I will talk about some recent work on the global linear and nonlinear asymptotic stability of two families of solutions of the 2D Euler equations: shear flows on bounded channels and vortices in the plane. This is joint work with Hao Jia.

3:00  4:30 pm EDTReceptionHemenway's Patio (weather permitting)
Tuesday, September 21, 2021

10:00  10:40 am EDTSmall scale formations in the incompressible porous media equation11th Floor Conference Room
 Yao Yao, Georgia Tech
Abstract
The incompressible porous media (IPM) equation describes the evolution of density transported by an incompressible velocity field given by Darcy’s law. Here the velocity field is related to the density via a singular integral operator, which is analogous to the 2D SQG equation. The question of global regularity vs finitetime blowup remains open for smooth initial data, although numerical evidence suggests that small scale formation can happen as time goes to infinity. In this talk, I will discuss rigorous examples of small scale formations in the IPM equation: we construct solutions to IPM that exhibit infiniteintime growth of Sobolev norms, provided that they remain globally smooth in time. As an application, this allows us to obtain nonlinear instability of certain stratified steady states of IPM. This is a joint work with Alexander Kiselev.

10:55  11:15 am EDTCoffee Break11th Floor Collaborative Space

11:15 am  12:00 pm EDTTBD11th Floor Lecture Hall
 George Karniadakis, Brown University

12:15  1:45 pm EDTLunch/Free Time

1:45  2:25 pm EDTThe second boundary value problem for a discrete MongeAmpere equation11th Floor Conference Room
 Gerard Awanou, University of Illinois, Chicago
Abstract
In this work we propose a natural discretization of the second boundary condition for the MongeAmpere equation of geometric optics and optimal transport. It is the natural generalization of the popular OlikerPrussner method proposed in 1988. For the discretization of the differential operator, we use a discrete analogue of the subdifferential. Existence, unicity and stability of the solutions to the discrete problem are established. Convergence results to the continuous problem are given.

2:40  3:45 pm EDTLightning Talks11th Floor Lecture Hall

3:45  4:15 pm EDTCoffee Break11th Floor Collaborative Space
Wednesday, September 22, 2021

10:00  10:40 am EDTEfficient and accurate structure preserving schemes for complex nonlinear systems11th Floor Conference Room
 Jie Shen, Purdue University
Abstract
Many complex nonlinear systems have intrinsic structures such as energy dissipation or conservation, and/or positivity/maximum principle preserving. It is desirable, sometimes necessary, to preserve these structures in a numerical scheme. I will present some recent advances on using the scalar auxiliary variable (SAV) approach to develop highly efficient and accurate structure preserving schemes for a large class of complex nonlinear systems. These schemes can preserve energy dissipation/conservation as well as other global constraints and/or are positivity/bound preserving, only require solving decoupled linear equations with constant coefficients at each time step, and can achieve higherorder accuracy.

10:55  11:15 am EDTCoffee Break11th Floor Collaborative Space

11:15 am  12:00 pm EDTEnergy growth for the Schrödinger map and the binormal flow11th Floor Lecture Hall
 Valeria Banica, Sorbonne Université
Abstract
In this talk I shall present a result of blow up of a density energy associated to the Schrödinger map and the binormal flow, a classical model for the dynamics of vortex filaments in Euler equations. This is a joint work with Luis Vega.

12:15  1:45 pm EDTLunch/Free Time

1:45  2:25 pm EDTWater Waves with Background Flow over Obstacles and Topography11th Floor Lecture Hall
 Jon Wilkening, University of California, Berkeley
Abstract
We present two accurate and efficient algorithms for solving the incompressible, irrotational Euler equations with a free surface in two dimensions with background flow over a periodic, multiplyconnected fluid domain that includes stationary obstacles and variable bottom topography. One approach is formulated in terms of the surface velocity potential while the other evolves the vortex sheet strength. Both methods employ layer potentials in the form of periodized Cauchy integrals to compute the normal velocity of the free surface. We also propose a new algorithm to dynamically vary the spacing of gridpoints on the free surface to efficiently resolve regions of high curvature as they develop. We study singularity formation and capillary effects and compare our numerical results with lab experiments.

