Schedule, Notes/Handouts & Videos

We construct critical trajectories in kinetic geometry, i.e. curves in (t,x,v) that are tangential to the transport and v-gradient, connecting any two given points, respecting the underlying kinetic scaling, and matching scaling properties of the stochastic trajectories near the starting point. The construction is based on solving the laws of motions with a forcing made up of desynchronised logarithmic oscillations. These critical trajectories provide an ''almost exponential map'' that allows to prove several functional analytic estimates. In particular they allow to extend to the kinetic setting the universal estimate for the logarithm of positive supersolutions by Moser 1971, and deduce optimal (weak) Harnack inequalities.

We consider linear kinetic equations of the form $\partial_t f + \frac{1}{\epsilon} v \nabla_x f = \frac{1}{\epsilon^2} L(f)$, for an unknown $f$ which depends on time $t$, position $x$ and velocity $v$, and where $L$ is a linear operator which acts only in the velocity variable, and which typically has a probability equilibrium in $v$. Important examples include the Fokker-Planck operator, nonlocal diffusion operators, linear BGK-type operators, or linear Boltzmann operators. This PDE typically represents a mesoscopic physical model, where we keep track of the probability distribution of the position and velocity of particles. It is well known that when $\epsilon$ tends to $0$, this type of equation has a macroscopic or diffusive limit for the density $\rho(t,x) := \int f(t,x,v) dv$, which is either the standard heat equation, or the fractional heat equation. As a new result, we show that for a fixed epsilon, the behaviour of this equation for large times also follows the standard or fractional heat equation, and that the long-time and small-epsilon limits are actually interchangeable in many cases. This is a work in collaboration with Stéphane Mischler (U. Paris-Dauphine) and Niccolò Tassi (U. Granada).

We study forward and inverse problems for a semilinear radiative transport model where the absorption coefficient depends on the angular average of the transport solution. Our first result is the well-posedness theory for the transport model with general boundary data, which significantly improves previous theories for small boundary data. For the inverse problem of reconstructing the nonlinear absorption coefficient from internal data, we develop stability results for the reconstructions and unify an $L^1$ stability theory for both the diffusion and transport regimes by introducing a weighted norm that penalizes the contribution from the boundary region. The problems studied here are motivated by applications such as photoacoustic imaging of multi-photon absorption of heterogeneous media.This is based on a joint work with Yimin Zhong of Auburn University.

Maintaining the stability and shape of a plasma is a crucial task in fusion energy. This is often challenging as plasma systems tend to be naturally unstable and kinetic effects can play an important role in the behavior of the instabilities. In this talk, I will introduce PDE-constrained optimization formulation that uses a kinetic description of plasma dynamics, particularly the Vlasov–Poisson system as the governing constraint. I will then discuss strategies to reduce computational costs through moment control. To further address the challenge of real-time control of plasma instabilities over long time horizons, we develop a dynamic feedback control strategy. This involves constructing an operator that maps state perturbations to an external control field. This operator is either approximated by a simple neural network, or directly constructed via energy estimates and a cancellation-based control approach that neutralizes the destabilizing components of the electric field.

The general relativistic Boltzmann equation is a fundamental physical model in astrophysics, for example in systems of galaxies, in supernova explosions, as a model of the early universe, and additionally as a model for hot gasses and plasmas.  The general relativistic Boltzmann equation with the Robertson-Walker metric in the massless case admits a family of non-stationary Equilibria of the form $J((t^q)^2 p) = \exp(-(t^q)^2 |p|)$.  The Robertson-Walker metric, or Friedmann–Lemaître–Robertson–Walker metric, is a widely used model describing a homogeneous isotropic and expanding universe.  In this work, for $0< q < 1$, we prove the global-in-time existence and uniqueness of suitably small perturbations of these Equilibria.  For $0< q < 1/3$ we prove that the perturbation has the superpolynomial large time-decay rate of $\exp(-t^{1-3q})$, and for $1/3< q < 1$ the perturbation has a slower time-decay rate of $t^{-3q}$.

This is a joint work with Martin Taylor and Renato Velozo Ruiz (both of Imperial College in London).

Computational expense is a bottleneck for many kinetic equations, due to the high-dimensionality of position-velocity phase space. As an approach to alleviating this issue, I will present results that use hp-adaptive mesh refinement in both position space and velocity space. As a specific application, I will show promising results for atmospheric radiative transfer, toward overcoming the 1D barrier and allowing 3D radiative transfer with clouds. More generally, I will describe our efforts to create a general-purpose software package for any kinetic equation. The aim of the software (called Jexpresso, work in progress) is to utilize continuous Galerkin or discontinuous Galerkin spectral element methods with adaptivity for efficiency, combined with a user-friendly interface to allow a user to easily specify and solve any kinetic equation.

In 1936 Landau introduced a modification of the Boltzmann equation – the Landau equation – which models a dilute hot plasma in which particles undergo Coulomb interactions. When particle velocities are close to the speed of light, which happens frequently in hot plasmas, effects of Einstein’s theory of special relativity become important. These effects are captured by the relativistic Landau equation, which was introduced by Budker and Beliaev in 1956. 

In this talk we will discuss recent results for the spatially inhomogeneous relativistic Landau equation in the far-from-equilibrium regime, including regularity and decay estimates. Some of the challenges in the analysis of the relativistic Landau equation come from the complex structure of the collision operator and the lack of scaling symmetries enjoyed by the classical counterpart. We will discuss tools and strategies that helped us overcome difficulties caused by the relativistic setting.

The talk is based on joint works with Henderson, Snelson and Tarfulea, and with Strain.

This project is related to the development of quantum algorithms for stochastic gradient descent. These algorithms will be based on the Wigner formulation of quantum mechanics, specifically related to open quantum systems, and with a quantum version of a stochastic equation as the continuous limit of a stochastic iteration. Thus, utilizing the Wigner-Fokker-Planck equation for quantum transport as the fundamental model under Markovian noise. By using previous estimates by Sparber et al., we study the dependance of the exponential convergence rate as a function of the problem dimensionality for the benchmark case of a harmonic potential. 

In this talk we will: 1. Introducing the most basic concepts in quantum computing. 2. Briefly highlight examples of quantum algorithms where the skills of an (numerical) analyst can be of use. 3. Describe one type of quantum computing hardware (a transmon) and how it is modeled. 4. Time permitting, we then turn to the simulation of open quantum systems and show how to design high order accurate methods that exploit low rank structure in the density matrix while respecting the essential structure of the Lindblad equation. Our methods preserve complete positivity and are trace preserving.

Abstract