2:45  3:15 pm EDTCoffee Break11th Floor Collaborative Space

3:15  3:55 pm EDTTopological Origin of Certain Fluid and Plasma Waves11th Floor Lecture Hall
 Brad Marston, Brown University
Abstract
Symmetries and topology play central roles in our understanding of physical systems. Topology, for instance, explains the precise quantization of the Hall effect and the protection of surface states in topological insulators against scattering from disorder or bumps. However discrete symmetries and topology have so far played little role in thinking about the fluid dynamics of oceans and atmospheres. In this talk I show that, as a consequence of the rotation of the Earth that breaks time reversal symmetry, equatorially trapped Kelvin and Yanai waves emerge as topologically protected edge modes. The nontrivial structure of the bulk Poincare ́ waves encoded through the first Chern number of value 2 guarantees the existence of these waves. Thus the oceans and atmosphere of Earth naturally share basic physics with topological insulators. As equatorially trapped Kelvin waves in the Pacific ocean are an important component of El Niño Southern Oscillation and other climate oscillations, these new results demonstrate that topology plays a surprising role in Earth’s climate system. We also predict that waves of topological origin will arise in magnetized plasmas. A planned experiment at UCLA’s Basic Plasma Science Facility to look for the waves is described.
Thursday, September 23, 2021

10:00  10:40 am EDTTBD11th Floor Lecture Hall
 Katharina Schratz, HeriotWatt University

10:55  11:15 am EDTCoffee Break11th Floor Collaborative Space

11:15 am  12:00 pm EDTEfficient timestepping methods for the rotating shallow water equations11th Floor Lecture Hall
 Thi Thao Phuong Hoang, Auburn University
Abstract
Numerical modeling of geophysical flows is challenging due to the presence of various coupled processes that occur at different spatial and temporal scales. It is critical for the numerical schemes to capture such a wide range of scales in both space and time to produce accurate and robust simulations over long time horizons.
In this talk, we will discuss efficient timestepping methods for the rotating shallow water equations discretized on spatial meshes with variable resolutions. Two different approaches will be considered: the first approach is a fully explicit local timestepping algorithm based on the strong stability preserving RungeKutta schemes, which allows different time step sizes in different regions of the computational domain. The second approach, namely the localized exponential time differencing method, is based on spatial domain decomposition and exponential time integrators, which makes possible the use of much larger time step sizes compared to explicit schemes and avoids solving nonlinear systems. Numerical results on various test cases will be presented to demonstrate the performance of the proposed methods. 
12:15  1:45 pm EDTLunch/Free Time

1:45  2:25 pm EDTDynamics in particle suspension flow11th Floor Conference Room
 Li Wang, University of Minnesota
Abstract
In this talk, I will consider two set up of particle suspension flow. One is a gravity driven flow down an incline, and the other is a pressure driven flow in a HeleShaw cell. In the former case, the interesting phenomena is the formation of singular shock that appears in the high particle concentration case that relates to the particlerich ridge observed in the experiments. We analyze the formation of the singular shock as well as its local structure. In the latter case, we rationalize a selfsimilar accumulation of particles at the interface between suspension and air. Our results demonstrate that the combination of the shear induced migration, the advancing fluidfluid interface, and Taylor dispersion yield the selfsimilar and gradual accumulation of particles.

2:45  3:15 pm EDTCoffee Break11th Floor Collaborative Space

3:15  3:55 pm EDTSeparatrix crossing and symmetry breaking in NLSElike systems due to forcing and damping11th Floor Conference Room
 Debbie Eeltink, MIT
Abstract
The nonlinear Schrödinger equation (NLSE) is a workhorse for many different fields (e.g. optical fibers, BoseEinstein condensates, water waves). It describes the evolution of the envelope of a field in time or space, taking into account the nonlinear interaction of the components of the spectrum of the envelope. While the NLSE is wellstudied in its conservative form, a relevant question to ask is how does it respond to damping and forcing? Limiting the spectrum to only three components allows one to construct a phasespace for the NLSE, spanned by the relative phase of the sidebands, and the energy fraction in the sidebands. Using wavetank measurements, we show that forcing and damping the NLSE induces separatrix crossing: switching from one solutiontype to the other in the phasespace. Our experiments are performed on deep water waves, which are better described by the higherorder NLSE, the Dysthe equation. We, therefore, extend our threewave analysis to this system. However, our conclusions are general as the dynamics are driven by the leading order terms. To our knowledge, it is the first phase evolution extraction from waterwave measurements. Furthermore, we observe a growth and decay cycle for modulated plane waves that are conventionally considered stable. Finally, we give a theoretical demonstration that forcing the NLSE system can induce symmetry breaking during the evolution.
Friday, September 24, 2021

10:00  10:40 am EDTExtreme Wave Events in Reflective Environments11th Floor Lecture Hall
 Amin Chabchoub, University of Sydney
Abstract
Waves dynamics in coastal zones is known to comprise incident and reflective wave motion. We report an experimental study in which several incident JONSWAP wave trains have been generated in a unidirectional water wave tank while the artificial beach inclination and its permeability have been varied to allow a variety of reflective wave conditions. Key statistical features obtained from an adaptive coupled nonlinear Schrödinger model simulations show an excellent agreement with the laboratory data collected near the beach.

10:55  11:15 am EDTCoffee Break11th Floor Collaborative Space

11:15  11:55 am EDTTBD11th Floor Lecture Hall
 Javier Gomez Serrano, Princeton University

12:10  12:50 pm EDTHigh order strong stability preserving multiderivative implicit and IMEX RungeKutta methods with asymptotic preserving properties11th Floor Lecture Hall
 Sigal Gottlieb, University of Massachusetts Dartmouth
Abstract
In this talk we present a class of high order unconditionally strong stability preserving (SSP) implicit twoderivative RungeKutta schemes, and SSP implicitexplicit (IMEX) multiderivative RungeKutta schemes where the timestep restriction is independent of the stiff term. The unconditional SSP property for a method of order $p>2$ is unique among SSP methods, and depends on a backwardintime assumption on the derivative of the operator. We show that this backward derivative condition is satisfied in many relevant cases where SSP IMEX schemes are desired. We devise unconditionally SSP implicit RungeKutta schemes of order up to $p=4$, and IMEX RungeKutta schemes of order up to $p=3$. For the multiderivative IMEX schemes, we also derive and present the order conditions, which have not appeared previously. The unconditional SSP condition ensures that these methods are positivity preserving, and we present sufficient conditions under which such methods are also asymptotic preserving when applied to a range of problems, including a hyperbolic relaxation system, the Broadwell model, and the BhatnagarGrossKrook (BGK) kinetic equation.
Wednesday, October 6, 2021
Hamiltonian Methods in Dispersive and Wave Evolution Equations

9:00  10:00 am EDTProfessional Development: Ethics IProfessional Development  11th Floor Lecture Hall
Wednesday, October 13, 2021
Hamiltonian Methods in Dispersive and Wave Evolution Equations

9:00  10:00 am EDTProfessional Development: Ethics IIProfessional Development  11th Floor Lecture Hall
Wednesday, November 3, 2021
Hamiltonian Methods in Dispersive and Wave Evolution Equations

9:00  10:00 am EDTProfessional Development: Hiring ProcessProfessional Development  11th Floor Lecture Hall
Wednesday, November 10, 2021
Hamiltonian Methods in Dispersive and Wave Evolution Equations

9:00  10:00 am ESTProfessional Development: Papers and JournalsProfessional Development  11th Floor Lecture Hall
Wednesday, November 17, 2021
Hamiltonian Methods in Dispersive and Wave Evolution Equations

9:00  10:00 am ESTProfessional Development: Grant ProposalsProfessional Development  11th Floor Lecture Hall
All event times are listed in ICERM local time in Providence, RI (Eastern Daylight Time / UTC4).
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